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In weighted graphs,",source:"@site/docs/algorithms/shortest-path/dijkstra.md",sourceDirName:"algorithms/shortest-path",slug:"/algorithms/shortest-path/dijkstra",permalink:"/graaf/docs/algorithms/shortest-path/dijkstra",draft:!1,editUrl:"https://github.com/bobluppes/graaf/tree/main/docs/algorithms/shortest-path/dijkstra.md",tags:[],version:"current",sidebarPosition:2,frontMatter:{sidebar_position:2},sidebar:"algorithmSidebar",previous:{title:"BFS Based Shortest Path",permalink:"/graaf/docs/algorithms/shortest-path/bfs-based-shortest-path"},next:{title:"A* Search Algorithm",permalink:"/graaf/docs/algorithms/shortest-path/a-star"}},p={},h=[{value:"Syntax",id:"syntax",level:2}],l={toc:h},c="wrapper";function d(e){let{components:t,...r}=e;return(0,n.kt)(c,(0,a.Z)({},l,r,{components:t,mdxType:"MDXLayout"}),(0,n.kt)("h1",{id:"dijkstra-shortest-path"},"Dijkstra Shortest Path"),(0,n.kt)("p",null,"Dijkstra's algorithm computes shortest paths between nodes in weighted and unweighted graphs. In weighted graphs,\nedge weights should be non-negative. Dijkstra's algorithm is implemented with a priority queue and runs\nin ",(0,n.kt)("inlineCode",{parentName:"p"},"O(|E|log|V|)")," for connected graphs, where ",(0,n.kt)("inlineCode",{parentName:"p"},"|E|")," is the number of edges and ",(0,n.kt)("inlineCode",{parentName:"p"},"|V|")," the number of vertices in the\ngraph."),(0,n.kt)("p",null,(0,n.kt)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm"},"wikipedia")),(0,n.kt)("h2",{id:"syntax"},"Syntax"),(0,n.kt)("p",null,"calculates the shortest path between on start_vertex and one end_vertex using Dijkstra's algorithm. Works on both\nweighted as well as unweighted graphs. For unweighted graphs, a unit weight is used for each edge."),(0,n.kt)("pre",null,(0,n.kt)("code",{parentName:"pre",className:"language-cpp"},"template <typename V, typename E, graph_type T, typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>\nstd::optional<graph_path<WEIGHT_T>> \ndijkstra_shortest_path(const graph<V, E, T>& graph, vertex_id_t start_vertex, vertex_id_t end_vertex);\n")),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"graph")," The graph to extract shortest path from."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"start_vertex")," Vertex id where the shortest path should start."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"end_vertex")," Vertex id where the shortest path should end."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"return")," An optional with the shortest path (list of vertices) if found.")),(0,n.kt)("p",null,"Find the shortest paths from a source vertex to all other vertices in the graph using Dijkstra's algorithm."),(0,n.kt)("pre",null,(0,n.kt)("code",{parentName:"pre",className:"language-cpp"},"template <typename V, typename E, graph_type T, typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>\n[[nodiscard]] std::unordered_map<vertex_id_t, graph_path<WEIGHT_T>>\ndijkstra_shortest_paths(const graph<V, E, T>& graph, vertex_id_t source_vertex);\n")),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"graph")," The graph we want to search."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"source_vertex")," The source vertex from which to compute shortest paths."),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"return")," A map containing the shortest paths from the source vertex to all other vertices. The map keys are target\nvertex IDs, and the values are instances of graph_path, representing the shortest distance and the path (list of\nvertex IDs) from the source to the target. If a vertex is not reachable from the source, its entry will be absent from\nthe map.")))