From dc7670867f85f50e41223582796e0806408cae9a Mon Sep 17 00:00:00 2001 From: Michal Patera <patermi2@fit.cvut.cz> Date: Thu, 11 May 2023 11:08:39 +0200 Subject: [PATCH] Adding thesis and LateX files. --- README.md | 92 --- patermi2-thesis.pdf | Bin 0 -> 642421 bytes thesis/.gitignore | 18 + thesis/.gitlab-ci.yml | 45 ++ thesis/LICENSE | 415 ++++++++++ thesis/ctufit-thesis.cls | 510 +++++++++++++ thesis/ctufit-thesis.tex | 219 ++++++ thesis/pictures/frama-c-BFS-1.PNG | Bin 0 -> 85171 bytes thesis/pictures/frama-c-BFS-2.png | Bin 0 -> 86535 bytes thesis/pictures/frama-c-BFS.PNG | Bin 0 -> 7304 bytes thesis/pictures/frama-c-DFS.PNG | Bin 0 -> 8336 bytes thesis/pictures/frama-c-dinitz.PNG | Bin 0 -> 7415 bytes thesis/pictures/frama-c-insert-verify.png | Bin 0 -> 5660 bytes thesis/pictures/frama-c-memory-messages.PNG | Bin 0 -> 26511 bytes thesis/pictures/frama-c-pop-vis.PNG | Bin 0 -> 67663 bytes thesis/pictures/frama-c-pop.PNG | Bin 0 -> 6776 bytes thesis/pictures/frama-c-ver-axiom.PNG | Bin 0 -> 37814 bytes thesis/text/appendix.tex | 2 + thesis/text/bib-database.bib | 275 +++++++ thesis/text/medium.tex | 11 + thesis/text/text.tex | 801 ++++++++++++++++++++ 21 files changed, 2296 insertions(+), 92 deletions(-) delete mode 100644 README.md create mode 100644 patermi2-thesis.pdf create mode 100644 thesis/.gitignore create mode 100644 thesis/.gitlab-ci.yml create mode 100644 thesis/LICENSE create mode 100644 thesis/ctufit-thesis.cls create mode 100644 thesis/ctufit-thesis.tex create mode 100644 thesis/pictures/frama-c-BFS-1.PNG create mode 100644 thesis/pictures/frama-c-BFS-2.png create mode 100644 thesis/pictures/frama-c-BFS.PNG create mode 100644 thesis/pictures/frama-c-DFS.PNG create mode 100644 thesis/pictures/frama-c-dinitz.PNG create mode 100644 thesis/pictures/frama-c-insert-verify.png create mode 100644 thesis/pictures/frama-c-memory-messages.PNG create mode 100644 thesis/pictures/frama-c-pop-vis.PNG create mode 100644 thesis/pictures/frama-c-pop.PNG create mode 100644 thesis/pictures/frama-c-ver-axiom.PNG create mode 100644 thesis/text/appendix.tex create mode 100644 thesis/text/bib-database.bib create mode 100644 thesis/text/medium.tex create mode 100644 thesis/text/text.tex diff --git a/README.md b/README.md deleted file mode 100644 index 7d5d5da..0000000 --- a/README.md +++ /dev/null @@ -1,92 +0,0 @@ -# Bakalarska Prace - Michal Patera - - - -## Getting started - -To make it easy for you to get started with GitLab, here's a list of recommended next steps. - -Already a pro? Just edit this README.md and make it your own. Want to make it easy? [Use the template at the bottom](#editing-this-readme)! - -## Add your files - -- [ ] [Create](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#create-a-file) or [upload](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#upload-a-file) files -- [ ] [Add files using the command line](https://docs.gitlab.com/ee/gitlab-basics/add-file.html#add-a-file-using-the-command-line) or push an existing Git repository with the following command: - -``` -cd existing_repo -git remote add origin https://gitlab.fit.cvut.cz/patermi2/bakalarska-prace-michal-patera.git -git branch -M master -git push -uf origin master -``` - -## Integrate with your tools - -- [ ] [Set up project integrations](https://gitlab.fit.cvut.cz/patermi2/bakalarska-prace-michal-patera/-/settings/integrations) - -## Collaborate with your team - -- [ ] [Invite team members and collaborators](https://docs.gitlab.com/ee/user/project/members/) -- [ ] [Create a new merge request](https://docs.gitlab.com/ee/user/project/merge_requests/creating_merge_requests.html) -- [ ] [Automatically close issues from merge requests](https://docs.gitlab.com/ee/user/project/issues/managing_issues.html#closing-issues-automatically) -- [ ] [Enable merge request approvals](https://docs.gitlab.com/ee/user/project/merge_requests/approvals/) -- [ ] [Automatically merge when pipeline succeeds](https://docs.gitlab.com/ee/user/project/merge_requests/merge_when_pipeline_succeeds.html) - -## Test and Deploy - -Use the built-in continuous integration in GitLab. - -- [ ] [Get started with GitLab CI/CD](https://docs.gitlab.com/ee/ci/quick_start/index.html) -- [ ] [Analyze your code for known vulnerabilities with Static Application Security Testing(SAST)](https://docs.gitlab.com/ee/user/application_security/sast/) -- [ ] [Deploy to Kubernetes, Amazon EC2, or Amazon ECS using Auto Deploy](https://docs.gitlab.com/ee/topics/autodevops/requirements.html) -- [ ] [Use pull-based deployments for improved Kubernetes management](https://docs.gitlab.com/ee/user/clusters/agent/) -- [ ] [Set up protected environments](https://docs.gitlab.com/ee/ci/environments/protected_environments.html) - -*** - -# Editing this README - -When you're ready to make this README your own, just edit this file and use the handy template below (or feel free to structure it however you want - this is just a starting point!). Thank you to [makeareadme.com](https://www.makeareadme.com/) for this template. - -## Suggestions for a good README -Every project is different, so consider which of these sections apply to yours. The sections used in the template are suggestions for most open source projects. Also keep in mind that while a README can be too long and detailed, too long is better than too short. If you think your README is too long, consider utilizing another form of documentation rather than cutting out information. - -## Name -Choose a self-explaining name for your project. - -## Description -Let people know what your project can do specifically. Provide context and add a link to any reference visitors might be unfamiliar with. A list of Features or a Background subsection can also be added here. If there are alternatives to your project, this is a good place to list differentiating factors. - -## Badges -On some READMEs, you may see small images that convey metadata, such as whether or not all the tests are passing for the project. You can use Shields to add some to your README. Many services also have instructions for adding a badge. - -## Visuals -Depending on what you are making, it can be a good idea to include screenshots or even a video (you'll frequently see GIFs rather than actual videos). Tools like ttygif can help, but check out Asciinema for a more sophisticated method. - -## Installation -Within a particular ecosystem, there may be a common way of installing things, such as using Yarn, NuGet, or Homebrew. However, consider the possibility that whoever is reading your README is a novice and would like more guidance. Listing specific steps helps remove ambiguity and gets people to using your project as quickly as possible. If it only runs in a specific context like a particular programming language version or operating system or has dependencies that have to be installed manually, also add a Requirements subsection. - -## Usage -Use examples liberally, and show the expected output if you can. It's helpful to have inline the smallest example of usage that you can demonstrate, while providing links to more sophisticated examples if they are too long to reasonably include in the README. - -## Support -Tell people where they can go to for help. It can be any combination of an issue tracker, a chat room, an email address, etc. - -## Roadmap -If you have ideas for releases in the future, it is a good idea to list them in the README. - -## Contributing -State if you are open to contributions and what your requirements are for accepting them. - -For people who want to make changes to your project, it's helpful to have some documentation on how to get started. Perhaps there is a script that they should run or some environment variables that they need to set. Make these steps explicit. These instructions could also be useful to your future self. - -You can also document commands to lint the code or run tests. These steps help to ensure high code quality and reduce the likelihood that the changes inadvertently break something. Having instructions for running tests is especially helpful if it requires external setup, such as starting a Selenium server for testing in a browser. - -## Authors and acknowledgment -Show your appreciation to those who have contributed to the project. - -## License -For open source projects, say how it is licensed. - -## Project status -If you have run out of energy or time for your project, put a note at the top of the README saying that development has slowed down or stopped completely. Someone may choose to fork your project or volunteer to step in as a maintainer or owner, allowing your project to keep going. 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ctufit-thesis-sk.tex + - sed 's/\[czech/\[english/' ctufit-thesis.tex > ctufit-thesis-en.tex + - latexmk -pdfxe ctufit-thesis.tex + - latexmk -pdfxe ctufit-thesis-en.tex + - latexmk -pdfxe ctufit-thesis-sk.tex + +pdfLaTeX: + script: + - sed 's/\[czech/\[slovak/' ctufit-thesis.tex > ctufit-thesis-sk.tex + - sed 's/\[czech/\[english/' ctufit-thesis.tex > ctufit-thesis-en.tex + - latexmk -pdf ctufit-thesis.tex + - latexmk -pdf ctufit-thesis-en.tex + - latexmk -pdf ctufit-thesis-sk.tex + artifacts: + paths: + - ctufit-thesis.pdf + - ctufit-thesis-en.pdf + - ctufit-thesis-sk.pdf + +LaTeX: + script: + - DEBIAN_FRONTEND=noninteractive apt-get update + - DEBIAN_FRONTEND=noninteractive apt-get -yq install ghostscript + - sed 's/\[czech/\[slovak/' ctufit-thesis.tex > ctufit-thesis-sk.tex + - sed 's/\[czech/\[english/' ctufit-thesis.tex > ctufit-thesis-en.tex + - latexmk -pdfps ctufit-thesis.tex + - latexmk -pdfps ctufit-thesis-en.tex + - latexmk -pdfps ctufit-thesis-sk.tex + artifacts: + paths: 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While you might intend that such +modifications will never be distributed, often this will happen by +accident -- you may forget that you have modified that component; or +it may not occur to you when allowing others to access the modified +version that you are thus distributing it and violating the conditions +of this license in ways that could have legal implications and, worse, +cause problems for the community. It is therefore usually in your +best interest to keep your copy of the Work identical with the public +one. Many works provide ways to control the behavior of that work +without altering any of its licensed components. + +How to Use This License +----------------------- + +To use this license, place in each of the components of your work both +an explicit copyright notice including your name and the year the work +was authored and/or last substantially modified. Include also a +statement that the distribution and/or modification of that +component is constrained by the conditions in this license. + +Here is an example of such a notice and statement: + + %% pig.dtx + %% Copyright 2005 M. Y. Name + % + % This work may be distributed and/or modified under the + % conditions of the LaTeX Project Public License, either version 1.3 + % of this license or (at your option) any later version. + % The latest version of this license is in + % http://www.latex-project.org/lppl.txt + % and version 1.3 or later is part of all distributions of LaTeX + % version 2005/12/01 or later. + % + % This work has the LPPL maintenance status `maintained'. + % + % The Current Maintainer of this work is M. Y. Name. + % + % This work consists of the files pig.dtx and pig.ins + % and the derived file pig.sty. + +Given such a notice and statement in a file, the conditions +given in this license document would apply, with the `Work' referring +to the three files `pig.dtx', `pig.ins', and `pig.sty' (the last being +generated from `pig.dtx' using `pig.ins'), the `Base Interpreter' +referring to any `LaTeX-Format', and both `Copyright Holder' and +`Current Maintainer' referring to the person `M. Y. Name'. + +If you do not want the Maintenance section of LPPL to apply to your +Work, change `maintained' above into `author-maintained'. +However, we recommend that you use `maintained', as the Maintenance +section was added in order to ensure that your Work remains useful to +the community even when you can no longer maintain and support it +yourself. + +Derived Works That Are Not Replacements +--------------------------------------- + +Several clauses of the LPPL specify means to provide reliability and +stability for the user community. They therefore concern themselves +with the case that a Derived Work is intended to be used as a +(compatible or incompatible) replacement of the original Work. If +this is not the case (e.g., if a few lines of code are reused for a +completely different task), then clauses 6b and 6d shall not apply. + + +Important Recommendations +------------------------- + + Defining What Constitutes the Work + + The LPPL requires that distributions of the Work contain all the + files of the Work. It is therefore important that you provide a + way for the licensee to determine which files constitute the Work. + This could, for example, be achieved by explicitly listing all the + files of the Work near the copyright notice of each file or by + using a line such as: + + % This work consists of all files listed in manifest.txt. + + in that place. In the absence of an unequivocal list it might be + impossible for the licensee to determine what is considered by you + to comprise the Work and, in such a case, the licensee would be + entitled to make reasonable conjectures as to which