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< p > We will delve into the details and ideas of the most common path-finding
algorithms. For the purpose of demonstrating some “features” of the improved
algorithms, we will use a 2D map with some rules that will allow us to show cons
and pros of the shown algorithms.< / p >
< p > Let' s have a look at the example map:< / p >
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< div class = "codeBlockContainer_Ckt0 theme-code-block" style = "--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)" > < div class = "codeBlockContent_biex" > < pre tabindex = "0" class = "prism-code language-text codeBlock_bY9V thin-scrollbar" style = "background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)" > < code class = "codeBlockLines_e6Vv" > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #############< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #..#..*.*.**#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > ##***.....**#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #..########.#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #...###...#.#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #..#...##.#.#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #..#.*.#..#.#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #....#....#.#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > ########*.*.#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #...........#< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #############< / span > < br > < / span > < / code > < / pre > < div class = "buttonGroup__atx" > < button type = "button" aria-label = "Copy code to clipboard" title = "Copy" class = "clean-btn" > < span class = "copyButtonIcons_eSgA" aria-hidden = "true" > < svg viewBox = "0 0 24 24" class = "copyButtonIcon_y97N" > < path fill = "currentColor" d = "M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z" > < / path > < / svg > < svg viewBox = "0 0 24 24" class = "copyButtonSuccessIcon_LjdS" > < path fill = "currentColor" d = "M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z" > < / path > < / svg > < / span > < / button > < / div > < / div > < / div >
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< p > We can see three different kinds of cells:< / p >
< ol >
< li > < code > #< / code > which represent walls, that cannot be entered at all< / li >
< li > < code > *< / code > which represent vortices that can be entered at the cost of 5 coins< / li >
< li > < code > .< / code > which represent normal cells that can be entered for 1 coin (which is the
base price of moving around the map)< / li >
< / ol >
< p > Let' s dissect a specific position on the map to get a better grasp of the rules:< / p >
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< div class = "codeBlockContainer_Ckt0 theme-code-block" style = "--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)" > < div class = "codeBlockContent_biex" > < pre tabindex = "0" class = "prism-code language-text codeBlock_bY9V thin-scrollbar" style = "background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)" > < code class = "codeBlockLines_e6Vv" > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > .< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > #S*< / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > .< / span > < br > < / span > < / code > < / pre > < div class = "buttonGroup__atx" > < button type = "button" aria-label = "Copy code to clipboard" title = "Copy" class = "clean-btn" > < span class = "copyButtonIcons_eSgA" aria-hidden = "true" > < svg viewBox = "0 0 24 24" class = "copyButtonIcon_y97N" > < path fill = "currentColor" d = "M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z" > < / path > < / svg > < svg viewBox = "0 0 24 24" class = "copyButtonSuccessIcon_LjdS" > < path fill = "currentColor" d = "M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z" > < / path > < / svg > < / span > < / button > < / div > < / div > < / div >
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< p > We are standing in the cell marked with < code > S< / code > and we have the following options< / p >
< ul >
< li > move to the north (< code > .< / code > ) with the cost of 1 coin,< / li >
< li > move to the west (< code > #< / code > ) < strong > is not< / strong > allowed because of the wall,< / li >
< li > move to the east (< code > *< / code > ) is allowed with the cost of 5 coins, and finally< / li >
< li > move to the south (< code > .< / code > ) with the cost of 1 coin.< / li >
< / ul >
< div class = "theme-admonition theme-admonition-info admonition_xJq3 alert alert--info" > < div class = "admonitionHeading_Gvgb" > < span class = "admonitionIcon_Rf37" > < svg viewBox = "0 0 14 16" > < path fill-rule = "evenodd" d = "M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z" > < / path > < / svg > < / span > info< / div > < div class = "admonitionContent_BuS1" > < p > Further on I will follow the same scheme for marking cells with an addition of
< code > D< / code > to denote the < em > destination< / em > to which we will be finding the shortest path.< / p > < / div > < / div >
< h2 class = "anchor anchorWithStickyNavbar_LWe7" id = "boilerplate" > Boilerplate< a href = "#boilerplate" class = "hash-link" aria-label = "Direct link to Boilerplate" title = "Direct link to Boilerplate" > < / a > < / h2 >
< p > For working with this map I have prepared a basic structure for the graph in C++
that will abstract some of the internal workings of our map, namely:< / p >
< ul >
< li > remembers the costs of moving around< / li >
< li > provides a simple function that returns price for moving < strong > directly< / strong > between
two positions on the map< / li >
< li > allows us to print the map out, just in case we' d need some adjustments to be
made< / li >
< / ul >
< p > We can see the < code > graph< / code > header here:< / p >