}d.isMDXComponent=!0}}]); \ No newline at end of file diff --git a/assets/js/49fbfbc8.2c00d934.js b/assets/js/49fbfbc8.2c00d934.js new file mode 100644 index 00000000..04366d95 --- /dev/null +++ b/assets/js/49fbfbc8.2c00d934.js @@ -0,0 +1 @@ +"use strict";(self.webpackChunkgraaf=self.webpackChunkgraaf||[]).push([[116],{3905:(t,e,r)=>{r.d(e,{Zo:()=>p,kt:()=>d});var o=r(7294);function a(t,e,r){return e in t?Object.defineProperty(t,e,{value:r,enumerable:!0,configurable:!0,writable:!0}):t[e]=r,t}function n(t,e){var r=Object.keys(t);if(Object.getOwnPropertySymbols){var o=Object.getOwnPropertySymbols(t);e&&(o=o.filter((function(e){return Object.getOwnPropertyDescriptor(t,e).enumerable}))),r.push.apply(r,o)}return r}function i(t){for(var e=1;e<arguments.length;e++){var r=null!=arguments[e]?arguments[e]:{};e%2?n(Object(r),!0).forEach((function(e){a(t,e,r[e])})):Object.getOwnPropertyDescriptors?Object.defineProperties(t,Object.getOwnPropertyDescriptors(r)):n(Object(r)).forEach((function(e){Object.defineProperty(t,e,Object.getOwnPropertyDescriptor(r,e))}))}return t}function l(t,e){if(null==t)return{};var r,o,a=function(t,e){if(null==t)return{};var r,o,a={},n=Object.keys(t);for(o=0;o<n.length;o++)r=n[o],e.indexOf(r)>=0||(a[r]=t[r]);return a}(t,e);if(Object.getOwnPropertySymbols){var n=Object.getOwnPropertySymbols(t);for(o=0;o<n.length;o++)r=n[o],e.indexOf(r)>=0||Object.prototype.propertyIsEnumerable.call(t,r)&&(a[r]=t[r])}return a}var s=o.createContext({}),c=function(t){var e=o.useContext(s),r=e;return t&&(r="function"==typeof t?t(e):i(i({},e),t)),r},p=function(t){var e=c(t.components);return o.createElement(s.Provider,{value:e},t.children)},g="mdxType",m={inlineCode:"code",wrapper:function(t){var e=t.children;return o.createElement(o.Fragment,{},e)}},u=o.forwardRef((function(t,e){var r=t.components,a=t.mdxType,n=t.originalType,s=t.parentName,p=l(t,["components","mdxType","originalType","parentName"]),g=c(r),u=a,d=g["".concat(s,".").concat(u)]||g[u]||m[u]||n;return r?o.createElement(d,i(i({ref:e},p),{},{components:r})):o.createElement(d,i({ref:e},p))}));function d(t,e){var r=arguments,a=e&&e.mdxType;if("string"==typeof t||a){var n=r.length,i=new Array(n);i[0]=u;var l={};for(var s in e)hasOwnProperty.call(e,s)&&(l[s]=e[s]);l.originalType=t,l[g]="string"==typeof t?t:a,i[1]=l;for(var c=2;c<n;c++)i[c]=r[c];return o.createElement.apply(null,i)}return o.createElement.apply(null,r)}u.displayName="MDXCreateElement"},1435:(t,e,r)=>{r.r(e),r.d(e,{assets:()=>s,contentTitle:()=>i,default:()=>m,frontMatter:()=>n,metadata:()=>l,toc:()=>c});var o=r(7462),a=(r(7294),r(3905));const n={sidebar_position:1},i="Topological sort algorithm",l={unversionedId:"algorithms/topological-sort/topological-sort",id:"algorithms/topological-sort/topological-sort",title:"Topological sort algorithm",description:"Topological sort algorithm processing DAG(directed acyclic graph) using DFS traversal.",source:"@site/docs/algorithms/topological-sort/topological-sort.md",sourceDirName:"algorithms/topological-sort",slug:"/algorithms/topological-sort/",permalink:"/graaf/docs/algorithms/topological-sort/",draft:!1,editUrl:"https://github.com/bobluppes/graaf/tree/main/docs/algorithms/topological-sort/topological-sort.md",tags:[],version:"current",sidebarPosition:1,frontMatter:{sidebar_position:1},sidebar:"algorithmSidebar",previous:{title:"Algorithms Overview",permalink:"/graaf/docs/algorithms/intro"},next:{title:"Traversal Algorithms",permalink:"/graaf/docs/category/traversal-algorithms"}},s={},c=[{value:"Syntax",id:"syntax",level:2}],p={toc:c},g="wrapper";function m(t){let{components:e,...r}=t;return(0,a.kt)(g,(0,o.Z)({},p,r,{components:e,mdxType:"MDXLayout"}),(0,a.kt)("h1",{id:"topological-sort-algorithm"},"Topological sort algorithm"),(0,a.kt)("p",null,"Topological sort algorithm processing DAG(directed acyclic graph) using DFS traversal.