files comprise + the Work. \ No newline at end of file diff --git a/thesis/ctufit-thesis.cls b/thesis/ctufit-thesis.cls new file mode 100644 index 0000000..316fbb0 --- /dev/null +++ b/thesis/ctufit-thesis.cls @@ -0,0 +1,510 @@ +%% This is the ctufit-thesis document class. It is used to produce theses +%% for submission to Czech Technical University, Faculty of Information Technology. +%% +%% Get the newest version from +%% https://gitlab.fit.cvut.cz/theses-templates/FITthesis-LaTeX +%% +%% +%% Copyright 2021, Eliska Sestakova and Ondrej Guth +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public Licenese, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% https://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The current maintainer of this work is Ondrej Guth. +%% Contact ondrej.guth@fit.cvut.cz for bug reports. +%% Alternatively, submit bug reports into the tracker at +%% https://gitlab.fit.cvut.cz/theses-templates/FITthesis-LaTeX/issues +%% +%% + +% identification + +\NeedsTeXFormat{LaTeX2e} +\ProvidesClass{ctufit-thesis}[2021/08/06 CTU FIT thesis template] + +% preliminary declarations + +% options +\DeclareOption{bachelor}{\def\@ctufitthesistype{B}} +\DeclareOption{master}{\def\@ctufitthesistype{M}} +\DeclareOption{dissertation}{\def\@ctufitthesistype{D}} +\DeclareOption{bw}{\def\@ctufitthesisblackandwhite} +\DeclareOption{czech}{ + \AtBeginDocument{\selectlanguage{czech}} + \PassOptionsToPackage{english,main=czech}{babel} + \def\@ctufitlang{C} +} +\DeclareOption{english}{ + \AtBeginDocument{\selectlanguage{english}} + \PassOptionsToPackage{czech,main=english}{babel} + \def\@ctufitlang{E} +} +\DeclareOption{slovak}{ + \AtBeginDocument{\selectlanguage{slovak}} + \PassOptionsToPackage{english,main=slovak}{babel} + \def\@ctufitlang{S} +} +\ExecuteOptions{unicode,twoside} +\ProcessOptions + +% main declarations, loading files + +\LoadClass[a4paper]{book}[2020/04/10] +\RequirePackage{babel}[2021/03/03] +\RequirePackage[bottom=4cm,footskip=4em]{geometry}[2020/01/02] %page layout +\RequirePackage{setspace}[2011/12/19] %line spacing in title page +\RequirePackage{xcolor}[2016/05/11] +\RequirePackage[labelsep=space,singlelinecheck=false,font={up,small},labelfont={sf,bf}]{caption}[2020/05/30] +\RequirePackage{fancyhdr}[2019/01/31] +\RequirePackage{multicol}[2019/12/09] % summary page +\RequirePackage{titlesec}[2019/10/16] + +\ifdefined\@ctufitthesisblackandwhite +\definecolor{decoration}{RGB}{0, 0, 0} %CTU blue +\definecolor{heading}{RGB}{0, 0, 0} +\definecolor{headbackgroundgray}{RGB}{208, 208, 208} %light blue +\definecolor{backgroundgray}{RGB}{240, 240, 240} %CTU light blue +\definecolor{headgray}{rgb}{0.50,0.50,0.50} +\definecolor{enumgray}{RGB}{0, 0, 0} %CTU blue +\else + \definecolor{decoration}{RGB}{0, 122, 195} %CTU blue + \definecolor{heading}{RGB}{0, 122, 195} + \definecolor{headbackgroundgray}{RGB}{199, 219, 241} %light blue + \definecolor{backgroundgray}{RGB}{199, 219, 241} %CTU light blue + \definecolor{headgray}{rgb}{0.50,0.50,0.51} + \definecolor{enumgray}{RGB}{0, 122, 195} %CTU blue +\fi +% colors end + +\DeclareRobustCommand{\ctufittitle}[1]{ + \def\thectufittitle{#1} + \title{#1} +} +\DeclareRobustCommand{\ctufitauthorfull}[1]{ + \def\thectufitauthorfull{#1} + \author{#1} +} +\DeclareRobustCommand{\ctufitauthorsurnames}[1]{ + \def\thectufitauthorsurnames{#1} + \author{#1} +} +\DeclareRobustCommand{\ctufitauthorgivennames}[1]{ + \def\thectufitauthorgivennames{#1} + \author{#1} +} +\DeclareRobustCommand{\ctufitsupervisor}[1]{ + \def\thectufitsupervisor{#1} +} +\DeclareRobustCommand{\ctufitdepartment}[1]{ + \def\thectufitdepartment{#1} +} +\DeclareRobustCommand{\ctufityear}[1]{ + \def\thectufityear{#1} + \date{#1} +} +\DeclareRobustCommand{\ctufitdeclarationplace}[1]{ + \def\thectufitdeclarationplace{#1} +} +\DeclareRobustCommand{\ctufitdeclarationdate}[1]{ + \def\thectufitdeclarationdate{#1} +} +\DeclareRobustCommand{\ctufitabstractCZE}[1]{ + \def\thectufitabstrakt{#1} +} +\DeclareRobustCommand{\ctufitabstractENG}[1]{ + \def\thectufitabstract{#1} +} +\DeclareRobustCommand{\ctufitkeywordsCZE}[1]{ + \def\thectufitklicovaslova{#1} +} +\DeclareRobustCommand{\ctufitkeywordsENG}[1]{ + \def\thectufitkeywords{#1} +} +\DeclareRobustCommand{\thectufitthesistype}{% + \if\@ctufitthesistype B{% + \if\@ctufitlang C{Bakal{\' a}{\v r}sk{\' a} pr{\' a}ce}\else\if\@ctufitlang S{Bakal{\' a}rska pr{\' a}ca}\else{Bachelor's thesis}\fi\fi + }\else\if\@ctufitthesistype M{% + \if\@ctufitlang C{Diplomov{\' a} pr{\' a}ce}\else\if\@ctufitlang S{Diplomov{\' a} pr{\' a}ca}\else{Master's thesis}\fi\fi + }\else\if\@ctufitthesistype D{Dissertation thesis}\else{\ClassError{ctufit-thesis}{Thesis type not specified}{Add either `bachelor', `master', or `dissertation' option to this document class}}\fi\fi\fi +} + +\DeclareRobustCommand{\thectufituniversity}{\if\@ctufitlang C{{\v C}esk{\' e} vysok{\' e} u{\v c}en{\' i} technick{\' e} v~Praze}\else\if\@ctufitlang S{{\v C}esk{\' e} vysok{\' e} u{\v c}en{\' i} technick{\' e} v~Praze}\else{Czech Technical University in Prague}\fi\fi} + +\DeclareRobustCommand{\thectufitfaculty}{\if\@ctufitlang C{Fakulta informa{\v c}n{\' i}ch technologi{\' i}}\else\if\@ctufitlang S{Fakulta informa{\v c}n{\' i}ch technologi{\' i}}\else{Faculty of Information Technology}\fi\fi} + +\DeclareRobustCommand{\thectufitsupervisorlabelcapital}{\if\@ctufitlang C{Vedouc{\' i}}\else\if\@ctufitlang S{Ved\'uci}\else{Supervisor}\fi\fi} + +\DeclareRobustCommand{\thectufitcitationlabel}{\if\@ctufitlang C{Odkaz na tuto pr{\' a}ci}\else\if\@ctufitlang S{Odkaz na t\'uto pr{\' a}cu}\else{Citation of this thesis}\fi\fi} + +\DeclareRobustCommand{\thectufitallrightsreservedlabel}{\if\@ctufitlang C{V{\v s}echna pr{\' a}va vyhrazena.}\else\if\@ctufitlang S{V{\v s}etky pr{\' a}va vyhraden\'e.}\else{All rights reserved.}\fi\fi} + +\DeclareRobustCommand{\thectufitlistinglabel}{\if\@ctufitlang C{V{\'y}pis k{\'o}du}\else\if\@ctufitlang S{V{\'y}pis k{\'o}du}\else{Code listing}\fi\fi} + +\DeclareRobustCommand{\thectufitlollabel}{\if\@ctufitlang C{Seznam v{\'y}pis{\r u} k{\'o}du}\else\if\@ctufitlang S{Zoznam v{\'y}pisov k{\'o}du}\else{List of code listings}\fi\fi} + +\DeclareRobustCommand{\thectufitdeclarationlabel}{\if\@ctufitlang C{ProhlášenÃ}\else\if\@ctufitlang E{Declaration}\else{Vyhlásenie}\fi\fi} + + +\DeclareRobustCommand{\thectufitcopyrighttext}{\if\@ctufitlang C{Tato pr{\' a}ce vznikla jako {\v s}koln{\' i} d{\' i}lo na {\v C}esk{\' e}m vysok{\' e}m u{\v c}en{\' i} technick{\' e}m v~Praze, Fakult{\v e} informa{\v c}n{\' i}ch technologi{\' i}. Pr{\' a}ce je chr{\' a}n{\v e}na pr{\' a}vn{\' i}mi p{\v r}edpisy a mezin{\' a}rodn{\' i}mi {\' u}mluvami o~pr{\' a}vu autorsk{\' e}m a pr{\' a}vech souvisej{\' i}c{\' i}ch s~pr{\' a}vem autorsk{\' y}m. K~jej\'\i{}mu u\v zit\'\i{}, s~v\'yjimkou bez\'uplatn\'ych z\'akonn\'ych licenc\'\i{} a nad r\'amec opr\'avn\v en\'\i{} uveden\'ych v~Prohl\'a\v sen\'\i{}, je nezbytn\'y souhlas autora.}\else\if\@ctufitlang S{T\'ato pr\'aca vznikla ako \v skolsk\'e dielo na FIT \v CVUT v Prahe. Pr\'aca je chr\'anen\'a medzin\'arodn\'ymi predpismi a zmluvami o autorskom pr\'ave a pr\'avach s\'uvisiacich s autorsk\'ym pr\'avom. Na jej vyu\v zitie, s~v\'ynimkou bezplatn\'ych z\'akonn\'ych licenci\'\i, je nutn\'y s\'uhlas autora.}\else{This thesis is school work as defined by Copyright Act of the Czech Republic. It has been submitted at Czech Technical University in Prague, Faculty of Information Technology. The thesis is protected by the Copyright Act and its usage without author's permission is prohibited (with exceptions defined by the Copyright Act).}\fi\fi} + +% title page +\renewcommand{\maketitle}{\begin{titlepage}% +\newgeometry{left=67mm,top=80mm,right=40mm}%\thispagestyle{empty}% +\noindent{\large\noindent\thectufitthesistype} +\vskip 3mm +\noindent{\noindent\huge\bfseries\color{black}\begin{flushleft}\begin{onehalfspace}\MakeUppercase{\thectufittitle}\end{onehalfspace}\end{flushleft}} +\vskip 35mm +\noindent{\large \bfseries \thectufitauthorfull} + +\vfill + +\noindent{\thectufitfaculty\\ +\thectufitdepartment\\ +\thectufitsupervisorlabelcapital: \thectufitsupervisor\\ +\today}\end{titlepage} +\restoregeometry +} +%title page end + +% IMPRINT +\DeclareRobustCommand{\imprintpage}{ + \clearpage + \thispagestyle{empty} + + + ~ + \vfill + + {\small + \noindent \thectufituniversity \\ + \noindent \thectufitfaculty \\ + \noindent \textcopyright{} \thectufityear{} \thectufitauthorfull. \thectufitallrightsreservedlabel\\ + \noindent \textit{\thectufitcopyrighttext} + + \vspace{1em} + + \noindent \thectufitcitationlabel: \thectufitauthorsurnames{} \thectufitauthorgivennames. \textit{\thectufittitle}. \thectufitthesistype. \thectufituniversity, \thectufitfaculty, \thectufityear. + } +} +% IMPRINT END + +% captions settings +\@ifpackagelater{hyperref}{2009/12/09} + {\captionsetup{compatibility=false}}%cf. http://groups.google.de/group/comp.text.tex/browse_thread/thread/db9310eb540fbbd8/42e30f3b7b3aa17a?lnk=raot + {} +\DeclareCaptionLabelFormat{boxed}{%0.61,0.61,0.61 + \kern0.05em{\color{decoration}\rule{0.73em}{0.73em}}% + \hspace*{0.67em}\bothIfFirst{#1}{~}#2} +\captionsetup{labelformat=boxed} +\captionsetup[table]{position=top} +% captions settings end + +% lists settings +\setlength\leftmargini \parindent +\setlength\leftmarginii {1.2em} +\setlength\leftmarginiii{1.2em} +\setlength\leftmarginiv {1.2em} +\setlength\leftmarginv {1.2em} +\setlength\leftmarginvi {1.2em} +\renewcommand\labelenumi{% + \textcolor{enumgray}{\bfseries\upshape\mathversion{bold}\theenumi.}} +\renewcommand\labelenumii{% + \textcolor{enumgray}{\bfseries\upshape\mathversion{bold}\theenumii.}} +\renewcommand\labelenumiii{% + \textcolor{enumgray}{\bfseries\upshape\mathversion{bold}\theenumiii.}} +\renewcommand\labelenumiv{% + \textcolor{enumgray}{\bfseries\upshape\mathversion{bold}\theenumiv.}} +% \makeatletter +\renewcommand\labelitemi{% + \textcolor{enumgray}{\ifnum\@listdepth=\@ne + \rule{0.67em}{0.33em}% + \else + \rule{0.45em}{0.225em}% + \fi}} +% \makeatother +\renewcommand\labelitemii{% + \textcolor{enumgray}{\rule{0.45em}{0.225em}}} +\renewcommand\labelitemiii{% + \textcolor{enumgray}{\bfseries\textasteriskcentered}} +\renewcommand\labelitemiv{% + \textcolor{enumgray}{\bfseries\textperiodcentered}} +\renewcommand*\descriptionlabel[1]{% + \hspace\labelsep\textcolor{enumgray}{\bfseries\mathversion{bold}#1}} +% lists settings end + +%%%%%%%%%%%%%% +% FRONTMATTER SETTINGS +% no need to modify this part +%%%%%%%%%%%%%% +\DeclareRobustCommand{\frontmatterinit}{ +\titleformat +{\chapter} % command +% [display] % shape +{} % format +{} % label +{} % sep +{\color{heading}\frontchapterfont \raggedleft} % before-code +[\vskip -2em] % after-code + +\titleformat +{\section} +{\frontsectionfont\color{heading}} +{} +{} +{} + + +% \titleformat{\subsection} +% {\frontsubsectionfont\color{heading}}{{{\color{black}\thesubsection}}}{1em}{}[\vskip -1em] + +% \titleformat{\subsubsection} +% {\frontsubsubsectionfont\color{heading}}{{{\color{black}\thesubsubsection}}}{1em}{}[\vskip -1em] + +% \makeatletter +\@openrightfalse +% \makeatother +} +%%%%%%%%%%%%%% +% FRONTMATTER SETTINGS END +%%%%%%%%%%%%%% + + +% frontmatter headings +\def\frontchapterfont{\Large \bfseries} +\def\frontsectionfont{\large\bfseries} +\def\frontsubsectionfont{\large} +\def\frontsubsubsectionfont{\bfseries} +% frontmatter headings end + +% frontmatter pseudochapters: named part without printing actual chapter heading +% \makeatletter +\newcommand{\frontchapternotprinted}[1]{% + \begingroup + \let\@makechapterhead\@gobble % make \@makechapterhead do nothing + \let\cleardoublepage\clearpage + \chapter{#1} + \endgroup +} +% \makeatother +% frontmatter pseudochapters end + +% table of contents colored chapters +% \makeatletter +\let\stdl@chapter\l@chapter +\renewcommand*{\l@chapter}[2]{% + \stdl@chapter{\textcolor{heading}{#1}}{\textcolor{heading}{#2}}} +% \makeatother +% table of contents colored chapters end + +% headers and footers +% \makeatletter +\def\ps@plain{%chapter beginning + \let\@evenhead\@empty% + \let\@oddhead\@empty% + \def\@evenfoot{\bfseries\color{headgray}\hfill\thepage\hfill}% + \def\@oddfoot{\bfseries\color{headgray}\hfill\thepage\hfill}} +% \makeatother +\renewcommand{\chaptermark}[1]{\markboth{#1}{}} +\renewcommand{\sectionmark}[1]{\markright{#1}} +\pagestyle{fancy} +\fancyhf{} +\fancyhead[LE]{\leavevmode\smash{\llap{\color{headgray} \bfseries \thepage \hspace*{4em}}}} +\fancyhead[RE]{\color{headgray}\bfseries\nouppercase{\leftmark}} +\fancyhead[RO]{\leavevmode\smash{\rlap{\hspace*{4em}\color{headgray}\bfseries\thepage}}} +\fancyhead[LO]{\color{headgray}\bfseries\nouppercase{\rightmark}} +\renewcommand{\headrulewidth}{0pt} +% headers and footers end + + +\newenvironment{abstrakt}{% + \vspace*{18mm} + \noindent + {{\frontchapterfont\begin{flushleft}{\color{heading}Abstrakt}\end{flushleft}}}% + \bigskip + \noindent\ignorespaces} + +\newenvironment{abstract}{% + \vspace*{18mm} + \noindent + {{\frontchapterfont\begin{flushleft}{\color{heading}Abstract}\end{flushleft}}}% + \bigskip + \noindent\ignorespaces} + +\newenvironment{declaration}{} + +% acknowledgment page +\newenvironment{acknowledgmentpage}{\frontchapternotprinted{\if\@ctufitlang C{PodÄ›kovánÃ}\else\if\@ctufitlang E{Acknowledgments}\else{PoÄakovanie}\fi\fi} +~ +\vfill +\hskip 0cm \begin{minipage}{0.7\textwidth}\itshape}{ +\end{minipage} + +\vfill + +\vfill} +% acknowledgment page END + +% declaration page +\newenvironment{declarationpage}{\frontchapternotprinted{\thectufitdeclarationlabel} +~ +\vfill + + {{\frontchapterfont\begin{flushright}{\color{heading}\thectufitdeclarationlabel}\end{flushright}}}% + \bigskip + \noindent\ignorespaces +}{\vskip 1cm +\noindent +\if\@ctufitlang E{In}\else{V}\fi~\thectufitdeclarationplace{} \if\@ctufitlang C{dne}\else\if\@ctufitlang S{dňa}\else{on}\fi\fi~\thectufitdeclarationdate{} \hspace{.3\textwidth} \dotfill} +% declaration page END + +% abstract page +\newenvironment{abstractpage}{\frontchapternotprinted{\if\@ctufitlang E{Abstract}\else{Abstrakt}\fi} +}{} +% abstract page END + +\newenvironment{summarypage}{\chapter{\if\@ctufitlang C{ShrnutÃ}\else\if\@ctufitlang E{Summary}\else{Zhrnutie}\fi\fi} +\setlength{\columnsep}{1cm} +\begin{multicols}{2}\small +}{\end{multicols}} + +%%%%%%%%%%%%%%%%%%%% +% MAINMATTER SETTINGS +% no need to modify this part +%%%%%%%%%%%%%%%%%%%% +\DeclareRobustCommand{\mainmatterinit}{ +% \makeatletter +\@openrighttrue +% \makeatother + +\titleformat +{\chapter} % command +[display] % shape +{ \LARGE \bfseries \raggedleft} % format +{ \textcolor{headbackgroundgray}{{\titlerule*[1pc]{\rule{0.25em}{0.25em}}}} \hspace{0.5ex} \color{black} \chaptername~\thechapter } % label +{-0.3cm} % sep +{\color{heading} \Huge \vskip 0.5cm} % before-code +[\vskip 1cm] % after-code + +%vzhled nadpisů sekcà +\titleformat{\section} + {\Large\bfseries\color{heading}}{\colorbox{headbackgroundgray}{{\color{black}\thesection}}}{1em}{}%[\vskip -1em] + +\titleformat{\subsection} + {\Large\bfseries\color{heading}}{{{\color{black}\thesubsection}}}{1em}{}%[\vskip -1em] + +\titleformat{\subsubsection} + {\large\bfseries\color{heading}}{{{\color{black}\thesubsubsection}}}{1em}{}%[\vskip -1em] +} +%%%%%%%%%%%%%%%%%%%% +% MAINMATTER SETTINGS END +%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%% +% APPENDIX SETTINGS +% no need to modify this part +%%%%%%%%%%%%%%%%%%%% +\DeclareRobustCommand{\appendixinit}{ +\titleformat +{\chapter} % command +[display] % shape +{ \LARGE \bfseries \raggedleft} % format +{ \textcolor{headbackgroundgray}{{\titlerule*[1pc]{\rule{0.25em}{0.25em}}}} \hspace{0.5ex} \color{black} \appendixname~\thechapter } % label +{-0.3cm} % sep +{\color{heading} \Huge \vskip 0.5cm} % before-code +[\vskip 1cm] % after-code +} +%%%%%%%%%%%%%%%%%%%% +% APPENDIX SETTINGS END +%%%%%%%%%%%%%%%%%%%% + +% theorems, proofs, definitions, etc. +\RequirePackage{amsthm}[2020/05/29] +\RequirePackage{mathtools}[2020/03/24] +\RequirePackage{amssymb}[2013/01/14] + +\thm@headfont{% + \textcolor{decoration}{$\blacktriangleright$}\nobreakspace\bfseries} +\def\th@remark{% + \thm@headfont{% + \textcolor{decoration}{$\blacktriangleright$}\nobreakspace}% + \normalfont % body font + \thm@preskip\topsep \divide\thm@preskip\tw@ + \thm@postskip\thm@preskip +} +\def\@endtheorem{\endtrivlist}%\@endpefalse +\renewcommand\qedsymbol{\textcolor{decoration}{\ensuremath{\blacktriangleleft}}} +\renewenvironment{proof}[1][\proofname]{\par + \pushQED{\qed}% + \normalfont \topsep6\p@\@plus6\p@\relax + \trivlist + \item[\hskip\labelsep + \color{black}\bfseries + #1\@addpunct{.}]\ignorespaces +}{% + \popQED\endtrivlist%\@endpefalse +} +% theorems, proofs, definitions, etc. end + +\DeclareRobustCommand{\printabstractpage}{% +\begin{abstractpage} +\if\@ctufitlang E{% +\begin{abstract}% +\thectufitabstract +\end{abstract} + +\vskip 0.5cm + +{\noindent\color{heading}\bfseries Keywords\hspace{1em}}{\thectufitkeywords}}\else{% +\begin{abstrakt}% +\thectufitabstrakt +\end{abstrakt} + +\vskip 0.5cm + +{\noindent\color{heading}\bfseries KlÃÄová slova\hspace{1em}}{\thectufitklicovaslova}}\fi + +\vskip 1cm + +\if\@ctufitlang E{% +\begin{abstrakt}% +\thectufitabstrakt +\end{abstrakt} + +\vskip 0.5cm + +{\noindent\color{heading}\bfseries KlÃÄová slova\hspace{1em}}{\thectufitklicovaslova}}\else{ +\begin{abstract}% +\thectufitabstract +\end{abstract} + +\vskip 0.5cm + +{\noindent\color{heading}\bfseries Keywords\hspace{1em}}{\thectufitkeywords}}\fi +\end{abstractpage} +} + +\newenvironment{chapterabstract}{\hfill\begin{minipage}{\dimexpr\textwidth-0.5cm}\itshape}{\end{minipage}\vskip 0.5cm\par\@afterindentfalse\@afterheading} + +\AtBeginDocument{% + \@ifpackageloaded{listings}{% + \DeclareRobustCommand{\lstlistingname}{\thectufitlistinglabel}% + \DeclareRobustCommand{\lstlistlistingname}{\thectufitlollabel}% + \lstset{basicstyle=\small\ttfamily,keywordstyle=\bfseries,% + backgroundcolor=\color{backgroundgray},% + frame=single,framerule=0pt,xleftmargin=\fboxsep,xrightmargin=\fboxsep}% + }{}% + \@ifpackageloaded{minted}{% + \DeclareRobustCommand{\listingscaption}{\thectufitlistinglabel}% + \DeclareRobustCommand{\listoflistingscaption}{\thectufitlollabel}% + \usemintedstyle{bw}% + \setminted{bgcolor=backgroundgray}% + \setmintedinline{bgcolor={}} + }{}% +} diff --git a/thesis/ctufit-thesis.tex b/thesis/ctufit-thesis.tex new file mode 100644 index 0000000..09fa9b6 --- /dev/null +++ b/thesis/ctufit-thesis.tex @@ -0,0 +1,219 @@ +%% This is the ctufit-thesis example file. It is used to produce theses +%% for submission to Czech Technical University, Faculty of Information Technology. +%% +%% Get the newest version from +%% https://gitlab.fit.cvut.cz/theses-templates/FITthesis-LaTeX +%% +%% +%% Copyright 2021, Eliska Sestakova and Ondrej Guth +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public Licenese, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% https://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The current maintainer of this work is Ondrej Guth. +%% Contact ondrej.guth@fit.cvut.cz for bug reports. +%% Alternatively, submit bug reports into the tracker at +%% https://gitlab.fit.cvut.cz/theses-templates/FITthesis-LaTeX/issues +%% +%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% CLASS OPTIONS +% language: czech/english/slovak +% thesis type: bachelor/master/dissertation +% colour: bw for black&white OR no option for default colour scheme +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\documentclass[czech,bachelor,unicode]{ctufit-thesis} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% FILL IN THIS INFORMATION +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\ctufittitle{Ověřená implementace Dinitzova algoritmu} % replace with the title of your thesis +\ctufitauthorfull{Michal Patera} % replace with your full name (first name(s) and then family name(s) / surname(s)) including academic degrees +\ctufitauthorsurnames{Patera} % replace with your surname(s) / family name(s) +\ctufitauthorgivennames{Michal} % replace with your first name(s) / given name(s) +\ctufitsupervisor{doc. RNDr. DuÅ¡an Knop, Ph.D.} % replace with name of your supervisor/advisor (include academic degrees) +\ctufitdepartment{Katedra informaÄnà bezpeÄnosti} % replace with the department of your defence +\ctufityear{2023} % replace with the year of your defence +\ctufitdeclarationplace{Praze} % replace with the place where you sign the declaration +\ctufitdeclarationdate{\today} % replace with the date of signature of the declaration +\ctufitabstractCZE{Tato práce se zabývá implementaci Dinitzova algoritmu pro hledánà maximálnÃho toku v sÃti a formálnÃmu důkazu jednotlivých použitÃch funkcÃ. Teoretická Äást seznámà Ätenáře s teorià toků v sÃtÃch a myÅ¡lenkou Dinitzova algoritmu. PÅ™edstavà Ätenáři také framework Frama-C pro formálnà analýzu kódu v programovacÃm jazyce C a anotaÄnà jazyk ACSL, který frameworku pomáhá s verifikacÃ. Praktická Äást seznámà Ätenáře s použitými metodami pro implementaci Dinitzova algoritmu a také objasnà použité anotace pro verifikaci algoritmu. } +\ctufitabstractENG{This thesis deals with the implementation of the Dinitz algorithm for finding maximum flow in the network and the formal proof of individual functions, that it uses. The theoretical part introduces the reader to the theory of flows in networks and the idea of the Dinitz algorithm. It will also introduce the reader to the Frama-C framework for formal code analysis in the C programming language and the ACSL annotation language, which helps the framework with the verification. The practical part introduces the reader to the methods used to implement the Dinitz algorithm and also clarifies the annotations used for the algorithm verification.} +\ctufitkeywordsCZE{Frama-C, WP, RTE, ACSL, Dinitzův algoritmus, ověřená implementace} +\ctufitkeywordsENG{Frama-C, WP, RTE, ACSL, Dinitz algorithm, verified implementation} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% END FILL IN +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% CUSTOMIZATION of this template +% Skip this part or alter it if you know what you are doing. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{iftex}[2020/03/06] +\iftutex % XeLaTeX and LuaLaTeX + \RequirePackage{ellipsis}[2020/05/22] %ellipsis workaround for XeLaTeX +\else + \RequirePackage[utf8]{inputenc}[2018/08/11] %this file encoding + \RequirePackage{lmodern}[2009/10/30] % vector flavor of Computer Modern font +\fi + +% hyperlinks +\RequirePackage[pdfpagelayout=TwoPageRight,colorlinks=false,allcolors=decoration,pdfborder={0 0 0.1}]{hyperref}[2020-05-15] + +% uncomment the following to hide all hyperlinks +% \RequirePackage[pdfpagelayout=TwoPageRight,hidelinks]{hyperref}[2020-05-15] + +\RequirePackage{pdfpages}[2020/01/28] + +\setcounter{secnumdepth}{4} % numbering sections; 4: subsubsection + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% CUSTOMIZATION of this template END +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%%%%%%%%% +% DEMO CONTENTS SETTINGS +% You may choose to modify this part. +%%%%%%%%%%%%%%%%%%%%%% +\usepackage{booktabs} +\usepackage{dirtree} +\usepackage{lipsum,tikz} +\usepackage{csquotes} +\usepackage{xcolor} +\usepackage[style=iso-numeric]{biblatex} +\usepackage{threeparttable} +\usepackage{enumitem} +\addbibresource{text/bib-database.bib} +\usepackage{listings} % typesetting of sources +% \usepackage{minted} % typesetting of sources + +%theorems, definitions, etc. +\theoremstyle{plain} +\newtheorem{theorem}{VÄ›ta} +\newtheorem{lemma}[theorem]{TvrzenÃ} +\newtheorem{corollary}[theorem]{Důsledek} +\newtheorem{proposition}[theorem]{Návrh} +\newtheorem{definition}[theorem]{Definice} +\theoremstyle{definition} +\newtheorem{example}[theorem]{PÅ™Ãklad} +\theoremstyle{remark} +\newtheorem{note}[theorem]{Poznámka} +\newtheorem*{note*}{Poznámka} +\newtheorem{remark}[theorem]{PozorovánÃ} +\newtheorem*{remark*}{PozorovánÃ} +\numberwithin{theorem}{chapter} +%theorems, definitions, etc. END +%%%%%%%%%%%%%%%%%%%%%% +% DEMO CONTENTS SETTINGS END +%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} +\frontmatter\frontmatterinit % do not remove these two commands + +\includepdf[pages={1-}]{patermi2-assignment.pdf} % replace that file with your thesis assignment provided by study office + +\thispagestyle{empty}\cleardoublepage\maketitle % do not remove these three commands + +\imprintpage % do not remove this command + +\tableofcontents % do not remove this command +%%%%%%%%%%%%%%%%%%%%%% +% list of other contents: figures, tables, code listings, algorithms, etc. +% add/remove commands accordingly +%%%%%%%%%%%%%%%%%%%%%% +\listoffigures % list of figures +\begingroup +\let\clearpage\relax +\listoftables % list of tables +\lstlistoflistings % list of source code listings generated by the listings package +% \listoflistings % list of source code listings generated by the minted package +\endgroup +%%%%%%%%%%%%%%%%%%%%%% +% list of other contents END +%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%% +% ACKNOWLEDGMENT +% FILL IN / MODIFY +% This is a place to thank people for helping you. It is common to thank your supervisor. +%%%%%%%%%%%%%%%%%%% +\begin{acknowledgmentpage} + ChtÄ›l bych podÄ›kovat doc. RNDr. DuÅ¡an Knop, Ph.D za vÅ¡echen Äas, který vÄ›noval mÄ› a mojà práci. +\end{acknowledgmentpage} +%%%%%%%%%%%%%%%%%%% +% ACKNOWLEDGMENT END +%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%%%%%% +% DECLARATION +% FILL IN / MODIFY +%%%%%%%%%%%%%%%%%%% +% INSTRUCTIONS +% ENG: choose one of approved texts of the declaration. DO NOT CREATE YOUR OWN. Find the approved texts at https://courses.fit.cvut.cz/SFE/download/index.html#_documents (document Declaration for FT in English) +% CZE/SLO: Vyberte jedno z fakultou schvalenych prohlaseni. NEVKLADEJTE VLASTNI TEXT. Schvalena prohlaseni najdete zde: https://courses.fit.cvut.cz/SZZ/dokumenty/index.html#_dokumenty (prohlášenà do ZP) +\begin{declarationpage} +ProhlaÅ¡uji, že jsem pÅ™edloženou práci vypracoval samostatnÄ› a že jsem uvedl veÅ¡keré použité +informaÄnà zdroje v souladu s Metodickým pokynem o dodržovánà etických principů pÅ™i pÅ™ÃpravÄ› +vysokoÅ¡kolských závÄ›reÄných pracÃ. +Beru na vÄ›domÃ, že se na moji práci vztahujà práva a povinnosti vyplývajÃcà ze zákona Ä. 121/2000 Sb., +autorského zákona, ve znÄ›nà pozdÄ›jÅ¡Ãch pÅ™edpisů, zejména skuteÄnost, že ÄŒeské vysoké uÄenà +technické v Praze má právo na uzavÅ™enà licenÄnà smlouvy o užità této práce jako Å¡kolnÃho dÃla podle § +60 odst. 1 citovaného zákona +\end{declarationpage} +%%%%%%%%%%%%%%%%%%% +% DECLARATION END +%%%%%%%%%%%%%%%%%%% + +\printabstractpage % do not remove this command + + +%%%%%%%%%%%%%%%%%%% +% ABBREVIATIONS +% FILL IN / MODIFY +% OR REMOVE ENTIRELY +% List the abbreviations in lexicography order. +%%%%%%%%%%%%%%%%%%% +\chapter{Seznam zkratek} + +\begin{tabular}{rl} +Frama-C & Framework for Modular Analysis of C programs\\ +ACSL & ANSI/ISO C Specification Language\\ +RTE & Runtime Error \\ +WP & Weakest Precondition\\ +\end{tabular} +%%%%%%%%%%%%%%%%%%% +% ABBREVIATIONS END +%%%%%%%%%%%%%%%%%%% + +\mainmatter\mainmatterinit % do not remove these two commands + +%%%%%%%%%%%%%%%%%%% +% THE THESIS +% MODIFY ANYTHING BELOW THIS LINE +%%%%%%%%%%%%%%%%%%% + +\include{text/text} % include `text.tex' from `text/' subdirectory + +\appendix\appendixinit % do not remove these two commands + +\include{text/appendix} % include `appendix.tex' from `text/' subdirectory + +\backmatter % do not remove this command + +\printbibliography % print out the BibLaTeX-generated bibliography list + +\include{text/medium} % include `medium.tex' from `text/' subdirectory + +\end{document} diff --git a/thesis/pictures/frama-c-BFS-1.PNG b/thesis/pictures/frama-c-BFS-1.PNG new file mode 100644 index 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b/thesis/text/appendix.tex @@ -0,0 +1,2 @@ +%\chapter{NÄ›jaká pÅ™Ãloha} +%Sem pÅ™ijde to, co nepatřà do hlavnà Äásti. diff --git a/thesis/text/bib-database.bib b/thesis/text/bib-database.bib new file mode 100644 index 0000000..7d4bc30 --- /dev/null +++ b/thesis/text/bib-database.bib @@ -0,0 +1,275 @@ +@InProceedings{compiler:analysis, +author={Cousot, Patrick and Cousot, Radhia}, +editor={Horspool, R. Nigel}, +title={Modular Static Program Analysis}, +booktitle={Compiler Construction}, +year={[online]. 