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< div class = "language-cpp codeBlockContainer_Ckt0 theme-code-block" style = "--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)" > < div class = "codeBlockContent_biex" > < pre tabindex = "0" class = "prism-code language-cpp codeBlock_bY9V thin-scrollbar" style = "background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)" > < code class = "codeBlockLines_e6Vv" > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > ifndef< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property expression" style = "color:hsl(5, 74%, 59%)" > _GRAPH_HPP< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > define< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property macro-name" style = "color:hsl(5, 74%, 59%)" > _GRAPH_HPP< / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" style = "display:inline-block" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < /span>< span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > include< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property string" style = "color:hsl(119, 34%, 47%)" > < cmath> < / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > include< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property string" style = "color:hsl(119, 34%, 47%)" > < limits> < / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > include< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property string" style = "color:hsl(119, 34%, 47%)" > < ostream> < / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > include< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property string" style = "color:hsl(119, 34%, 47%)" > < utility> < / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" > < / span > < span class = "token macro property directive-hash" style = "color:hsl(5, 74%, 59%)" > #< / span > < span class = "token macro property directive keyword" style = "color:hsl(301, 63%, 40%)" > include< / span > < span class = "token macro property" style = "color:hsl(5, 74%, 59%)" > < / span > < span class = "token macro property string" style = "color:hsl(119, 34%, 47%)" > < vector> < / span > < span class = "token plain" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)" > < span class = "token plain" style = "display:inline-block" > < / span > < br > < / span > < span class = "token-line" style = "color:hsl(230, 8%, 24%)"
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< div class = "theme-admonition theme-admonition-info admonition_xJq3 alert alert--info" > < div class = "admonitionHeading_Gvgb" > < span class = "admonitionIcon_Rf37" > < svg viewBox = "0 0 14 16" > < path fill-rule = "evenodd" d = "M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z" > < / path > < / svg > < / span > Source code< / div > < div class = "admonitionContent_BuS1" > < p > You can find all the source code referenced in this series
< a href = "/files/algorithms/paths/bf-to-astar.tar.gz" target = "_blank" rel = "noopener noreferrer" > here< / a > .< / p > < / div > < / div >
< p > Let' s finally start with some algorithms!< / p > < / div > < footer class = "theme-doc-footer docusaurus-mt-lg" > < div class = "theme-doc-footer-tags-row row margin-bottom--sm" > < div class = "col" > < b > Tags:< / b > < ul class = "tags_jXut padding--none margin-left--sm" > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/cpp/" > cpp< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/brute-force/" > brute force< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/bellman-ford/" > bellman ford< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/dynamic-programming/" > dynamic programming< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/dijkstra/" > dijkstra< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/greedy/" > greedy< / a > < / li > < li class = "tag_QGVx" > < a class = "tag_zVej tagRegular_sFm0" href = "/algorithms/tags/a-star/" > a star< / a > < / li > < / ul > < / div > < / div > < div class = "theme-doc-footer-edit-meta-row row" > < div class = "col" > < a href = "https://github.com/mfocko/blog/tree/main/algorithms/11-paths/2024-01-01-bf-to-astar/index.md" target = "_blank" rel = "noopener noreferrer" class = "theme-edit-this-page" > < svg fill = "currentColor" height = "20" width = "20" viewBox = "0 0 40 40" class = "iconEdit_Z9Sw" aria-hidden = "true" > < g > < path d = "m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z" > < / path > < / g > < / svg > Edit this page< / a > < / div > < div class = "col lastUpdated_vwxv" > < span class = "theme-last-updated" > Last updated<!-- --> on < b > < time datetime = "2024-01-01T00:00:00.000Z" > Jan 1, 2024< / time > < / b > < / span > < / div > < / div > < / footer > < / article > < nav class = "pagination-nav docusaurus-mt-lg" aria-label = "Docs pages" > < a class = "pagination-nav__link pagination-nav__link--prev" href = "/algorithms/category/paths-in-graphs/" > < div class = "pagination-nav__sublabel" > Previous< / div > < div class = "pagination-nav__label" > Paths in Graphs< / div > < / a > < a class = "pagination-nav__link pagination-nav__link--next" href = "/algorithms/paths/bf-to-astar/bf/" > < div class = "pagination-nav__sublabel" > Next< / div > < div class = "pagination-nav__label" > BF< / div > < / a > < / nav > < / div > < / div > < div class = "col col--3" > < div class = "tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop" > < ul class = "table-of-contents table-of-contents__left-border" > < li > < a href = "#intro" class = "table-of-contents__link toc-highlight" > Intro< / a > < / li > < li > < a href = "#boilerplate" class = "table-of-contents__link toc-highlight" > Boilerplate< / a > < / li > < / ul > < / div > < / div > < / div > < / div > < / main > < / div > < / div > < / div > < footer class = "footer footer--dark" > < div class = "container container-fluid" > < div class = "row footer__links" > < div class = "col footer__col" > < div class = "footer__title" > Git< / div > < ul class = "footer__items clean-list" > < li class = "footer__item" > < a href = "https://github.com/mfocko" target = "_blank" rel = "noopener noreferrer" class = "footer__link-item" > GitHub< svg width = "13.5" height = "13.5" aria-hidden = "true" viewBox = "0 0 24 24" class = "iconExternalLink_nPIU" > < path fill = "currentColor" d = "M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z" > < / path > < / svg > < / a > < / li > < li class = "footer__item" > < a href = "https://gitlab.com/mfocko" target = "_blank" rel = "noopener noreferrer" class = "footer__link-item" > GitLab< svg width = "13.5" height = "13.5" aria-hidden = "true" viewBox = "0 0 24 24" class = "iconExternalLink_nPIU" > < path fill = "currentColor" d = "M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z" > < / path > < / svg > < / a > < / li > < li class = "footer__item" > < a href = "https://git.mfocko.xyz/mfocko" target = "_blank" rel = "noopener noreferrer" class = "footer__link-item" > Gitea (self-hosted)< svg width = "13.5" height = "13.5" aria-hidden = "true" viewBox = "0 0 24 24" class = "iconExternalLink_nPIU" > < path fill = "currentColor" d = "M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z" > < / path > < / svg > < / a > < / li > < / ul > < /div
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