\nEach vertex is visited only after all its dependencies are visited.\nThe runtime of the algorithm is ",(0,a.kt)("inlineCode",{parentName:"p"},"O(|V|+|E|)")," and the memory consumption is ",(0,a.kt)("inlineCode",{parentName:"p"},"O(|V|)"),"."),(0,a.kt)("p",null,(0,a.kt)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Topological_sorting"},"wikipedia")),(0,a.kt)("h2",{id:"syntax"},"Syntax"),(0,a.kt)("pre",null,(0,a.kt)("code",{parentName:"pre",className:"language-cpp"},"template <typename V, typename E>\n[[nodiscard]] std::optional<std::vector<vertex_id_t>> topological_sort(\n const graph<V, E, graph_type::DIRECTED>& graph);\n")),(0,a.kt)("ul",null,(0,a.kt)("li",{parentName:"ul"},(0,a.kt)("strong",{parentName:"li"},"graph")," The directed graph to traverse."),(0,a.kt)("li",{parentName:"ul"},(0,a.kt)("strong",{parentName:"li"},"return")," Vector of vertices sorted in topological order. If the graph contains cycles, it returns std::nullopt.")))}m.isMDXComponent=!0}}]); \ No newline at end of file diff --git a/assets/js/4e71c965.e333dc90.js b/assets/js/4e71c965.e333dc90.js new file mode 100644 index 00000000..d174ea5b --- /dev/null +++ b/assets/js/4e71c965.e333dc90.js @@ -0,0 +1 @@ +"use strict";(self.webpackChunkgraaf=self.webpackChunkgraaf||[]).push([[646],{3905:(e,t,a)=>{a.d(t,{Zo:()=>h,kt:()=>u});var r=a(7294);function n(e,t,a){return t in e?Object.defineProperty(e,t,{value:a,enumerable:!0,configurable:!0,writable:!0}):e[t]=a,e}function i(e,t){var a=Object.keys(e);if(Object.getOwnPropertySymbols){var r=Object.getOwnPropertySymbols(e);t&&(r=r.filter((function(t){return Object.getOwnPropertyDescriptor(e,t).enumerable}))),a.push.apply(a,r)}return a}function l(e){for(var t=1;t<arguments.length;t++){var 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r.createElement(r.Fragment,{},t)}},m=r.forwardRef((function(e,t){var a=e.components,n=e.mdxType,i=e.originalType,s=e.parentName,h=o(e,["components","mdxType","originalType","parentName"]),c=p(a),m=n,u=c["".concat(s,".").concat(m)]||c[m]||d[m]||i;return a?r.createElement(u,l(l({ref:t},h),{},{components:a})):r.createElement(u,l({ref:t},h))}));function u(e,t){var a=arguments,n=t&&t.mdxType;if("string"==typeof e||n){var i=a.length,l=new Array(i);l[0]=m;var o={};for(var s in t)hasOwnProperty.call(t,s)&&(o[s]=t[s]);o.originalType=e,o[c]="string"==typeof e?e:n,l[1]=o;for(var p=2;p<i;p++)l[p]=a[p];return r.createElement.apply(null,l)}return r.createElement.apply(null,a)}m.displayName="MDXCreateElement"},8469:(e,t,a)=>{a.r(t),a.d(t,{assets:()=>s,contentTitle:()=>l,default:()=>d,frontMatter:()=>i,metadata:()=>o,toc:()=>p});var r=a(7462),n=(a(7294),a(3905));const i={sidebar_position:2},l="Depth First Search (DFS)",o={unversionedId:"algorithms/traversal/depth-first-search",id:"algorithms/traversal/depth-first-search",title:"Depth First Search (DFS)",description:"Depth First Search (DFS) Algorithm",source:"@site/docs/algorithms/traversal/depth-first-search.md",sourceDirName:"algorithms/traversal",slug:"/algorithms/traversal/depth-first-search",permalink:"/graaf/docs/algorithms/traversal/depth-first-search",draft:!1,editUrl:"https://github.com/bobluppes/graaf/tree/main/docs/algorithms/traversal/depth-first-search.md",tags:[],version:"current",sidebarPosition:2,frontMatter:{sidebar_position:2},sidebar:"algorithmSidebar",previous:{title:"Breadth First Search (BFS)",permalink:"/graaf/docs/algorithms/traversal/breadth-first-search"},next:{title:"Shortest Path Algorithms",permalink:"/graaf/docs/category/shortest-path-algorithms"}},s={},p=[{value:"Depth First Search (DFS) Algorithm",id:"depth-first-search-dfs-algorithm",level:2},{value:"DFS is particularly