2002 [cit. 2023-04-30]}, +publisher={Springer Berlin Heidelberg}, +address={Berlin, Heidelberg}, +pages={159--179}, +isbn={978-3-540-45937-8}, +doi={10.1007/3-540-45937-5_13} +} + +@book{PruvodceLab, + author = {MAREÅ , Martin and VALLA, Tomáš}, + title = {Průvodce labyrintem algoritmů}, + publisher = {CZ.NIC, z.s.p.o.}, + address = {Praha}, + year = {2022}, + edition = {2. vydánÃ}, + isbn = {978-80-88168-66-9}, +} + +@misc{predAG2, + title = {SÃtÄ›, toky v sÃtÃch, Fordův-Fulkersonův algoritmus.}, + author = {SUCHÃ, OndÅ™ej and VALLA, Tomáš}, + year = {[online prezentace]. 2020 [cit. 2023-04-28]}, + accessed = {2023-04-28}, + url = {https://courses.fit.cvut.cz/BI-AG2/media/lectures/bi-ag2-p03.pdf} +} + + +@Inbook{Dinitz2006, +author={Dinitz, Yefim}, +editor={GOLDREICH, Oded +and ROSENBERG, Arnold L. +and SELMAN, Alan L.}, +title={Dinitz' Algorithm: The Original Version and Even's Version}, +bookTitle={Theoretical Computer Science: Essays in Memory of Shimon Even}, +year={[online]. 2006 [cit. 2023-05-02]}, +publisher={Springer Berlin Heidelberg}, +address={Berlin, Heidelberg}, +pages={218--240}, +isbn={978-3-540-32881-0}, +doi={10.1007/11685654_10}, +url={https://doi.org/10.1007/11685654_10} +} + + +@misc{FordFulEx, +author = {Neto, Euclides and Callou, Gustavo}, +year = {[online]. 2015 [cit. 2023-05-02]}, +pages = {76-79}, +title = {An Approach Based on Ford-Fulkerson Algorithm to Optimize Network Bandwidth Usage}, +doi = {10.1109/SBESC.2015.21} +} + +@article{EdKarpEx, +author = {Mallick, Kalyan and Khan, Aminur and Ahmed, M. and Arefin, Md and Uddin, Md}, +year = {[online]. 2016 [cit. 2023-05-02]}, +month = {02}, +pages = {131-140}, +title = {Modified EDMONDS-KARP Algorithm to Solve Maximum Flow Problems}, +volume = {6}, +journal = {Open Journal of Applied Sciences} +} + +@article{GoldEx, +author = {Goldberg, Andrew V. and Tarjan, Robert E.}, +title = {A New Approach to the Maximum-Flow Problem}, +year = {[online]. 1988 [cit. 2023-05-02]}, +issue_date = {Oct. 1988}, +publisher = {Association for Computing Machinery}, +address = {New York, NY, USA}, +volume = {35}, +number = {4}, +issn = {0004-5411}, +url = {https://doi.org/10.1145/48014.61051}, +doi = {10.1145/48014.61051}, +journal = {J. ACM}, +month = {oct}, +pages = {921–940}, +numpages = {20} +} + +@article{MKMEX, +author = {Malhotra, Vishv and Kumar, M.Pramodh and Maheshwari, S.N.}, +year = {[online]. 1978 [cit. 2023-05-02]}, +month = {10}, +pages = {277–278}, +title = {An O(V3) Algorithm for Finding Maximum Flows in Networks}, +volume = {7}, +journal = {Information Processing Letters}, +doi = {10.1016/0020-0190(78)90016-9} +} + +@inproceedings{KRTEX, +author = {King, Valerie and Rao, Satish and Tarjan, Robert}, +year = {[online]. 1992 [cit. 2023-05-02]}, +month = {01}, +pages = {157-164}, +title = {A Faster Deterministic Maximum Flow Algorithm} +} + +@misc{KatEX, + title={A Potential Reduction Inspired Algorithm for Exact Max Flow in Almost $\widetilde{O}(m^{4/3})$ Time}, + author={Tarun Kathuria}, + year={[online]. 2020 [cit. 2023-05-02]}, + eprint={2009.03260}, + archivePrefix={arXiv}, + primaryClass={cs.DS} +} + +@article{DijkEx, +author = {Javaid, Adeel}, +year = {[online]. 2013 [cit. 2023-05-03]}, +month = {01}, +pages = {}, +title = {Understanding Dijkstra Algorithm}, +journal = {SSRN Electronic Journal}, +doi = {10.2139/ssrn.2340905} +} + +@article{BFSEx, +author = {Holdsworth, Jason}, +year = {[online]. 1999 [cit. 2023-05-03]}, +month = {02}, +pages = {}, +title = {The Nature of Breadth-First Search}, +url = {https://www.researchgate.net/publication/273264449_Understanding_Dijkstra_Algorithm} +} + +@inproceedings{DFSEx, +author = {Putri, Sheila Eka and Tulus, Tulus and Napitupulu, Normalina}, +year = {[online]. 2011 [cit. 2023-05-03]}, +month = {08}, +pages = {}, +title = {Implementation and Analysis of Depth-First Search (DFS) Algorithm for Finding The Longest Path}, +doi = {10.13140/2.1.2878.2721} +} + +@online{ACSLMan, + title = {ACSL: ANSI/ISO C Specification Language}, + subtitle = {Version 1.18}, + urldate = {2023-05-05}, + url = {https://frama-c.com/download/frama-c-acsl-implementation.pdf}, +} + +@online{WPMan, + title = {WP Plug-in Manual}, + subtitle = {Frama-C 26.1}, + urldate = {2023-05-05}, + url = {https://frama-c.com/download/frama-c-wp-manual.pdf}, +} + + +@online{RTEMan, + title = {Frama-C’s RTE plug-in}, + subtitle = {for Frama-C 26.1}, + urldate = {2023-05-05}, + url = {https://frama-c.com/download/frama-c-rte-manual.pdf}, +} + +@online{ERTEMan, + title = {E-RTE plug-in Manual}, + subtitle = {for Frama-C 26.1}, + urldate = {2023-05-05}, + url = {https://frama-c.com/download/e-acsl/e-acsl-manual.pdf}, +} + +@online{altEx, + title = {Alt-Ergo Documentation}, + urldate = {2023-05-05}, + url = {https://ocamlpro.github.io/alt-ergo/}, +} + +@online{cvcEx, + title = {CVC4 Documentation}, + urldate = {2023-05-05}, + url = {https://cvc4.github.io/documentation.html/}, +} + + +@online{z3Ex, + title = {Z3 Theorem Prover}, + urldate = {2023-05-05}, + url = {https://github.com/z3prover/}, +} + + + +@online{why3Ex, + title = {Why3 Documentation}, + urldate = {2023-05-05}, + url = {https://why3.lri.fr/#documentation}, +} + +@online{sparkEx, + title = {Spark About Page}, + urldate = {2023-05-05}, + url = {https://www.adacore.com/about-spark}, +} +@online{Atelier, + title = {Atelier B Main Page}, + urldate = {2023-05-05}, + url = {https://www.atelierb.eu/en/}, +} +@online{dafnyEx, + title = {Dafny Reference Manual}, + urldate = {2023-05-05}, + url = {https://dafny.org/latest/DafnyRef/DafnyRef}, +} + +@online{symbiEx, + title = {SymbiYosis Documentation}, + urldate = {2023-05-05}, + url = {https://symbiyosys.readthedocs.io/en/latest/index.html}, +} + +@online{viperEx, + title = {Viper Research}, + urldate = {2023-05-05}, + url = {https://www.pm.inf.ethz.ch/research/viper.html}, +} + +@article{Aceto_2013, + doi = {10.4204/eptcs.120.8}, + url = {https://doi.org/10.4204\%2Feptcs.120.8}, + year = 2013, + month = {jul}, + publisher = {Open Publishing Association}, + volume = {120}, + pages = {93--107}, + author = {Luca Aceto and Eugen-Ioan Goriac and Anna Ingolfsdottir}, + title = {Meta {SOS} - A Maude Based {SOS} Meta-Theory Framework}, + journal = {Electronic Proceedings in Theoretical Computer Science} +} + +@inbook{dinitzRoads, +author = {Bhadra, Somasree and Kundu, Anirban and Khatua, Sunirmal}, +year = {[online]. 2021 [cit. 2023-05-03]}, +month = {04}, +pages = {372-379}, +title = {Optimization of Road Traffic Congestion in Urban Traffic Network Using Dinic’s Algorithm}, +isbn = {978-3-030-73602-6}, +doi = {10.1007/978-3-030-73603-3_34} +} + +@inbook{dinitzNet, +author = {Haque, Md and Isla, Md}, +year = {[online]. 2020 [cit. 2023-05-03]}, +month = {09}, +pages = {}, +title = {Traffic model of LTE using maximum flow algorithm with binary search technique} +} + +@misc{dinitzSit, + title={Near-Optimal Distributed Maximum Flow}, + author={Mohsen Ghaffari and Andreas Karrenbauer and Fabian Kuhn and Christoph Lenzen and Boaz Patt-Shamir}, + year={[online]. 2015 [cit. 2023-05-03]}, + eprint={1508.04747}, + archivePrefix={arXiv}, + primaryClass={cs.DS} +} + +@article{dinitzProc, +author = {Shiloach, Yossi and Vishkin, Uzi}, +year = {[online]. 1982 [cit. 2023-05-03]}, +month = {06}, +pages = {128-146}, +title = {An O(n2 log(n)) Parallel Max-Flow Algorithm}, +volume = {3}, +journal = {J. Algorithms}, +doi = {10.1016/0196-6774(82)90013-X} +} \ No newline at end of file diff --git a/thesis/text/medium.tex b/thesis/text/medium.tex new file mode 100644 index 0000000..cc35b95 --- /dev/null +++ b/thesis/text/medium.tex @@ -0,0 +1,11 @@ +\chapter{Obsah pÅ™iloženého média} + + + \dirtree{% + .1 patermi2-thesis.pdf\DTcomment{text práce ve formátu PDF}. + .1 src + .2 dinitz.c\DTcomment{zdrojový kód Dinitzova algoritmu}. + .2 dinitz.h\DTcomment{hlaviÄkový soubor Dinitzova algoritmu}. + .2 main.c\DTcomment{zdrojové kód pro spuÅ¡tÄ›nà Dinitzova algoritmu }. + .1 thesis\DTcomment{zdrojová forma práce ve formátu \LaTeX{}}. + } diff --git a/thesis/text/text.tex b/thesis/text/text.tex new file mode 100644 index 0000000..971c3c9 --- /dev/null +++ b/thesis/text/text.tex @@ -0,0 +1,801 @@ +% Do not forget to include Introduction +%--------------------------------------------------------------- +\chapter*{Úvod}\addcontentsline{toc}{chapter}{Úvod}\markboth{Úvod}{Úvod} +%--------------------------------------------------------------- +\setcounter{page}{1} +V dneÅ¡nà dobÄ› se programy staly nedÃlnou souÄástà naÅ¡Ã spoleÄnosti, Å™Ãdà naÅ¡Ã infrastrukturu poÄÃnaje od základnÃch lidských potÅ™eb, napÅ™Ãklad dostupnost pitné vody, zemnÃho plynu až po provoz jaderných elektráren. StejnÄ› tak programy spravujà mnoho dalÅ¡Ãch systémů, které spoleÄnost využÃvá, mezi jinými peněžnà systémy jako jsou platby, bankovnà úÄty, danÄ› nebo také dopravnà systémy, aÅ¥ už jsou to semafory na silnicÃch nebo Å™Ãzenà vlaků, lodà Äi letadel. + +Ve zkratce se programy vyskytujà vÅ¡ude kolem nás a o to vÄ›tÅ¡Ã je poté problém, když v nÄ›kterém z tÄ›chto programů nastane chyba. Tyto chyby mohou zavinit peněžnà ztráty od desÃtek, nebo stovek tisÃc korun až po trvalé následky, Äi dokonce ztráty na životech. A proto se s každým rokem klade vÄ›tÅ¡Ã důraz na prověřovánà tÄ›chto programů, abychom se chybám vyhnuli. + +Jednou z nejÄastÄ›ji použÃvaných metod je testovánà programu. Základnà myÅ¡lenkou tohoto testovánà je vytvoÅ™enà velkého množstvà vstupů, které jsou náhodné a poté menÅ¡Ã Äást vstupů, které byly vytvoÅ™eny programátorem nebo testerem a jsou zaměřené na krajnà Äi zranitelnÄ›jÅ¡Ã Äásti programu. Tento postup nám dává jistotu, že ve vÄ›tÅ¡inÄ› pÅ™Ãpadů program bude fungovat správnÄ›, ale nezaruÄuje nám bezchybnost daného programu a jak už bylo zmÃnÄ›no i jedna malá chyba může vést na katastrofálnà následky. + +Z tohoto důvodu se tato práce bude zabývat statickou a hlavnÄ› formálnà analýzou programu ~\cite{compiler:analysis}. Na rozdÃl od testovánÃ, statická analýza kontroluje program bez nutnosti spuÅ¡tÄ›nà a dokáže verifikovat, zda program je korektnÃ. Kvůli nerozhodnutelnosti nÄ›kterých problémů, ne vÅ¡echny chyby lze oznaÄit za vyÅ™eÅ¡ené, nebo nemožné a tedy malé procento chyb zůstává možnou hrozbou, ale dosahujeme zde dostateÄných procent korektnosti a bezchybnosti, abychom mohli daný program využÃvat v důležitých mÃstech. + +Formálnà analýza je rozÅ¡ÃÅ™enou statickou analýzou, která využÃvá matematických operacà a tvrzenÃ, aby vyÅ™eÅ¡ila složité problémy v programu. ÚÄelem formálnà analýzy je tedy dokázat, aÅ¥ už pomocà vlastnÃch vygenerovaných důkazů, nebo programátorem dodaných, že program je korektnÃ. V této práci budeme využÃvat formálnà analýzu k testovánà korektnosti implementace Dinitzova algoritmu. + +Dinitzův algoritmus je jednÃm z postupů, jak zÃskat maximálnà tok v sÃtÃch. Maximálnà tok nám pÅ™edstavuje množstvà média napÅ™. vody, elektÅ™iny nebo i lidÃ, které dokážeme pÅ™enést Äi pÅ™evést z bodu A do bodu B. Mnoho situacà z naÅ¡eho života lze tedy pÅ™eformulovat na problém nalezenà maximálnÃho toku, jak už bylo zmÃnÄ›no doruÄenà média, pÅ™eprava lidà nebo Å™Ãzenà leteckého provozu. + + + + +\newpage{} + +HlavnÃm cÃlem práce je vytvoÅ™it implementaci Dinitzova algoritmu, pro hledánà maximálnÃho toku v orientované grafové sÃti, napsáném v programovacÃm jazyce C a poté ověřenà implementace Dinitzova algoritmu pomocà anotaÄnÃho jazyka ACSL a prostÅ™edà Frama-C. + +ZámÄ›rem teoretické Äásti práce je seznámenà Ätenáře se zakládanými pojmy z oblasti toků v orientovaných grafových sÃtÃch a dále popsánà a vysvÄ›tlenà Dinitzova algoritmu. + +ÚÄelem praktické Äásti práce je následné popsánà a vysvÄ›tlenà konceptů využitých pro implementaci, ověřenà Dinitzova algoritmu a diskuze, které Äásti nebo aspekty nebylo možné ověřit. + + + + +%--------------------------------------------------------------- +\chapter{Toky v sÃtÃch} +%--------------------------------------------------------------- +V této kapitole uvedeme základnà pojmy a definice teorie grafů a toků v sÃtÃch, které budou dalÅ¡Ã kapitoly využÃvat. Poté pÅ™edstavÃme problém maximálnÃho toku, způsoby Å™eÅ¡enà a situace z reálného života, které můžeme pÅ™evést na problém maximálnÃho toku. HlavnÃm zdrojem informacà jsou materiály z pÅ™edmÄ›tu BI-AG2~\cite{predAG2} a kniha \textit{Průvodce labyrintem algoritmů}~\cite{PruvodceLab} + +\section{Teorie a definice} + +\begin{definition}[Orientovaný graf~\cite{PruvodceLab}] \label{def:orientGraf} + \emph{Orientovaný graf} G je uspořádaná dvojice $(V, E)$, kde V je neprázdná koneÄná +množina vrcholů a $E \subseteq V \times V$ je množina orientovaných hran. \emph{Orientovaná hrana} $(u, v) \in{} E$ je uspořádaná dvojice různých vrcholů $u, v \in{} V$, +nÄ›kdy ji zkrácenÄ› znaÄÃme uv. +\end{definition} + +\begin{definition} [SÃÅ¥~\cite{predAG2}] \label{def:Sit} + \emph{SÃtÃ} nazveme ÄtveÅ™ici $(G, z, s, c)$, kde $G = (V,E)$ je orientovaný graf, $z$ a $s$ dva různé vrcholy grafu G (Å™Ãkáme jim $zdroj$ a $stok$) a kapacita $c\colon{}E \rightarrow{} \mathbb{R} _Q^+$ je funkce ohodnocujÃcà hrany nezápornými reálnými ÄÃsly. +\end{definition} + +\begin{definition} [Tok v sÃti~ \cite{predAG2}] \label{def:vlastToku} + \emph{Tok v sÃti $(G, z, s, c)$} je každá funkce $f\colon{}E \rightarrow{} \mathbb{R} _0^+$ splňujÃcÃ: + \begin{itemize} + \item Pro každou hranu e \in{} E platà 0 \le{} f(e) \le{} f(c). + \item Pro každý vrchol u \in{} V mimo zdroj a stok platÃ: + \end{itemize} + \[ \sum_{(x, u)\in{}E} f(x, u) = \sum_{(u, y)\in{}E} f(u,y) \] +\end{definition} + +\begin{definition} [Velikost toku~\cite{predAG2}] \label{def:velikToku} + \emph{Velikost toku} je: + \[ w(f) = \sum_{(z, x)\in{}E} f(z, x) - \sum_{(x, z)\in{}E} f(x, z) \] +\end{definition} + +Pozor na to, že u orientovaných grafů se pojem zdroj použÃvá pro libovolný vrchol, do nÄ›jž nevedou žádné hrany. Zdrojem v sÃti se oznaÄuje právÄ› jeden vrchol a hrany do nÄ›j dovolujeme. Analogicky pro stok.~\cite{PruvodceLab} + +SÃÅ¥ si lze pÅ™edstavovat jako soustavu jednosmÄ›rných vodovodnÃch trubek, kde každá má svojà šÃÅ™ku, která urÄuje kolik vody lze danou trubkou transportovat. Každá trubka tedy může být různÄ› veliká a propouÅ¡tÄ›t různé množstvà vody, zároveň nÄ›kolik trubek se může setkat v jednom mÃstÄ›, takovému mÃstu budeme Å™Ãkat vrchol. Vrchol má tedy za úkol pouze trubky spojit tak, aby žádná voda nevytekla a sám žádnou vodu neudržoval, sÃÅ¥ tedy ,,tÄ›snÃ"\footnote{Také známe jako Kirchhoffův zákon.}. Dále má dvÄ› vyznaÄená mÃsta -- zdroj a stok. Ve zdroji pouÅ¡tÃme vodu dovnitÅ™ a ve stoku voda vytéká ven. Tok je poté rozvrženÃ, kterými trubkami voda proudà a jakou intenzitou. + +\begin{definition} [Průtok v sÃti~\cite{PruvodceLab}] \label{def:prutSit} + Každé hranÄ› uv pÅ™iÅ™adÃme jejà \emph{průtok} f*(uv) = f(uv) − f(vu). +\end{definition} +\begin{remark} \label{poz:vlastPrut} + Průtoky majà následujÃcà vlastnosti: + \begin{enumerate} + \item f*(uv) = −f*(vu) + \item f*(uv) \le{} c(uv) + \item f*(uv) \geq{} -c(uv) + \item pro vÅ¡echny vrcholy v, kde v \neq{} z,s platÃ: \[ \sum_{u:uv\in{}E}f^*(uv) = 0 \] + \end{enumerate} +\end{remark} + +\begin{definition} [Rezerva hrany~\cite{PruvodceLab}] \label{def:rezHra} + \emph{Rezerva hrany} uv k toku f v sÃti S = $(G, z, s, c)$ je ÄÃslo r(uv) := c(uv) − f(uv) + f(vu). HranÄ› s nulovou rezervou budeme Å™Ãkat \emph{nasycená}, hranÄ› s kladnou rezervou \emph{nenasycená}. O cestÄ› Å™ekneme, že je nasycená, pokud je nasycená alespoň jedna jejà hrana; jinak majà vÅ¡echny hrany kladné rezervy a cesta je nenasycená. +\end{definition} + +\begin{definition} [SÃÅ¥ rezerv~\cite{PruvodceLab}] \label{def:sitRezer} + \emph{SÃÅ¥ rezerv} k toku f v sÃti S = (G, z, s, c) je sÃÅ¥ R(S, f)~:=~(G,~z,~s,~r), kde r(e) je rezerva hrany e pÅ™i toku f. +\end{definition} + +\section{Problém maximálnÃho toku} + +Problém maximálnÃho toku si tedy můžeme zadefinovat jako nalezenà způsobu jak nejlépe smÄ›rovat médium ze zdroje do stoku, tak abychom dosáhli maximálnÃho toku v dané sÃti. VÅ¡imnÄ›me si, že může existovat nÄ›kolik způsobů jak maximálnÃho toku dosáhnout obrázek~\ref{fig:sitVicToku}. Zároveň maximálnà tok se může rovnat nule, pokud neexistuje cesta složená z hran o kladné kapacitÄ› ze zdroje do stoku obrázek~\ref{fig:sitTokuNula}. +\begin{figure}[ht] + \centering + \begin{minipage}[c]{0.4\textwidth} + \centering + \begin{tikzpicture}[node distance={22mm}, thick, main/.style = {draw, circle}] + \node[main] (2) {$z$}; + \node[main] (3) [below right of=2] {$B$}; + \node[main] (4) [above right of=2] {$A$}; + \node[main] (5) [above right of=3] {$s$}; + \draw[->] (2) -- node[midway, above, sloped] {15} (3); + \draw[->] (2) -- node[midway, above, sloped] {15} (4); + \draw[->] (3) -- node[midway, above, sloped] {10} (5); + \draw[->] (4) -- node[midway, above, sloped] {5} (5); + \draw[->] (3) to [bend right=20] node[midway, above, sloped, rotate=180] (TextNode1) {15} (4); + \draw[->] (4) to [bend right=20] node[midway, above, sloped] (TextNode2) {15} (3); + \end{tikzpicture} + \end{minipage} + \begin{minipage}[c]{0.5\textwidth} + \centering + Prvnà z možných maximálnÃch toků: z\rightarrow{}A(15), A\rightarrow{}s(5), A\rightarrow{}B(10), B\rightarrow{}s(10) \\ + Druhý z možných maximálnÃch toků: z\rightarrow{}B(15), B\rightarrow{}s(10), B\rightarrow{}A(5), A\rightarrow{}s(5) + \end{minipage} + \caption{Ukázka sÃtÄ› s vÃce možnými maximálnÃmi toky} + \label{fig:sitVicToku} +\end{figure} + + + +\begin{figure}[ht] + \centering + \begin{tikzpicture}[node distance={15mm}, thick, main/.style = {draw, circle}] + \node[main] (2) {$z$}; + \node[main] (3) [below right of=2] {$B$}; + \node[main] (4) [above right of=2] {$A$}; + \node[main] (5) [above right of=3] {$s$}; + \draw[->] (2) -- node[midway, above, sloped] {5} (3); + \draw[->] (2) -- node[midway, above, sloped] {8} (4); + \draw[->] (4) to [bend left=20] node[midway, above, sloped] (TextNode2) {1} (3); + \end{tikzpicture} + \caption{Ukázka sÃtÄ› s maximálnÃm tokem o velikosti 0} + \label{fig:sitTokuNula} +\end{figure} +\newpage{} +\section{Algoritmy pro nalezenà maximálnÃho toku} +\begin{itemize} + \item Ford–Fulkersonův algoritmus~\cite{FordFulEx} + + Ford–Fulkersonův algoritmus je založen na jednoduché myÅ¡lence, mÄ›jme nulový tok a zlepÅ¡ujme ho, dokud je to možné a tÃm dostaneme maximálnà tok. Jinými slovy tento algoritmus vyhledává nenasycenou cestu z-s, pÅ™idá minimálnà tok, který tuto cestu nasytà a poté proces opakuje dokud dalÅ¡Ã cestu z-s nenajde. To ale pÅ™Ãmo nezaruÄuje správnost, může se stát, že zasytà cestu, která v jiném poÅ™adà by zvÄ›tÅ¡ila aktuálnà tok a je zapotÅ™ebà pro nalezenà maximálnÃho toku. Algoritmus tento problém Å™eÅ¡Ã pomocà jednoduchého triku, a to možnost zmenÅ¡ovat aktuálnà tok pÅ™i průchodu proti smÄ›ru hrany. Neboli pÅ™idáme do naÅ¡Ã sÃtÄ› rezervy hrany podle definicÃ~\ref{def:rezHra}~a~\ref{def:sitRezer}. VÃce si o Ford–Fulkersonův algoritmu a jeho využità si můžete pÅ™eÄÃst v této práci + \item Edmonds-Karpův algoritmus~\cite{EdKarpEx} + + Edmonds-Karpův algoritmus je velice podobný Ford–Fulkersonovu algoritmu, zaÄÃná stejnÄ› s nulovým tokem a hledá zlepÅ¡ujÃcà cesty, jenže tentokrát nevybÃrá náhodnou cestu, ale cestu nejkratÅ¡Ã. + \item Dinitzův algoritmus + + Dinitzovu algoritmu se budeme detailnÄ›ji zabývat v dalÅ¡Ã kapitole \ref{dinitz}, proto ho zde pÅ™eskoÄÃme. + + \item Goldbergův algoritmus~\cite{GoldEx} + + Goldbergův algoritmus na zaÄátku nastavà vÅ¡em vrcholům ,,výšku" a pÅ™ebytek, výšku si můžeme pÅ™edstavit jako tÅ™età osu pro naÅ¡Ã sÃÅ¥, tedy doopravdy si můžeme pÅ™edstavit sÃÅ¥ jako 3D objekt, výška vÅ¡ech vrcholů se ze zaÄátku rovná poÄtu vrcholů. PÅ™ebytek znázorňuje kolik média aktuálnÄ› vrchol~v~sobÄ› držÃ. Algoritmus běžà dokud existuje vrchol s pÅ™ebytkem. V průbÄ›hu algoritmus posouvá pÅ™ebytek po hranách které nevedou do vrcholů které jsou výš. + \item a mnoho dalÅ¡Ãch jako jsou napÅ™Ãklad algoritm třà indů (MKM)~\cite{MKMEX}, KRT~\cite{KRTEX} nebo algoritmus Kathuria-Liu-Sidford~\cite{KatEX} +\end{itemize} + +\begin{table}[ht!] +\centering +\begin{threeparttable} +\begin{tabular}{@{}lll@{}} +\toprule +Název algoritmu & ÄŒasová složitost & Paměťová složitost \\ \midrule +Ford-Fulkerson & $O(E \cdot \mathrm{max\_flow})$ & $O(V + E)$ \\ +Edmonds-Karp & $O(V \cdot E^2)$ & $O(V + E)$ \\ +Dinitz's & $O(V^2 \cdot E)$ & $O(V^2 + E)$ \\ +Push-relabel & $O(V^2 \cdot E)$ & $O(V^2)$ \\ +Push–relabel with FIFO vertex selection & $O(V^3)$ & $O(V^2)$ \\ +Karp-Stein (capacity scaling) & $O(V \cdot E \cdot \log C)$ & $O(V + E)$ \\ +FIFO Preflow Push & $O(V^2 \cdot \sqrt{E})$ & $O(V^2)$ \\ +Boykov-Kolmogorov & $O(V \cdot E^2)$ & $O(V^2 + E)$ \\ +Malhotra-Kumar-Maheshwari & $O(V^3)$ & $O(V^2)$ \\ +Ahuja-Orlin & $O(V^2 \cdot E)$ & $O(V^2)$ \\ +Goldberg-Rao & $O(\min(V^{\frac{2}{3}}, E^{\frac{1}{2}}) \cdot E \cdot \log C)$ & $O(V + E)$ \\ +Cheriyan-Maheshwari-Mehlhorn & $O(E \cdot \log^2 V)$ & $O(V + E)$ \\ +King-Rao-Tarjan & $O(V^{\frac{3}{2}} \cdot \log V)$ & $O(V^2)$ \\ +\bottomrule +\end{tabular} +\caption{Porovnánà algoritmů pro Å™eÅ¡enà problému maximálnÃho toku} +\label{tab:maxFlowTable} +\begin{tablenotes} +\item \textbf{Poznámka:} E znaÄà poÄet hran v grafu, V znaÄà poÄet vrcholů v grafu a C znaÄà kapacitu hran v grafu. +\end{tablenotes} +\end{threeparttable} +\end{table} + +%--------------------------------------------------------------- +\chapter{Dinitzův algoritmus} \label{dinitz} +%--------------------------------------------------------------- + +Dinitzův algoritmus, také znám jako Dinicův algoritmus je efektivnÃm způsobem pro Å™eÅ¡enà problému maximálnÃho toku v sÃti S = (G, z, s, c). Tento algoritmus byl vyvinut Yefimem Dinitzem v roce 1970 a je založen na myÅ¡lence použÃvánà blokujÃcÃch toků a proÄiÅ¡tÄ›né sÃtÄ›. +\begin{definition} [BlokujÃcà tok] \label{def:blokTok} + \emph{BlokujÃcÃm tokem} (ang. Blocking Flow) nazýváme tok, který na každé cestÄ› uv obsahuje alespoň jednu hranu nasycenou, neboli můžeme blokujÃcà tok také brát jako tok o minimálnà velikosti, který nasytà cesty uv z definice \ref{def:rezHra}. +\end{definition} +\begin{definition} [ProÄiÅ¡tÄ›ná sÃÅ¥~\cite{PruvodceLab} ] \label{def:cistaSit} + \emph{ProÄiÅ¡tÄ›nou sÃtÃ} (ang. Layered network) budeme nazývat sÃÅ¥ T = (H, z, s, c), kde H je podgrafem grafu G sÃtÄ› S = (G, z, s, c) a podgraf H obsahuje pouze vrcholy a hrany, které ležà na nejkratÅ¡Ã cestách zs. +\end{definition} +Jak tedy Dinitzův algoritmus využÃvá blokujÃcà toky a proÄiÅ¡tÄ›né sÃtÄ›? Pokud bychom chtÄ›li najÃt blokujÃcà tok na celé sÃti, tak podle definice najdeme tok, který nasytà urÄité hrany tak, aby neexistovala žádná dalÅ¡Ã nenasycená cesta ze zdroje do stoku, neboli naÅ¡li bychom maximálnà tok v sÃti. Z tohoto důvodu nebudeme hledat blokujÃcà tok na celé sÃti, ale na proÄiÅ¡tÄ›né sÃti, pokud se zase zamyslÃme co to znamená, tak zjistÃme, že dostaneme tok o velikosti rovném souÄtu kapacit hran tak, aby vÅ¡echny nejkratÅ¡Ã cesty zs byly nasyceny. + +Pokud znova aplikujeme proÄiÅ¡tÄ›nà sÃtÄ› s novými hranami, které byly nasyceny, tak dostaneme dalÅ¡Ã proÄiÅ¡tÄ›nou sÃÅ¥ s novými nenasycenými nejkratÅ¡Ãmi cestami, na které znova najdeme blokujÃcà tok a tento postup opakujeme dokud existuje nenasycená cesta zs v sÃti. Pomocà tÄ›chto blokujÃcÃch toků poté zlepÅ¡ujeme maximálnà tok.\\ +Postup Dinitzova algoritmu~\cite{PruvodceLab}: \\ +\textit{Vstup: SÃÅ¥ $S = (G, z, s, c)$} +\begin{itemize} + \item $f \leftarrow$ nulový tok + \item Opakujeme: + \begin{enumerate} + \item SestrojÃme proÄiÅ¡tÄ›nou sÃÅ¥ H ze sÃtÄ› S. + \item Najdeme nejkratÅ¡Ã cesty v sÃti H. + \item Pokud žádná taková cesta neexistuje, zastavÃme se a vrátÃme tok $f$. + \item $g \leftarrow$ blokujÃcà tok v $R$. + \item ZlepÅ¡Ãme tok $f$ pomocà $g$. + \item UpravÃme aktuálnà sÃÅ¥ S podle toku $g$. + \end{enumerate} +\end{itemize} +Výstup: Maximálnà tok $f$. + + +DovysvÄ›tlÃme jeÅ¡tÄ› ÄiÅ¡tÄ›nà sÃtÄ›, jak už jsme zmÃnili, jedná se o sÃÅ¥ tvoÅ™enou podgrafem H z grafu G. Způsob jakým můžeme jednoduÅ¡e proÄiÅ¡tÄ›nou sÃÅ¥ vytvoÅ™it se zakládá na myÅ¡lence rozdÄ›lenà vrcholů sÃtÄ› do skupin podle vzdálenosti od zdroje. MÄ›jme tedy rozdÄ›lené vrcholy do vhodných vrstev, poté odeberme vÅ¡echny vrstvy, které majà vzdálenost vÄ›tÅ¡Ã než vzdálenost stoku, poté odeberme vÅ¡echny hrany které vedly do tÄ›chto vrcholů, odeberme také vÅ¡echny hrany které vedou mezi vrcholy stejné vrstvy a hrany které vedou do vrstev bližšÃch ke zdroji. Na konec odeberme vÅ¡echny vrcholy, z kterých nevede žádná hrana.\footnote{Nasycené hrany, budeme považovat za smazané.