useful for:",id:"dfs-is-particularly-useful-for",level:3},{value:"Limitations of DFS:",id:"limitations-of-dfs",level:4},{value:"Complexity and Performance:",id:"complexity-and-performance",level:3},{value:"Syntax",id:"syntax",level:2},{value:"Explanation of Parameters:",id:"explanation-of-parameters",level:3}],h={toc:p},c="wrapper";function d(e){let{components:t,...a}=e;return(0,n.kt)(c,(0,r.Z)({},h,a,{components:t,mdxType:"MDXLayout"}),(0,n.kt)("h1",{id:"depth-first-search-dfs"},"Depth First Search (DFS)"),(0,n.kt)("h2",{id:"depth-first-search-dfs-algorithm"},"Depth First Search (DFS) Algorithm"),(0,n.kt)("p",null,"Depth First Search (DFS) is a fundamental graph traversal algorithm used to explore and analyze graphs, whether they are directed or undirected.\nDFS traverses deeper into the graph before backtracking to explore other branches."),(0,n.kt)("p",null,"The DFS algorithm can be succinctly described using the following steps:"),(0,n.kt)("ol",null,(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},"Begin by selecting a source node as the starting point of the traversal and push it onto a stack data structure.")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},"While the stack is not empty, repeat the following steps:"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"a. Pop a node from the top of the stack."),(0,n.kt)("li",{parentName:"ul"},"b. Process the popped node, which may involve examining its attributes, marking it as visited, or performing other relevant operations."),(0,n.kt)("li",{parentName:"ul"},"c. Push all unvisited neighbors of the popped node onto the stack."))),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},"Continue this process until the stack becomes empty, indicating that all reachable nodes have been visited."))),(0,n.kt)("p",null,"The main difference to the BFS is the use of a stack instead of a queue."),(0,n.kt)("h3",{id:"dfs-is-particularly-useful-for"},"DFS is particularly useful for:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Topological Sorting:"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Finds a linear ordering of nodes that respects the partial order imposed by directed edges in a directed acyclic graph."))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Pathfinding:"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Can be used to find paths between nodes, although it may not always find the shortest path."))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Solving Mazes:"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Navigates through maze-like structures to find a way from a starting point to an end point."))),(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("p",{parentName:"li"},"Detecting Cycles:"),(0,n.kt)("ul",{parentName:"li"},(0,n.kt)("li",{parentName:"ul"},"Helps identify cycles in a graph, which is valuable for various applications.")))),(0,n.kt)("h4",{id:"limitations-of-dfs"},"Limitations of DFS:"),(0,n.kt)("ol",null,(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("strong",{parentName:"p"},"Completeness:")," DFS may not explore all nodes in disconnected graphs unless modifications are made to the algorithm.")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("strong",{parentName:"p"},"Infinite Graphs:")," DFS can get stuck in an infinite loop if applied to graphs with infinite branches.")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("strong",{parentName:"p"},"Performance on Dense Graphs:")," In dense graphs, DFS might explore many nodes before reaching a solution, making it less efficient compared to other algorithms.")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("strong",{parentName:"p"},"No Guarantee of Optimality:")," Like BFS, DFS may not always find the optimal solution, especially in cases where the graph has weighted edges or other complexities.")