} +\section{Implementace} +Existuje nÄ›kolik způsobů implementace Dinitzova algoritmu a volba konkrétnà implementace může záviset na programovacÃm jazyce, použitých datových strukturách a konkrétnÃch požadavcÃch problému. +\begin{enumerate} + \item \textit{VýbÄ›r datových struktur}: Pro efektivnà implementaci Dinicova algoritmu je důležité vybrat vhodné datové struktury pro reprezentaci grafu, kapacit hran a toku. BěžnÄ› se použÃvajà matice sousednosti, seznamy sousedů nebo seznamy hran. + \item \textit{Algoritmus pro hledánà nejkratÅ¡Ã cesty}: Pro konstrukci vrstevnicového grafu je potÅ™eba najÃt nejkratÅ¡Ã cesty od zdroje k ostatnÃm vrcholům v sÃti. Můžeme použÃt algoritmy, jako je BFS~\cite{BFSEx} nebo Dijkstrův algoritmus~\cite{DijkEx}, v závislosti na naÅ¡ich požadavcÃch a omezenÃch. + \item \textit{Hledánà blokujÃcÃho toku}: Pro hledánà blokujÃcÃho toku v proÄÃÅ¡tÄ›né sÃti lze použÃt DFS~\cite{DFSEx} nebo BFS~\cite{BFSEx}z s vhodnými úpravami. V nÄ›kterých implementacÃch se použÃvajà také augmentaÄnà cesty nebo push-relabel metoda. + \item \textit{Aktualizace toku a kapacity hran}: Po nalezenà blokujÃcÃho toku je potÅ™eba aktualizovat tok v sÃti a kapacity hran. To lze provést pomocà jednoduchých cyklů nebo vektorizovaných operacÃ, v závislosti na použitých datových strukturách a programovacÃm jazyce. +\end{enumerate} +\newpage{} +\section{PoužitÃ} +Dinitzův algoritmus má Å¡irokou Å¡kálu praktických aplikacà v reálném svÄ›tÄ›. NásledujÃcà pÅ™Ãklady ilustrujà nÄ›které z jeho souÄasných použitÃ: +\begin{enumerate} + \item \textit{Optimalizace dopravnÃch sÃtà \cite{dinitzRoads}}: Algoritmus můžeme použit k optimalizaci dopravnÃch sÃtÃ, napÅ™Ãklad pro Å™Ãzenà dopravy ve velkých mÄ›stech nebo zajiÅ¡tÄ›nà efektivnÃho rozdÄ›lovánà zásob. Algoritmus může pomoci najÃt nejlepÅ¡Ã rozloženà zdrojů a cest, aby byl zajiÅ¡tÄ›n maximálnà tok zbožà Äi dopravy s minimálnÃmi náklady a zpoždÄ›nÃm. + \item \textit{ŘÃzenà vodnÃch zdrojů}: Algoritmus lze aplikovat na Å™Ãzenà vodnÃch zdrojů, jako jsou Å™eky, pÅ™ehrady a zavlažovacà systémy. Algoritmus může pomoci urÄit optimálnà rozdÄ›lenà vody mezi různé oblasti, aby byla zajiÅ¡tÄ›na maximálnà úÄinnost a minimalizace ztrát. + \item \textit{Telekomunikace \cite{dinitzNet}}: V oblasti telekomunikacà se algoritmus použÃvá pro optimalizaci pÅ™enosu dat v sÃtÃch a pro Å™Ãzenà zdrojů, jako je Å¡ÃÅ™ka pásma nebo kapacita spojenÃ. Algoritmus může napomoci pÅ™i navrhovánà sÃtà tak, aby bylo zajiÅ¡tÄ›no maximálnà propustnost a minimálnà zpoždÄ›nÃ. + \item \textit{Plánovánà a analýza sÃtÄ› \cite{dinitzSit}}: Algoritmus můžeme použÃt pro plánovánà a analýzu sÃtÄ› v různých oborech, jako je napÅ™Ãklad výroba, logistika nebo energetika. Algoritmus může pomoci identifikovat úzká hrdla, optimalizovat zdroje a zlepÅ¡it celkovou efektivitu sÃtÄ›. + \item \textit{Optimalizace zpracovánà informacÃ\cite{dinitzProc}}: V oblasti zpracovánà informacà může být algoritmus použit pro optimalizaci paralelnÃho zpracovánà úkolů, distribuci zdrojů nebo plánovánà procesů. Algoritmus může napomoci pÅ™i navrhovánà systémů tak, aby bylo dosaženo maximálnÃho výkonu a minimálnÃho Äasu zpracovánÃ. +\end{enumerate} + + +%--------------------------------------------------------------- +\chapter{VerifikaÄnà prostÅ™edà Frama-C} +%--------------------------------------------------------------- +Frama-C je sofistikovaný analytický nástroj založený na frameworku, který je navržen pro analýzu zdrojového kódu napsaného v jazyce C. Tento nástroj byl nejdÅ™Ãve vyvinut pro zlepÅ¡enà bezpeÄnosti a provozu jaderných elektráren spoleÄnostà Inria, francouzským národnÃm institutem pro výzkum v oblasti informatiky a matematického modelovánÃ, a jeho hlavnÃm cÃlem dnes je zlepÅ¡it kvalitu, bezpeÄnost a spolehlivost softwaru. Frama-C poskytuje sadu modulárnÃch a rozÅ¡iÅ™itelných pluginů, které umožňujà provádÄ›t různé druhy analýz, jako je formálnà analýza, statická analýza, abstraktnà interpretace, kontrola bÄ›hových chyb nebo generovánà testovacÃch dat. My budeme framework Frama-C využÃvat hlavnÄ› pro statickou a formálnà analýzu programu napsaného v jazyce C, budeme také využÃvat dva z mnoha možných pluginů -- plugin WP a plugin RTE. +\section{Plugin WP} +Plugin WP (Weakest Precondition)~\cite{WPMan} je souÄástà Frama-C a sloužà k formálnà verifikaci zdrojového kódu napsaného v jazyce C. WP využÃvá techniku nejslabÅ¡Ãch pÅ™edpokladů (ang. weakest preconditions) pro ověřenÃ, zda daný program splňuje formálnà specifikace zapsané v jazyce ACSL (ANSI/ISO C Specification Language). Tato technika spoÄÃvá ve vytvářenà matematických důkazů, které zachycujà chovánà programu a umožňujà ověřit splnÄ›nà požadavků specifikovaných v ACSL. + +WP podporuje různé druhy nástrojů pro automatické i interaktivnà důkazy, jako jsou SMT (Satisfiability Modulo Theories) solvery nebo interaktivnà důkazové asistenty, napÅ™Ãklad Coq. Plugin WP je navržen tak, aby byl modulárnà a rozÅ¡iÅ™itelný, což umožňuje snadno pÅ™idávat podporu pro dalÅ¡Ã důkazové nástroje nebo techniky. +\section{Plugin RTE} \label{rte} +Plugin RTE (Runtime Error) je dalÅ¡Ã komponenta Frama-C, která se zaměřuje na kontrolu bÄ›hových chyb v programu napsaném v jazyce C. Tento plugin analyzuje zdrojový kód a automaticky generuje ACSL anotace pro kontrolu bÄ›hových chyb, jako jsou pÅ™eteÄenà ÄÃsel, dÄ›lenà nulou, neplatné ukazatele nebo pÅ™Ãstup mimo rozsah pole. + +RTE~\cite{RTEMan} využÃvá Frama-C framework pro propagaci tÄ›chto anotacà v kódu a ověřenà jejich splnÄ›nà pomocà dostupných analýz a pluginů, jako je WP~\cite{WPMan} nebo E-ACSL~\cite{ERTEMan}. Plugin RTE umožňuje identifikovat potenciálnà bÄ›hové chyby a nebezpeÄné chovánà programu, což pÅ™ispÃvá ke zlepÅ¡enà kvality, bezpeÄnosti a spolehlivosti softwaru. +\newpage{} +Ověřenà programu potÅ™ebuje nejen framework Frama-C, zvolené pluginy, ale i takzvané Å™eÅ¡iÄe (ang. solver) založené na teorii splnitelnosti modulo teorie (SMT -- satisfiability modulo theories). Tyto Å™eÅ¡iÄe umožňujà automatické dokazovánà vlastnostà programu, které vycházejà z anotacà ve zdrojovém kódu a matematických teoriÃ. Frama-C podporuje nÄ›kolik SMT Å™eÅ¡iÄů, jako jsou Alt-Ergo, CVC4 a Z3. Každý Å™eÅ¡iÄ má své specifické vlastnosti a výhody, což nám umožňuje si vybrat nejvhodnÄ›jÅ¡Ã Å™eÅ¡iÄ pro konkrétnà úlohu ověřovánÃ. My budeme v naÅ¡Ã práci použÃvat právÄ› zmÃnÄ›né tÅ™i Å™eÅ¡iÄe Alt-Ergo, CVC4 a Z3. +\section{Solver Alt-Ergo} +\emph{Alt-Ergo}~\cite{altEx} je open-source automatický Å™eÅ¡iÄ teorià (SMT solver) vyvinutý ve francouzském výzkumném institutu INRIA. Je založen na Satisfiability Modulo Theories (SMT) a je navržen tak, aby podporoval různé datové typy a struktury, vÄetnÄ› celoÄÃselných a reálných aritmetik, polà a typů výÄtů. ŘeÅ¡iÄ je napsán v OCaml a lze jej snadno integrovat do ověřovacÃch platforem softwaru, jako je Frama-C. PoužÃvá se v různých aplikacÃch, jako je napÅ™Ãklad verifikaÄnà software Why3~\cite{why3Ex}, programovacà jazyk SPARK~\cite{sparkEx} nebo software pro vývoj programů Atelier-B~\cite{Atelier}. +\section{Solver CVC4} +CVC4~\cite{cvcEx} je Å¡iroce použÃvaný open-source SMT Å™eÅ¡iÄ vyvinutý týmem výzkumnÃků ze Stanfordské university a Iowa univerzity. Je Ätvrtým v Å™adÄ› Å™eÅ¡itelů Cooperating Validity Checker (CVC). CVC4 byl navržen se zaměřenÃm na výkon, Å¡kálovatelnost a robustnost a byl úspěšnÄ› aplikován v různých projektech ověřovánà softwaru a formálnÃch metod, mezi jinými byl použit pro ověřovaÄ programů Dafny vyvinutý spoleÄnostà Microsoft Research~\cite{dafnyEx} nebo pro ověřenà hardwaru sadou nástrojů SymbiYosys~\cite{symbiEx}. +\section{Solver Z3} +Z3~\cite{z3Ex} je vysoce výkonný Å™eÅ¡iÄ vyvinutý spoleÄnostà Microsoft Research. ŘeÅ¡iÄ je známý svou efektivitou a snadnou integracÃ, dÃky Äemuž je oblÃbenou volbou pro mnoho ověřovacÃch úloh. Z3 je Å¡iroce použÃván pÅ™i ověřovánà softwaru, analýze programů a kontrole modelů a byl integrován do mnoha nástrojů a rámců, napÅ™. ověřovacà infrastruktura Viper~\cite{viperEx} nebo Metatheory of Algebraic Process Theories (Maude) system~\cite{Aceto_2013}, který se použÃvá ke specifikaci a analýze Å¡iroké Å¡kály formálnÃch modelů. +%--------------------------------------------------------------- +\chapter{AnotaÄnà jazyk ACSL} +%--------------------------------------------------------------- +JednÃm z klÃÄových aspektů Frama-C je jeho schopnost provádÄ›t formálnà analýzu pomocà specifikace ACSL (ANSI/ISO C Specification Language)~\cite{ACSLMan}. ACSL je bohatý a výkonný jazyk pro formálnà analýzu, který umožňuje popsat chovánà programu a jeho oÄekávané vlastnosti na základÄ› matematických konstrukcà a logických výrazů. Frama-C dokáže analyzovat zdrojový kód spolu se specifikacà ACSL a ověřit, zda kód splňuje dané požadavky. TÃmto způsobem je možné odhalit potenciálnà chyby, nekonzistence nebo nebezpeÄné chovánÃ, které by mohly vést k selhánà softwaru nebo bezpeÄnostnÃm problémům. +\section{Anotace jazyka ACSL} +Jazyk ACSL umožňuje popsánà oÄekávaného chovánà programu pomocà kontraktů. Kontrakty se skládajà z pÅ™edpodmÃnek (ang. preconditions), popisujÃcà oÄekávaný stav pÅ™ed volánÃm funkce a z následujÃcÃch podmÃnek (ang. postconditions), které popisujà oÄekávaný stav po dokonÄenà funkce. Frama-C tak může analyzovat a ověřovat správnost programu, ale i správnost jednotlivých funkcÃ. + +Jazyk ACSL použÃvá mnoho různých klÃÄových slov (ang. keywords), kterými kategorizuje tvrzenÃ, která se za nimi nacházÃ. Rychle zde uvedeme a lehce vysvÄ›tlÃme nÄ›která klÃÄová slova, které budeme použÃvat v naÅ¡em ověřenà Dinitzova algoritmu použÃvat. + + + \begin{itemize} \label{frama:keywords} + \item \emph{@requires}: Specifikuje pÅ™edpoklady, které musà být splnÄ›ny pÅ™ed zavolánÃm funkce. + \item \emph{@ensures}: Popisuje podmÃnky, které musà být splnÄ›ny po úspěšném ukonÄenà funkce. + \item \emph{@assigns}: Udává, jaké promÄ›nné mohou být zmÄ›nÄ›ny funkcÃ. + \item \emph{@allocates}: Popisuje paměť, kterou funkce alokuje. + \item \emph{@frees}: Popisuje paměť, kterou funkce uvolňuje. + \item \emph{@loop\_invariant}: Popisuje podmÃnky, které musà být splnÄ›ny pÅ™ed a po každé iteraci smyÄky. Pokud bychom vzali ukázkový kód \ref{ex:loopC}, tak jako invariant cyklu bychom napsali $0 \leq a < 10 $, tedy celkem jako /*@ loop invariant 0 <= a <= 10; @*/. Jak tedy vidÃme, invarianta musà také obsahovat hodnotu i po skonÄenà cyklu. + \item \emph{@loop\_assigns}: Udává, jaké promÄ›nné mohou být zmÄ›nÄ›ny smyÄkou. Z ukázkového kódu \ref{ex:loopC}, by se jednalo o dvÄ› promÄ›nné: \emph{a} a \emph{i}. + \item \emph{@loop\_variant}: Udává funkci, která klesá pÅ™i každé iteraci smyÄky a je nezáporná. Pro náš ukázkový kód \ref{ex:loopC} máme vÃce možnostÃ, napÅ™.\\ /*@ loop variant 10 - i;@*/ nebo + /*@ loop variant 10 - a;@*/ + \item \emph{@axiomatic}: Definuje soubor axiómů, které popisujà logické vlastnosti, na kterých lze stavÄ›t důkazy. + \item \emph{@axiom}: Definuje jednotlivý axiom, který je souÄástà axiomatického bloku a popisuje logickou vlastnost. + \item \emph{@predicate}: Definuje logickou funkci, která může být použita jako nástroj pro ověřovánÃ. + \item \emph{@assert}: Kontroluje, zda je daná podmÃnka splnÄ›na, a pokud ne, generuje chybu ověřovánÃ. + \item \emph{@logic}: Definuje logickou funkci, která může být použita v axiomatických bloků, predikátů a dalÅ¡Ãch logických výrazů. + \item \emph{@reads}: UrÄuje, které promÄ›nné nebo paměťové oblasti může logická funkce ÄÃst. + \end{itemize} + +\begin{lstlisting}[language=C, caption={Ukázkový for cyklus v jazyce C}, backgroundcolor=\color{white}, label={ex:loopC}] +int a = 0; +for(int i = 0; i < 10; ++i) { + a++; +} +\end{lstlisting} + +%--------------------------------------------------------------- +\chapter{Implementace a ověřenà algoritmu} +%--------------------------------------------------------------- +V této kapitole si popÃÅ¡eme implementaci a ověřenà Dinitzova algoritmu pro nalezenà maximálnÃho toku v sÃti, který jsme v minule kapitole\ref{dinitz} popsali z teoretické stránky a nynà ho pÅ™edstavÃme z praktické. Náš Dinitzův algoritmus jsme implementovali v programovacÃm jazykem C a to z důvodu pozdÄ›jÅ¡Ã verifikace frameworkem Frama-C, který toto vyžaduje. + +NaÅ¡i implementaci jsme rozdÄ›lili do třà souborů. Soubor main.c, který obsahuje naÄÃtánà vstupu od uživatele a uloženà dat do správné struktury, s kterou dále náš algoritmus pracuje. HlaviÄkový soubor dinitz.h, který obsahuje definice pomocných metod a pomocných struktur a jako poslednà máme soubor dinitz.c, ve kterém je kód, který implementuje dÅ™Ãve definované funkce z dinitz.h. + +\section{Implementace} +Nynà si ukážeme kód implementace a pÅ™esnÄ›ji popÃÅ¡eme rozhodnutÃ, které vedly k vybraným technikám, které jsme využili. ZaÄneme od souboru main.c, abychom si uvedli jaké data náš algoritmus pÅ™ijÃmá a s jakými podmÃnkami pracujeme. + +\subsection{Main.c} +NejdůležitÄ›jÅ¡Ã Äásti souboru main.c je funkce readInput z výpisu kódu \ref{prac:main.c}, tato funkce Äte uživatelem zadané data, a poté je ukládá do pro nás vhodné struktury -- matice sousednosti. Zároveň tato funkce hlÃdá námi pÅ™edurÄené požadavky, neboli chceme aby graf mÄ›l alespoň 2 vrcholy, abychom mohli mÃt zdroj a stok jako dva různé vrcholy. \footnote{Vždycky považujeme prvnà vrchol jako zdroj a poslednà vrchol jako stok.} + +Dále jsme si stanovili, že budeme uvažovat grafy pouze o maximálnÃm poÄtů vrcholů rovném, nebo menÅ¡Ãm než 300 000. Tuto velikost grafů jsme si vybrali, aby tento kód mÄ›l stále využità v reálném svÄ›te, ale zároveň aby doba bÄ›hu programu byla stále akceptovatelná. + +Zároveň nám, ale toto rozhodnutà pÅ™inášà jiné následky. Znamená to pro nás využità dynamicky alokované pamÄ›ti, abychom mohli graf takové velikosti uložit. S tÃm se váže nepÅ™Ãjemná vlastnost frameworku Frama-C a to, že klÃÄová slova pro ověřovánà frees a alocates z kapitoly \ref{frama:keywords} nejsou implementovány. A proto, jak jeÅ¡tÄ› uvidÃme Äásti kódu spojené, nebo úzce spojené s dynamickou alokacà pamÄ›ti nelze plnÄ› ověřit. + +Soubor main.c také obsahuje funkci cleanUp, která uvolnà paměť, výpis výsledku naÅ¡eho algoritmu. +\newpage{} +\begin{lstlisting}[language=C, caption={Soubor main.c}, backgroundcolor=\color{white}, label={prac:main.c}] +int readInput( double ***graph) +{ + int nodes; + // Reading the amount of nodes and allocating + printf("Insert the number of nodes.\n"); + if(scanf("%d", &nodes) != 1){ + printf("Invalid number of nodes.\n"); + return -1; + } + if (nodes < 2 || nodes > 300000){ + printf("Invalid number of nodes.