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("strong",{parentName:"p"},"Memory Usage:")," DFS on deep graphs may lead to excessive recursion and memory consumption due to the call stack.")),(0,n.kt)("li",{parentName:"ol"},(0,n.kt)("p",{parentName:"li"},(0,n.kt)("strong",{parentName:"p"},"Biased Exploration:")," DFS can lead to biased exploration when some branches are deeper than others, potentially missing relevant solutions."))),(0,n.kt)("h3",{id:"complexity-and-performance"},"Complexity and Performance:"),(0,n.kt)("p",null,"The DFS algorithm is implemented with a stack and runs in ",(0,n.kt)("inlineCode",{parentName:"p"},"O(|V| + |E|)")," time complexity for connected graphs, where ",(0,n.kt)("inlineCode",{parentName:"p"},"|E|")," is the number of edges and ",(0,n.kt)("inlineCode",{parentName:"p"},"|V|")," the number of vertices in the graph."),(0,n.kt)("p",null,"In summary, Depth First Search is a powerful and versatile algorithm for exploring graphs, but its limitations in handling weighted graphs and negative edge weights should be considered. It provides a straightforward way to explore a graph layer by layer and is particularly useful for unweighted graph scenarios and connectivity analysis."),(0,n.kt)("p",null,(0,n.kt)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Depth-first_search"},"wikipedia")),(0,n.kt)("h2",{id:"syntax"},"Syntax"),(0,n.kt)("p",null,"The dfs_termination_strategy returns true when a certain condition is met, causing to terminate. The dfs_edge_callback is a function that is used as a callback during the DFS traversal to perform some action whenever an edge is traversed."),(0,n.kt)("pre",null,(0,n.kt)("code",{parentName:"pre",className:"language-cpp"},"template <\n typename V, typename E, graph_type T,\n typename EDGE_CALLBACK_T = detail::noop_callback,\n typename SEARCH_TERMINATION_STRATEGY_T = detail::exhaustive_search_strategy>\n requires std::invocable<EDGE_CALLBACK_T &, edge_id_t &> &&\n std::is_invocable_r_v<bool, SEARCH_TERMINATION_STRATEGY_T &,\n vertex_id_t>\nvoid depth_first_traverse(\n const graph<V, E, T> &graph, vertex_id_t start_vertex,\n const EDGE_CALLBACK_T &edge_callback,\n const SEARCH_TERMINATION_STRATEGY_T &search_termination_strategy =\n SEARCH_TERMINATION_STRATEGY_T{});\n")),(0,n.kt)("h3",{id:"explanation-of-parameters"},"Explanation of Parameters:"),(0,n.kt)("ul",null,(0,n.kt)("li",{parentName:"ul"},(0,n.kt)("strong",{parentName:"li"},"graph"),": The graph to traverse. 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Dijkstra's algorithm or the Bellman-Ford algorithm are better suited for such cases."))),(0,n.kt)("h3",{id:"complexity-and-performance"},"Complexity and Performance:"),(0,n.kt)("p",null,"The BFS algorithm is implemented with a priority queue and runs in ",(0,n.kt)("inlineCode",{parentName:"p"},"O(|V| + |E|)")," time complexity for connected graphs, where ",(0,n.kt)("inlineCode",{parentName:"p"},"|E|")," is the number of edges and ",(0,n.kt)("inlineCode",{parentName:"p"},"|V|")," the number of vertices in the graph."),(0,n.kt)("p",null,"In summary, Breadth First Search is a powerful and versatile algorithm for exploring graphs, but its limitations in handling weighted graphs and negative edge weights should be considered. 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dir="ltr" class="docs-wrapper docs-doc-page docs-version-current plugin-docs plugin-id-default docs-doc-id-algorithms/coloring/greedy-graph-coloring"> +<head> +<meta charset="UTF-8"> +<meta name="generator" content="Docusaurus v2.4.1"> +<title data-rh="true">Greedy Graph Coloring Algorithm | Graaf lib + + + + +

+

Greedy Graph Coloring Algorithm

Greedy Graph Coloring computes a coloring of the vertices of a (simple, connected) graph such that no two adjacent vertices have the same color.

If the graph has different connected components, each component will be treated as a separate simple connected graph.

The algorithm is heuristic and does not guarantee an optimal number of different colors (that is, equal to the chromatic number of a simple, connected graph).

Colors are represented by the numbers 0, 1, 2,... The greedy algorithm considers the vertices of the graph in sequence and assigns each vertex its first available color, i.e. the color with the smallest number that is not already used by one of its neighbors.

The overall worst-case time complexity of the algorithm is O(n^2). In cases where the graph has a fixed degree (a constant number of neighbors for each vertex), the time complexity can be approximated as O(n). However, if the graph is highly connected (dense) and approaches a complete graph, the time complexity could approach O(n^2).

If no coloring is possible, an empty unordered_map is returned. This is the case when the graph contains no vertices.

wikipedia

Syntax​

template <typename GRAPH>
std::unordered_map<vertex_id_t, int> greedy_graph_coloring(const GRAPH& graph);
  • graph A graph to perform graph coloring on.
  • return An unordered_map where keys are vertex identifiers and values are their respective colors. If no coloring is possible, an empty unordered_map is returned.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/cycle-detection/dfs-based.html b/docs/algorithms/cycle-detection/dfs-based.html new file mode 100644 index 00000000..a6650ac9 --- /dev/null +++ b/docs/algorithms/cycle-detection/dfs-based.html @@ -0,0 +1,23 @@ + + + + + +DFS Based Cycle Detection | Graaf lib + + + + +
+

DFS Based Cycle Detection

A DFS based cycle detection algorithm is used to identify cycles in graphs, both directed and undirected. +The algorithm can be used to detect cycles in the structure of a graph, as it does not consider edge weights.

Directed graph

The key idea is that when a vertex is processed, mark it as: UNVISITED, VISITED and NO_CYCLE. +By default all vertices marked as UNVISITED. During the traversal, we label vertices as VISITED. At the exit of the recursion, we label the vertex as NO CYCLE. +If we met a vertex labeled VISITED, we found a cycle in the graph.

Undirected graph

The key idea is to store the parent of each vertex during the traversal. So when we check neighboring vertices, we skip back edge. +During the traversal we mark the vertex as visited and continue the traversal. In case a vertex was visited before and vertices have different parents, we found a cycle.

The runtime of the algorithm is O(|V| + |E|) and memory consumption is O(|V|). Where V is the number of vertices in the graph and E the number of edges. +The algorithm uses DFS traversal and therefore suffers the same limitations (see depth-first-search.md).

Use cases

  • Resource dependencies:
    • Redundant connections.
    • Deadlocks in concurrent systems.
    • Deadlocks in concurrent systems.
  • Logical dependencies:
    • Data base relation.
    • Dependency management.
    • Circuit design.
  • Infinity loops.

wikipedia

Syntax​

Cycle detection for directed graph.

template <typename V, typename E>
[[nodiscard]] bool dfs_cycle_detection(
    const graph<V, E, graph_type::DIRECTED> &graph);

Cycle detection for unidrected graph.

template <typename V, typename E>
[[nodiscard]] bool dfs_cycle_detection(
    const graph<V, E, graph_type::UNDIRECTED> &graph);
  • graph The graph to traverse.
  • return Returns true in case of cycle otherwise returns false.

Similar algorithms​

There are many algorithms for cycle detection or algorithms with specific cycle conditions. +See wikipedia

+ + + + \ No newline at end of file diff --git a/docs/algorithms/intro.html b/docs/algorithms/intro.html new file mode 100644 index 00000000..c6a02704 --- /dev/null +++ b/docs/algorithms/intro.html @@ -0,0 +1,17 @@ + + + + + +Algorithms Overview | Graaf lib + + + + +
+

Algorithms Overview

This section provides an overview of the algorithms currently implemented in Graaf.

+ + + + \ No newline at end of file diff --git a/docs/algorithms/minimum-spanning-tree/kruskal.html b/docs/algorithms/minimum-spanning-tree/kruskal.html new file mode 100644 index 00000000..c57c3120 --- /dev/null +++ b/docs/algorithms/minimum-spanning-tree/kruskal.html @@ -0,0 +1,22 @@ + + + + + +Kruskal's Algorithm | Graaf lib + + + + +
+

Kruskal's Algorithm

Kruskal's algorithm finds the minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, +it finds a minimum spanning tree. +The algorithm is implemented with disjoint set union and finding minimum weighted edges. +Worst-case performance is O(|E|log|V|), where |E| is the number of edges and |V| is the number of vertices in the +graph. Memory usage is O(V+E) for maintaining vertices (DSU) and edges.

wikipedia

Syntax​

Calculates the shortest path with the minimum edge sum.

template <typename V, typename E>
[[nodiscard]] std::vector<edge_id_t> kruskal_minimum_spanning_tree(
    const graph<V, E, graph_type::UNDIRECTED>& graph);
  • graph The graph to extract MST or MSF.
  • return Returns a vector of edges that form MST if the graph is connected, otherwise it returns the minimum +spanning forest.