\n"); + return -1; + } + + *graph = (double**) malloc((nodes) * sizeof(double *)); + if (*graph == NULL){ + return -1; + } + + for (int i = 0; i < (nodes); ++i) + { + (*graph)[i] = (double*) calloc((nodes), sizeof(double)); + if ((*graph)[i] == NULL){ + return -1; + } + } + + // Reading all the edges and saving them to the 2D array + int start, end; + double cap; + printf("Insert the edges of the graph.\n"); + while (scanf("%d %d %lf", &start, &end, &cap) == 3){ + if (start < 0 || end < 0 || cap <= 0 || start >= (nodes) + || end >= (nodes)){ + printf("This edge is invalid.\n"); + } + else if ((*graph)[start][end] != 0.0){ + printf("This edge already exists.\n"); + } + else{ + (*graph)[start][end] = cap; + } + } + return nodes; +} +\end{lstlisting} +\newpage{} +\subsection{Dinitz.h/Dinitz.c} +Nynà si popÃÅ¡eme hlavnà soubory naÅ¡eho programu. ZaÄneme od nejdůležitÄ›jÅ¡Ã funkce dinitzMaxFlow \ref{prac:dinitz} a budeme pomalu pÅ™edstavovat dalÅ¡Ã pomocné funkce. NaÅ¡e hlavnà funkce jako prvnà pÅ™ipravà dalÅ¡Ã pole urÄené pro pamatovánà si úrovnà vrcholů a zavolá BFS \ref{prac:BFS}, kterému toto pole pÅ™edá. + +BFS má za úkol prohledat do Å¡ÃÅ™ky zadaný graf, nastavit úrovnÄ› vÅ¡ech dostupných vrcholů a~tedy~i~zjistit zda existuje cesta zs. BFS také využÃvá naÅ¡Ã pomocnou strukturu Node \ref{prac:Node} a funkce insertHead \ref{prac:insert} a popHead \ref{prac:pop}, které spoleÄnÄ› simulujà frontu pomocà spojového seznamu. + +Po návratu BFS, pokud existuje cesta zs, tak opakovanÄ› voláme funkci DFS \ref{prac:DFS}. NaÅ¡e prohledávánà do hloubky je trochu upravené, dovolujeme DFS cestovat pouze po hranách, které vedou z vrcholu o nižšà úrovni do vrcholu s úrovni o jedna vyÅ¡Å¡Ã. Zároveň si pamatujeme cestu, kterou jsme pÅ™iÅ¡li, navÅ¡tÃvené vrcholy a velikost toku, který se rovná nejmenÅ¡Ã kapacitÄ› hrany na cestÄ› zs. DFS poté upravuje aktuálnà graf a maže hrany, které jsou nasyceny. + +V naÅ¡Ã implementaci tedy nevytvářÃme proÄiÅ¡tÄ›nou sÃÅ¥, ale pouze pÅ™iÅ™azujeme úrovnÄ› a mažeme nasycené hrany. Pro DFS to znamená, že může prohledávat delÅ¡Ã dobu, protože nemažeme slepé uliÄky, ale zvolili jsme tento postup, abychom se vyhnuli dalÅ¡Ãm alokacÃm dynamické pamÄ›ti, které vytvářà problémy s ověřovacÃm frameworkem. + + +\begin{lstlisting}[language=C, caption={kód funkce Dinitz Max Flow}, backgroundcolor=\color{white}, label={prac:dinitz}] +double dinitzMaxFlow(int nodes, double **graph){ + double total = 0; + int *vertexLevel = (int*) malloc(nodes * sizeof(int)); + int *visited; + // Run BFS to check if flow is possible and set the levels of nodes + while (BFS(0, nodes - 1, graph, vertexLevel)) + { + visited = (int*) calloc(nodes, sizeof(int)); + double flow; + while (1) + { + flow = DFS(0, nodes - 1, graph, vertexLevel, visited, DBL_MAX); + if (flow == 0.0) + { + break; + } + else + { + if (DBL_MAX - total < flow) { + total = DBL_MAX; + break; + } else { + total += flow; + } + } + // Reset visited array + memset(visited, 0, nodes * sizeof(int)); + } + free(visited); + } + free(vertexLevel); + // Return max flow + return total; +} +\end{lstlisting} + +\begin{lstlisting}[language=C, caption={kód funkce BFS}, backgroundcolor=\color{white}, label={prac:BFS}] +int BFS(int start, int end, double **graph, int *levels){ + for (int i = 0; i <= end; ++i) { + levels[i] = -1; + } + levels[start] = 0; + Node* queueHead = NULL; + insertHead(&queueHead, start); + int cur; + while(queueHead != NULL) + { + cur = queueHead->data; + popHead(&queueHead); + for (int i = 0; i <= end; ++i) + { + if(graph[cur][i] > 0) + { + if(levels[i] < 0){ + levels[i] = levels[cur] + 1; + insertHead(&queueHead, i); + } + } + } + } + if (levels[end] < 0){ + return 0; + } + else { + return 1; + } +} +\end{lstlisting} + +\begin{lstlisting}[language=C, caption={kód funkce DFS}, backgroundcolor=\color{white}, label={prac:DFS}] +double DFS (int cur, int end, double **graph, +int *levels, int *visited, double flow){ + if (cur == end){ + return flow; + } + visited[cur] = 1; + for (int i = 0; i <= end; ++i) { + if (!visited[i] && graph[cur][i] > 0 + && levels[i] == levels[cur] + 1){ + double augmentingFlow = DFS(i, end, graph, + levels, visited, + min(flow, graph[cur][i])); + if (augmentingFlow > 0){ + graph[cur][i] -= augmentingFlow; + graph[i][cur] += augmentingFlow; + return augmentingFlow; + } + } + } + return 0; +} +\end{lstlisting} + +\begin{lstlisting}[language=C, caption={kód pomocné struktury Node}, backgroundcolor=\color{white}, label={prac:Node}] +typedef struct Node { + int data; + struct Node *next; +} Node; +\end{lstlisting} + +\begin{lstlisting}[language=C, caption={kód funkce popHead}, backgroundcolor=\color{white}, label={prac:pop}] +void popHead(Node** head_ref) { + Node* nextHead = (*head_ref)->next; + free(*head_ref); + (*head_ref) = nextHead; +} +\end{lstlisting} + +\begin{lstlisting}[language=C, caption={kód funkce insertHead}, backgroundcolor=\color{white}, label={prac:insert}] +void insertHead(Node** head_ref, int new_data) { + Node* next = (*head_ref); + (*head_ref) = (Node*) malloc(sizeof(Node)); + (*head_ref)->data = new_data; + (*head_ref)->next = next; +} +\end{lstlisting} +\section{OvěřenÃ} +Nynà už máme naÅ¡Ã implementaci kódu a chceme jà ověřit, zavedeme tedy do kódu anotace jazyka ACSL, takové anotace uvádÃme pomocà znaků /*@ a @*/. VlastnÄ› se jedná o blokové komentáře programovacÃho jazyka C doplnÄ›né o znak @. Pokud anotace jsou na vÃce řádků tak nemusÃme na zaÄátek každého řádku uvádÄ›t @, ale my to tak budeme dÄ›lat, aby bylo jasné kde máme anotace pro framework Frama-C. + +Nynà už zaÄneme s uvádÄ›nÃm funkcà a jejich kontraktů, zaÄneme od menÅ¡Ãch funkcÃ, kde kontrakty bývajà jednoduÅ¡Å¡Ã a budeme se posouvat k funkcÃm složitÄ›jÅ¡Ãm, které Äasto závisà na pÅ™edeÅ¡lých funkcÃch. + +\subsection{Funkce insertHead} +\begin{lstlisting}[language=C, caption={anotace insertHead}, backgroundcolor=\color{white}, label={ver:insert}] +/*@ requires \valid(head_ref) && (*head_ref) != \null; + @ ensures *head_ref != \null && (*head_ref)->data == new_data + && (*head_ref)->next == \old(*head_ref) && \fresh(*head_ref, sizeof(Node)) + && \separated(*head_ref, \old(*head_ref)); + @ allocates *head_ref; + @ assigns *head_ref, (*head_ref)->data, (*head_ref)->next; + @*/ +void insertHead(Node **head_ref, int new_data){ +... +} +\end{lstlisting} + +PopiÅ¡me anotace funkce insertHead, funkce požaduje, aby ukazatel head\_ref byl validnà a zároveň se nerovnal null, to pro nás znamená že chceme, aby paměť, na kterou ukazatel ukazuje byla validnà ke Ätenà a zapisovánÃ. + +Dále funkce zajiÅ¡Å¥uje, že hodnota head\_ref nebude null a zároveň že složka data v Node, na který ukazuje se bude rovnat parametru new\_data, který tato funkce dostala a složka next se bude rovnat staré hodnotÄ› *head\_ref, neboli bude to ukazatel na dalÅ¡Ã prvek v seznamu. + +KlÃÄová slova fresh a separated nám pouze Å™ÃkajÃ, že head\_ref ukazuje na novÄ› alokovaný blok o velikosti Node, a že bloky pamÄ›ti na které ukazujà head\_ref a starý head\_ref se nepÅ™ekrývajÃ. + +Dále máme informaci o tom, že tato funkce alokuje paměť, na kterou ukazuje head\_ref a jako poslednà máme klauzuli assigns, která nám Å™Ãká, že tato funkce manipuluje s pamÄ›tÃ, v tomto pÅ™ÃpadÄ› promÄ›nných *head\_ref, (*head\_ref)->data a (*head\_ref)->next. + +Pokud se pokusÃme ověřit pouze tuto Äást programu tak dostaneme upozornÄ›nà z obrázku \ref{img:dynMemErr} a Äást naÅ¡eho kódu nebude možné z tohoto důvodu ověřit. \ref{img:verIns} +\begin{figure}[ht!] + \centering + \includegraphics[width=14cm]{pictures/frama-c-memory-messages.PNG} + \caption{Problémy s dynamickou alokacà pamÄ›ti ve Frama-C} + \label{img:dynMemErr} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.75]{pictures/frama-c-insert-verify.png} + \caption{Ověřenà insertHead} + \label{img:verIns} +\end{figure} + +\subsection{Funkce popHead} +\begin{lstlisting}[language=C, caption={anotace popHead}, backgroundcolor=\color{white}, label={ver:pop}] +/*@ predicate is_valid_linked_list(Node* head) = + @ head == \null || (is_valid_linked_list(head->next) && \valid(head)); + @*/ + +/*@ requires \valid(head_ref) && *head_ref != \null + && is_valid_linked_list(*head_ref); + @ assigns *head_ref; + @ ensures \at(*head_ref,Post) == \old((*head_ref)->next) + && !\valid(\old(*head_ref)); + @*/ +void popHead(Node** head_ref) { +... +} +\end{lstlisting} + +Nynà si ukažme anotace funkce popHead, na prvnà pohled vidÃme Äást anotacÃ, které se v minulé funkci neobjevily. Jedná se o anotace predicate, jak už jsme zmÃnili dÅ™Ãve v kapitole \ref{frama:keywords} tato anotace nás informuje o funkci, která může být použitá k ověřovánÃ. + +V~tomto pÅ™ÃpadÄ› se jedná o funkci, která dostane hlaviÄku naÅ¡Ã fronty v podobÄ› spojového seznamu a rekurzivnÄ› se volá na dalÅ¡Ã prvky dokud nenarazà na null a v každém kroku kontroluje, zda je možné ÄÃst a zapisovat do daného prvku fronty. + +Dále máme již nám známe anotace ohlednÄ› požadovánà validnÃch vstupnÃch parametrů funkce a použità zmÃnÄ›né predicate funkce ke kontrole fronty. Anotace nám také Å™ÃkajÃ, že funkce mÄ›nà promÄ›nnou *head\_ref, uvolňuje paměť na kterou ukazovala dÅ™ÃvÄ›jÅ¡Ã hodnota *head\_ref. + +V Äásti anotacÃ, kde máme, co funkce zaruÄuje dostáváme nové klÃÄové slovo at, které nám Å™Ãká, že se koukneme na hodnotu dané promÄ›nné až po daném bodÄ› v kódu, který oznaÄÃme pomocà štÃtku, v tomto pÅ™ÃpadÄ› Post. + +Slovo at, ale už tak doopravdy známe, pokud se podÃváme na slovo old(x), tak se vlastnÄ› jedná pouze o zkrácenou formu at(x, Old). Slovo old už jsme vysvÄ›tlili, že se jedná o hodnotu promÄ›nné pÅ™ed provedenÃm kódu funkce a slovÃÄko Post je jejÃm opakem, tedy hodnota po skonÄenà funkce. CelkovÄ› tedy zaruÄujeme, že poté co funkce skonÄà tak hodnota *head\_ref se rovná ukazateli na dalÅ¡Ã prvek z minulé hlaviÄky. + +A jako poslednÃ, že paměť, na kterou ukazovala stará hodnota *head\_ref nynà už nenà validnÃ, neboli nemůžeme jà ÄÃst ani do nà psát. V programovacÃm jazyce C, vÅ¡echny tyto vlastnosti vyjadÅ™ujeme funkcà free. + +Nynà už můžeme ověřit naÅ¡Ã funkci popHead obrázek~\ref{img:verPop}, oÄekáváme podobné výsledky jako s funkcà insertHead, protože nadále pracujeme s dynamickou alokacà pamÄ›ti. Zároveň si ukážeme jak Frama-C transformuje námi dodané anotace a také jak plugin RTE~\ref{rte} pÅ™idává vlastnà ověřovacà anotace obrázek~\ref{img:popFramaCode}. + +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.9]{pictures/frama-c-pop.png} + \caption{Ověřenà popHead} + \label{img:verPop} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.65]{pictures/frama-c-pop-vis.PNG} + \caption{Generovaný kód Frama-C pro funkci popHead} + \label{img:popFramaCode} +\end{figure} + +Na obrázku~\ref{img:popFramaCode} je vidÄ›t jak pÅ™edpoklady pro funkci jsou brané jako správné, ale nebyly dokazované, protože aktuálnÄ› ověřujeme osamocený kód. V tÄ›le funkce bylo dokázáno, že vstupnà parametr head\_ref ukazuje na paměť, kterou lze ÄÃst a také do nà zapisovat. To nám vycházà z pÅ™edpokladů, ale jen co se dostaneme na ověřenà uvolnÄ›nà pamÄ›ti, tak nám Frama-C oznamuje, že se o ověřenà pokusilo, ale nedokáže vrátit úspÄ›ch ani neúspÄ›ch. A tedy ani dalÅ¡Ã notace závisejÃcà na správnosti tÄ›chto výroků nemohou býti dokázány. + +\subsection{Funkce DFS} +\begin{lstlisting}[language=C, caption={anotace DFS}, backgroundcolor=\color{white}, label={ver:dfs}] +/*@ requires 0 <= cur <= end && end >= 0 && end < 300000; + @ requires 0 <= flow <= DBL_MAX; + @ requires graph != \null && \valid(levels + (0..end)) + && \valid(visited + (0..end)); + @ requires \forall int i; 0 <= i <= end ==> \valid(graph[i] + (0..end)); + @ assigns visited[0..end], graph[0..end][0..end]; + @ ensures \result >= 0.0 && (\result > 0.0 ==> + (\forall int i; 0 <= i <= end + ==> graph[cur][i] <= \old(graph[cur][i]))); + @*/ +double DFS (int cur, int end, double **graph, int *levels, int *visited, + double flow){ +... + visited[cur] = 1; + /*@ loop invariant 0 <= i <= end + 1; + @ loop invariant 0 <= flow <= DBL_MAX; + @ loop assigns i, graph[0..end][0..end], visited[0..end]; + @ loop variant end - i; + @*/ + for (int i = 0; i <= end; ++i) { +... +} +\end{lstlisting} +ZaÄÃnáme se dostávat ke složitÄ›jÅ¡Ãm funkcÃm, jednou z nich je DFS výpis kódu~\ref{prac:DFS}, která je volaná z naÅ¡Ã hlavnà funkce dinitzMaxFlow. ZaÄneme klasicky od požadavků funkce, DFS od nás vyžaduje aby index aktuálnÃho vrcholu byl menÅ¡Ã, nebo roven indexu stoku, který má být menÅ¡Ã než 300 000\footnote{Jakožto pole se ÄÃsluje od 0, tak pro nás maximálnà index se bude rovnat 299 999.}, což byla námi stanovena hornà hranice poÄtu vrcholů v grafu. Funkce také oÄekává nezáporný tok a validnà matici sousednosti, která zobrazuje náš graf, validnà pole držÃcà úrovnÄ› vrcholů a také validnà pole již navÅ¡tÃvených vrcholů. + +Dále Å™Ãkáme, že funkce DFS pouze mÄ›nà obsah pole s naÅ¡Ãm grafem a pole navÅ¡tÃvených vrcholů. DFS v naÅ¡em pÅ™ÃpadÄ› zaruÄuje, že výsledek bude nezáporný a pokud je vÄ›tÅ¡Ã jak nula, tak pro vÅ¡echny hrany z aktuálnÃho vrcholů platÃ, že jejich hodnota bude menÅ¡Ã nebo rovná minulé. Neboli pokud toto Å™ekneme vÃce se znalostà naÅ¡eho algoritmu, tak Å™Ãkáme, že pokud nalezneme zlepÅ¡ujÃcà tok zs, tak rezervy hran vrcholů, kterými tento tok vede, se může pouze zmenÅ¡it. + +Nynà nám už v této funkci zbývajà pouze anotace pro cyklus, cykly bývajà pro framework frama-C velice těžké na ověřenÃ, proto pÅ™idáváme anotace, které s tÃm pomáhajÃ, jednotlivá klÃÄová slova s pÅ™Ãkladem jsme již trochu vysvÄ›tlovali na ukázkovém kódu \ref{ex:loopC}, tedy zde budeme psát jejich význam pro ověřenÃ. + +PrvnÄ› uvádÃme promÄ›nné, které se bÄ›hem cyklu mohou mÄ›nit nebo musà splňovat jisté podmÃnky, tyto podmÃnky budou kontrolovány vždy na zaÄátku a na konci jednoho cyklu. Zde tedy uvádÃme promÄ›nnou i, která je Å™ÃdÃcà promÄ›nnou cyklu a promÄ›nnou flow, od které požadujeme aby byla nezáporná. + +Dále Å™Ãkáme jaké promÄ›nné cyklus může mÄ›nit a jako poslednà uvádÃme ÄÃselnou hodnotu, která se z každým cyklem bude zmenÅ¡ovat a je rovná, nebo vÄ›tÅ¡Ã maximálnÃmu poÄtu iteracà daného cyklu. My jsme si zde vybrali poÄet vrcholů, protože je to pÅ™esný odhad, ale stejnÄ› tak bychom mohli Å™Ãci, že varianta cyklu bude 300 000 - i, protože vÃme, že naÅ¡e implementace nebude mÃt vÃce jak 300 000 vrcholů a tento cyklus procházà vÅ¡echny hrany daného vrcholu. + +Ověřme tedy naÅ¡Ã implementaci funkce DFS obrázek~\ref{img:verDFS} a zjistÃme, že vÄ›tÅ¡ina této funkce bude úspěšnÄ› ověřená, stalo se tak, protože ve funkci DFS pracujeme velice minimálnÄ› s promÄ›nnými, které jsou dynamicky alokované. + +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.95]{pictures/frama-c-DFS.PNG} + \caption{Ověřenà DFS} + \label{img:verDFS} +\end{figure} + +\subsection{Funkce BFS} +\begin{lstlisting}[language=C, caption={anotace BFS}, backgroundcolor=\color{white}, label={ver:bfs}] +/*@ requires start >= 0 && end >= 0 && end < 300000; + @ requires graph != \null && \valid(levels + (0..end)); + @ requires \forall int i; 0 <= i < end + 1 ==> + \valid(graph[i] + (0..end)); + @ assigns levels[0..end]; + @ ensures \result == 0 || \result == 1; + @ ensures \forall int i; 0 <= i <= end ==> levels[i] >= -1 + && levels[i] <= end; + @ ensures \forall int i; 0 <= i <= end && levels[i] == -1 + ==> (\forall int j;0 <= j <= end && levels[j] != -1 + ==> graph[j][i] == 0.0 && graph[i][j] == 0.0); + @ ensures levels[end] > 0 ==> + (\forall int i; \exists int j; \exists int k; 0 <= i < levels[end] + && 0 <= j <= end && 0 <= k <= end && (levels[j] == i + && levels[k] == i + 1) && graph[j][k] > 0); + @*/ +int BFS(int start, int end, double **graph, int *levels){ +... + /*@ loop invariant 0 <= i <= end + 1; + @ loop invariant \forall int j; 0 <= j < i ==> levels[j] == -1; + @ loop assigns i, levels[0..end]; + @ loop variant end - i; + @*/ + for (int i = 0; i <= end; ++i) { + levels[i] = -1; + } +... + /*@ loop invariant \valid(queueHead) || queueHead == \null; + @ loop invariant \forall int i; 0 <= i <= end ==> levels[i] >= -1 + && levels[i] <= end; + @ loop assigns cur, queueHead, levels[0..end]; + @*/ + while(queueHead != NULL){ + cur = queueHead->data; + popHead(&queueHead); + /*@ loop invariant \forall int i; 0 <= i <= end ==> levels[i] >= -1 + && levels[i] <= end; + @ loop assigns queueHead, levels[0..end]; + @*/ + for (int i = 0; i <= end; ++i) { + if(graph[cur][i] > 0){ + if(levels[i] < 0){ + levels[i] = levels[cur] + 1; + insertHead(&queueHead, i); + } + } + } + } +... +} +\end{lstlisting} +Funkci BFS už urÄitÄ› musÃme rozebrat po Äástech, zaÄnÄ›me od vrchu, funkce BFS má menÅ¡Ã požadavky než funkce DFS, požaduje pouze, aby startovnà vrchol neboli zdroj mÄ›l menÅ¡Ã index než index stoku, a aby stok byl menÅ¡Ã než 300 000. Klasicky vyžadujeme validnà graf a pole kam můžeme ukládat úrovnÄ› vrcholů. + +NaÅ¡e BFS pouze upravuje hodnoty pro úrovnÄ› vrcholů a poté hlavnà funkcionalitou je co vÅ¡e nám BFS zaruÄuje. ZaÄneme jednoduÅ¡e, BFS nám vracà 0 pro nenalezenà cesty zs, nebo 1 pokud jsme cestu naÅ¡li. Druhá vlastnost, kterou nám BFS dává je, že každý vrchol bude mÃt nastavenou úroveň od -1 až po poÄet vrcholů - 1. + +TÅ™età vlastnost nám Å™Ãká, že pro každý vrchol, pokud jeho úroveň je -1 po skonÄenà funkce, tak neexistuje hrana mezi vrcholem, který má úroveň vyÅ¡Å¡Ã než -1. Neboli BFS nám v teorii grafů zaruÄuje, že pokud nÄ›jaký vrchol nebyl nalezen, tak neexistuje cesta mezi zdrojem a daným vrcholem. + +A poslednà vlastnost nám Å™Ãká, že pokud stok má úroveň vyÅ¡Å¡Ã než -1, tak pro každou úroveň existuje vrchol který má hranu, která vede z tohoto vrcholu do vrcholu, který má úroveň o jedna vyÅ¡Å¡Ã. Jinými slovy tato vlastnost nám zaruÄuje, že pokud stok má úroveň vyÅ¡Å¡Ã než -1 tak existuje cesta o délce úrovnÄ› stoku ze zdroje do stoku. + +Tedy na rozdÃl od pÅ™edchozÃho tvrzenÃ, Å™Ãkáme nejenom, že cesta existuje, ale také tvrdÃme, že je dané délky a v tomto pÅ™ÃpadÄ› to také bude délka nejkratÅ¡Ã cesty. + +Jakožto ve funkci BFS podle výpisu kódu~\ref{prac:BFS} využÃváme také funkce popHead z výpisu kódu~\ref{prac:pop} a insertHead z výpisu kódu~\ref{prac:insert}, tak už nemůžeme prověřit BFS jako osamocený kód, ale musÃme ho prověřit spoleÄnÄ› s tÄ›mito funkcemi, tedy vÄ›tÅ¡ina kódu bude možné verifikovat jak vidÃme na obrázku~\ref{img:verBFS}, ale jejich neověřené Äásti se budou propagovat do pozdÄ›jÅ¡Ãch Äásti BFS a znemožňovat dalÅ¡Ã verifikaci.\\ +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.95]{pictures/frama-c-BFS.PNG} + \caption{Ověřenà BFS} + \label{img:verBFS} +\end{figure} +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.5]{pictures/frama-c-BFS-1.PNG} + \centering + \includegraphics[scale=0.5]{pictures/frama-c-BFS-2.PNG} + \caption{Generovaný kód Frama-C pro funkci BFS } + \label{img:genBFS} +\end{figure} + + +Jak již jsme vidÄ›li na obrázku~\ref{img:popFramaCode}, tak i zde Frama-C byla schopná ověřit vÅ¡e, až do bodu, kdy se stÅ™etne s alokacà dynamické pamÄ›ti obrázek~\ref{img:genBFS} a poté nemá možnost ověřit naÅ¡e tvrzenà a pÅ™iÅ™azovánÃ, protože si nenà jistá, zda paměť lze ÄÃst a modifikovat. +\newpage{} +\subsection{Funkce dinitzMaxFlow} + +\begin{lstlisting}[language=C, caption={anotace dinitzMaxFlow}, backgroundcolor=\color{white}, label={ver:dinitz}] +/*@ axiomatic SumCapacity { + @ logic double sum_capacity(double **graph, integer nodes, + integer index) reads graph[0][0..nodes-1]; + @ axiom sum_capacity_base: + @ \forall double **graph, integer nodes; + @ 0 <= nodes ==> sum_capacity(graph, nodes, 0) == 0; + @ axiom sum_capacity_step: + @ \forall double **graph, integer nodes, integer index; + @ 0 <= index < nodes - 1 ==> + @ sum_capacity(graph, nodes, index + 1) == sum_capacity + (graph, nodes, index) + graph[0][index]; + @ } + @*/ + +/*@ predicate valid_flow(double result, double **graph, integer nodes) = + result >= -1 && result <= sum_capacity(graph, nodes, nodes - 1); + @*/ + +/*@ requires nodes > 0 && nodes <= 300000; + @ requires graph != \null && \forall integer i; 0 <= i < nodes + ==> \valid(graph[i] + (0..nodes-1)); + @ ensures valid_flow(\result, graph, nodes); + @ assigns graph[0..nodes-1][0..nodes-1]; + @*/ + +double dinitzMaxFlow(int nodes, double **graph) + ... + // Run BFS to check if flow is possible and set the levels of nodes + /*@ loop invariant 0 <= total && total <= sum_capacity + (graph, nodes, nodes - 1); + @ loop assigns total, vertexLevel[0..nodes-1], + *visited, graph[0..nodes-1][0..nodes-1]; + @ loop variant nodes - 1 - vertexLevel[nodes-1]; + @*/ + while (BFS(0, nodes - 1, graph, vertexLevel)) + { + // Initialize visited array + visited = (int*) calloc(nodes, sizeof(int)); + double flow; + /*@ loop invariant 0 <= total; + @ loop assigns flow, total, visited[0..nodes-1], + graph[0..nodes-1][0..nodes-1]; + @ loop variant (int)(DBL_MAX - total) * 1000; + @*/ + while (1) + { + flow = DFS(0, nodes - 1, graph, vertexLevel, visited, DBL_MAX); + ... + } + free(visited); + } + ... +} +\end{lstlisting} +Poslednà funkcÃ, kterou musÃme popsat je naÅ¡e hlavnà funkce dinitzMaxFlow, zaÄneme nejdÅ™Ãv axiomem, který pozdÄ›ji využÃváme dál. Axiomatic SumCapacity nám definuje nový axiomatický blok s názvem SumCapacity, poté definujeme logickou funkci sum\_capacity, která bere tÅ™i argumenty: dvojrozmÄ›rné pole, které udržuje náš graf, ÄÃslo nodes, které nám Å™Ãká poÄet vrcholů a jako poslednà index. Tato logická funkce Äte hodnoty z naÅ¡eho grafu od indexu 0 až po nodes - 1. + +Nynà již definujeme axiom sum\_capacity\_base, který nám dává základnà krok pro funkci sum\_capacity, samotný axiom Å™Ãká, že pokud nodes je vÄ›tÅ¡Ã nebo rovno nule, tak sum\_capacity pro index 0 je rovno 0. Druhý axiom sum\_capacity\_step, nám definuje rekurzivnà krok pro funkci sum\_capacity, kde tvrdÃme, že pro každý index, který je mezi 0 a nodes - 1, tak sum\_capacity pro index + 1 je rovná sum\_capacity pro index + hodnota hrany v grafu z vrcholu o indexu 0 do vrcholu o indexu index. + +Pokud tento axiom spojÃme s naÅ¡Ã reprezentacà grafu pomocà matice sousednosti a také toho, že pro nás vrchol o indexu 0 je vždy zdroj, tak tento axiom nám dá souÄet kapacit vÅ¡ech hran vycházejÃcÃch ze zdroje. Axiomy můžeme také ověřit jak je vidÄ›t na obrázku~\ref{img:verAxiom}. + +Dále tento souÄet použÃváme v námi definovaném predikátu valid\_flow, kde Å™Ãkáme, že výsledek se musà nacházet mezi -1 a souÄtem kapacit hran vÄetnÄ› z obou stran. Tento predikát poté použÃváme jako vlastnost kterou dinicMaxFlow funkce splňuje. + +Zbylé anotace už jsou oproti minulé triviálnÃ, pouze požadujeme, aby poÄet vrcholů byl vÄ›tÅ¡Ã než 0 a menÅ¡Ã než 300 000 a také požadujeme aby graf byl validnÃ. Nakonec Å™Ãkáme, že tato funkce může zmÄ›nit obsah matice udržované v promÄ›nné graph. + +TeÄ nám už zbývajà pouze anotace u cyklů, zaÄneme anotacemi vnitÅ™nÃho cyklu. NejdÅ™Ãve uvádÃme podmÃnku na promÄ›nnou total, jedná se o náš výsledek, který by mÄ›l být vždycky nezáporný. Poté Å™Ãkáme, že tento cyklus může mÄ›nit hodnoty promÄ›nných flow a total, prvky pole visited a prvky matice graph. + +Jako poslednà chceme urÄit hodnotu varianty grafu, tedy jak omezit shora iterace cyklu, v tomto pÅ™ÃpadÄ› jsme se rozhodli o odeÄÃtánà aktuálnÃho celkového toku od maximálnà hodnoty typu double zvÄ›tÅ¡ená o 1000. Dovolili jsme totiž hrany, které nemusà být celá ÄÃsla a proto chceme zachytávat i tyto malé zmÄ›ny. Zároveň si myslÃme, že tato hodnota bude s každým cyklem klesat, protože DFS najde pokaždé hodnotu o kterou zvÄ›tÅ¡Ã náš tok, nebo vrátà nulu, která cyklus ukonÄÃ. + +Od vnÄ›jÅ¡Ãho cyklu vyžadujeme, aby promÄ›nná flow byla pÅ™ed a po každé iteraci vÄ›tÅ¡Ã, nebo rovno nule a menÅ¡Ã, nebo rovno sumÄ› kapacit hran vycházejÃcÃch ze zdroje. Tento cyklus také může mÄ›nit promÄ›nnou total a prvky polà vertexLevel, graph a visited\footnote{Pole visited zde jeÅ¡tÄ› nenà alokované, takže nemůžeme urÄit rozmÄ›ry stejnÄ› jako u ostatnÃch polÃ.}. + +TeÄ už nám zbývá pouze variant vnÄ›jÅ¡Ãho cyklu, který tentokrát bude celkový poÄet vrcholů - 1 - úroveň stoku. Pokud se trochu nad tÃmto zamyslÃme, tak v nejhorÅ¡Ãm pÅ™ÃpadÄ› naÅ¡e nejkratÅ¡Ã cesty se budou s každou iteracà prodlužovat o jeden vrchol a to až do doby kdy vÅ¡echny vrcholy budou potÅ™eba k nalezenà nejkratÅ¡Ã cesty zs. V této iteraci naÅ¡e DFS nasytà i tyto cesty a nezbudou již žádné cesty zs a algoritmus skonÄÃ. Tedy variant se z každou iteracà zmenÅ¡Ã alespoň o jedna a bude vždy vetÅ¡Ã nebo roven 0. + +Zbývá nám poslednà verifikace a to celého kódu obrázek~\ref{img:verDinitz}, jakožto dinitzMaxFlow využÃvá vÅ¡ech ostatnÃch funkcÃ. +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.8]{pictures/frama-c-ver-axiom.PNG} + \caption{Ověřenà Axiomů} + \label{img:verAxiom} +\end{figure} + +\begin{figure}[ht!] + \centering + \includegraphics[scale=0.95]{pictures/frama-c-dinitz.PNG} + \caption{Ověřenà dinitzMaxFlow} + \label{img:verDinitz} +\end{figure} + +%--------------------------------------------------------------- +\chapter*{ZávÄ›r} +\addcontentsline{toc}{chapter}{ZávÄ›r} +\markboth{ZávÄ›r}{ZávÄ›r} +%--------------------------------------------------------------- + +V této práci jsme se vÄ›novali problematice formálnÃho ověřenà a implementace Dinitzova algoritmu s dynamicky alokovanou pamÄ›tÃ, pro hledánà maximálnÃho toku v orientované grafové sÃti. PodaÅ™ilo se nám plnÄ› implementovat Dinitzův algoritmus v programovacÃm jazyce C s ACSL anotacemi a také ověřit Äást kódu, která nepotÅ™ebuje prozatÃm neimplementované výrazy frameworku Frama-C týkajÃcÃch se dynamické alokace pamÄ›ti. + +V prvnÃch dvou kapitolách jsme také Ätenáře seznámili s oblastà toků v orientovaných grafech, pÅ™edstavili jsme jim mnoho základnÃch pojmů a definic a také plnÄ› popsali myÅ¡lenku Dinitzova algoritmus pro hledánà maximálnÃho toku. Poskytli jsme mu také dalÅ¡Ã zdroje a práce, pokud by se chtÄ›l v tomto tématu dále vzdÄ›lávat. + +PozdÄ›ji jsme také uvedli nÄ›které možnosti, z mnoha dalÅ¡Ãch, frameworku Frama-C a doprovázejÃcÃmu mu anotaÄnÃmu jazyku ACSL, který jsme zde lehce popsali. Zavedli jsme si nástroje, které jsme použÃvali v naÅ¡Ã práci, mimo frameworku Frama-C také pluginy WP, RTE a také Å™eÅ¡iÄe Alt-Ergo, CVC4 a Z3. Ke každému z nich jsme dodali lehký popisek a také projekty v kterých byli využiti. + +V praktické Äásti jsme se již plnÄ› zaměřili na popsánà naÅ¡Ã implementace Dinitzova algoritmu a objasnÄ›nà naÅ¡Ãch rozhodnutà ohlednÄ› výbÄ›rů prostÅ™edků pro funkÄnost naÅ¡eho algoritmu. Poté jsme se vrhli na vysvÄ›tlenà a zaÅ™azenà anotacà jazyka ACSL do tématu a problému nalezenà maximálnÃho toku v sÃtÃch. Řekli jsme si zde také problémy svázané s použitÃm dynamicky alokované pamÄ›ti a pozdÄ›jÅ¡Ã verifikacà pomocà frameworku Frama-C. + +RozÅ¡ÃÅ™enÃm této práce by mohlo být porovnánà rychlostà ověřeného Dinitzova algoritmu pro hledánà maximálnÃho toku a neověřeného Dinitzova algoritmu s využitÃm programovacÃch taktik, sloužÃcÃm ke zrychlenà algoritmu. Takové taktiky Äasto nemůžeme pÅ™i verifikaci použÃvat, protože je velice obtÞné, nebo i nemožné je poté ověřit. \ No newline at end of file -- GitLab