Special case​

In case of multiply edges with same weight leading to a vertex, prioritizing vertices with lesser vertex number.

std::sort(edges_to_process.begin(), edges_to_process.end(),
              [](detail::edge_to_process<E>& e1,
                 detail::edge_to_process<E>& e2) {
                       if (e1 != e2)
                           return e1.get_weight() < e2.get_weight();
          return e1.vertex_a < e2.vertex_a || e1.vertex_b < e2.vertex_b;
        });

For custom type edge, we should provide < and != operators

struct custom_edge : public graaf::weighted_edge<int> {
 public:
  int weight_{};

  [[nodiscard]] int get_weight() const noexcept override { return weight_; }
 
  custom_edge(int weight): weight_{weight} {};
  custom_edge(){};
  ~custom_edge(){};

  // Providing '<' and '!=' operators for sorting edges
  bool operator<(const custom_edge& e) const noexcept {
    return this->weight_ < e.weight_;
  }
  bool operator!=(const custom_edge& e) const noexcept {
    return this->weight_ != e.weight_;
  }
};
+ + + + \ No newline at end of file diff --git a/docs/algorithms/minimum-spanning-tree/prim.html b/docs/algorithms/minimum-spanning-tree/prim.html new file mode 100644 index 00000000..e54184eb --- /dev/null +++ b/docs/algorithms/minimum-spanning-tree/prim.html @@ -0,0 +1,20 @@ + + + + + +Prim's Algorithm | Graaf lib + + + + +
+

Prim's Algorithm

Prim's algorithm computes the minimum spanning tree (MST) of a connected, undirected graph with weighted edges. Starting +with an arbitrary vertex, the algorithm iteratively selects the edge with the smallest weight that connects a +vertex in the tree to a vertex outside the tree, adding it to the MST.

The algorithm's worst-case time complexity is O(∣E∣log∣V∣).

Unlike Kruskal's algorithm, Prim's algorithm works efficiently on dense graphs. A limitation is that it requires the +graph to be connected and does not handle disconnected graphs or graphs with negative-weight cycles.

Prim's MST is often used in network design, such as electrical wiring and telecommunications.

wikipedia

Syntax​

template <typename V, typename E>
[[nodiscard]] std::optional<std::vector<edge_id_t> > prim_minimum_spanning_tree(
    const graph<V, E, graph_type::UNDIRECTED>& graph, vertex_id_t start_vertex);
  • graph The undirected graph for which we want to compute the MST.
  • start_vertex The vertex ID which should be the root of the MST.
  • return Returns a vector of edges that form MST if the graph is connected, otherwise returns an empty optional.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/shortest-path/a-star.html b/docs/algorithms/shortest-path/a-star.html new file mode 100644 index 00000000..ef6e6124 --- /dev/null +++ b/docs/algorithms/shortest-path/a-star.html @@ -0,0 +1,19 @@ + + + + + +A* Search Algorithm | Graaf lib + + + + +
+

A* Search Algorithm

A* computes the shortest path between a starting vertex and a target vertex in weighted and unweighted graphs. +It can be seen as an extension of Dijkstra's classical shortest paths algorithm. The implementation of A* also tries to follow dijkstra_shortest_path closely where appropriate. Compared to Dijkstra's algorithm, A* only finds the shortest path from a start vertex to a target vertex, and not the shortest path to all possible target vertices. Another difference is that A* uses a heuristic function to achieve better performance.

At each iteration of its main loop, A* needs to determine which of its paths to extend. It does so by minimizing the so-called f_score.

In A*, the f_score represents the estimated total cost of the path from the start vertex to the goal vertex through the current vertex. It's a combination of two components:

  1. g_score: The actual cost of the path from the start vertex to the current vertex.
  2. h_score (heuristic score): An estimate of the cost required from the current vertex to the goal vertex.

A* tries to minimize the f_score for each vertex as it explores the graph. The idea is to prioritize exploring vertices that have lower f_score values, as they are expected to lead to potentially better paths.

Mathematically, f_score is often defined as:

f_score = g_score + h_score

Where:

  • g_score is the cost of the path from the start vertex to the current vertex.
  • h_score is the heuristic estimate of the cost from the current vertex to the goal vertex.

In the implementation, the heuristic function heuristic provides an estimate of h_score for each vertex, and the actual cost of the path from the start vertex to the current vertex is stored in the g_score unordered map, as the algorithm progresses.

In the implementation, dist_from_start from path_vertex represents the f_score of the path.

The time complexity of A* depends on the provided heuristic function. In the worst case of an unbounded search space, the number of nodes expanded is exponential in the depth of the solution (the shortest path) d. This can be expressed as O(b^d), where b is the branching factor (the average number of successors per state) per stage.

In weighted graphs, edge weights should be non-negative. Like in the implementation of Dijkstra's algorithm, A* is implemented with the priority queue provided by C++, to perform the repeated selection of minimum (estimated) cost nodes to expand. This is the open_set. If the shortest path is not unique, one of the shortest paths is returned.

Syntax​

calculates the shortest path between on start_vertex and one end_vertex using A* search. +Works on both weighted as well as unweighted graphs. For unweighted graphs, a unit weight is used for each edge.

template <typename V, typename E, graph_type T, typename HEURISTIC_T, typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>
  requires std::is_invocable_r_v<WEIGHT_T, HEURISTIC_T&, vertex_id_t>
std::optional<graph_path<WEIGHT_T>> a_star_search(
    const graph<V, E, T> &graph, vertex_id_t start_vertex, vertex_id_t target_vertex,
    const HEURISTIC_T &heuristic);
  • graph The graph to extract shortest path from.
  • start_vertex The vertex id where the shortest path should should start.
  • target_vertex The vertex id where the shortest path should end.
  • heuristic A heuristic function estimating the cost from a vertex to the target.
  • return An optional containing the shortest path (a list of vertices) if found, or std::nullopt if no such path exists.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/shortest-path/bellman-ford.html b/docs/algorithms/shortest-path/bellman-ford.html new file mode 100644 index 00000000..4f8498b3 --- /dev/null +++ b/docs/algorithms/shortest-path/bellman-ford.html @@ -0,0 +1,20 @@ + + + + + +Bellman-Ford Shortest Path | Graaf lib + + + + +
+

Bellman-Ford Shortest Path

Bellman-Ford's algorithm computes shortest paths from a single source vertex to all of the other vertices in weighted graph and unweighted graphs. In weighted graphs, edge weights are allowed to be negative. Bellman-Ford's algorithm runs in O(|E||V|) for connected graphs, where |E| is the number of edges and |V| the number of vertices in the +graph.

A limitation is that this implementation doesn't check for negative-weight cycles.

wikipedia

Syntax​

Find the shortest paths from a source vertex to all other vertices using the Bellman-Ford algorithm.

template <typename V, typename E, graph_type T, 
          typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>
std::unordered_map<vertex_id_t, graph_path<WEIGHT_T>>
bellman_ford_shortest_paths(const graph<V, E, T>& graph, vertex_id_t start_vertex);
  • graph The graph to extract shortest path from.
  • start_vertex The source vertex for the shortest paths.
  • return A map of target vertex IDs to shortest path structures. +Each value contains a graph_path object representing the shortest path from the source vertex to the respective vertex. +If a vertex is unreachable from the source, its entry will be absent from the map.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/shortest-path/bfs-based-shortest-path.html b/docs/algorithms/shortest-path/bfs-based-shortest-path.html new file mode 100644 index 00000000..8f026211 --- /dev/null +++ b/docs/algorithms/shortest-path/bfs-based-shortest-path.html @@ -0,0 +1,23 @@ + + + + + +BFS Based Shortest Path | Graaf lib + + + + +
+

BFS Based Shortest Path

Breadth-First Search (BFS) is a graph traversal algorithm that efficiently finds the shortest +path between two vertices in an unweighted graph by exploring vertices level by level, +guaranteeing the shortest path, and has a time complexity of O(|E| + |V|), +where |V| is the number of vertices and |E| is the number of edges in the graph. +BFS uses a queue to iteratively visit neighboring vertices from the source +vertex, ensuring that the shortest path is discovered before longer paths.

wikipedia

Syntax​

Calculates the shortest path between one start_vertex and one +end_vertex using BFS. This does not consider edge weights.

template <typename V, typename E, graph_type T, typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>
std::optional<graph_path<WEIGHT_T>> bfs_shortest_path(
    const graph<V, E, T>& graph, vertex_id_t start_vertex, vertex_id_t end_vertex);
  • graph The graph to extract shortest path from.
  • start_vertex Vertex id where the shortest path should start.
  • end_vertex Vertex id where the shortest path should end.
  • return An optional with the shortest path (list of vertices) if found.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/shortest-path/dijkstra.html b/docs/algorithms/shortest-path/dijkstra.html new file mode 100644 index 00000000..fccff711 --- /dev/null +++ b/docs/algorithms/shortest-path/dijkstra.html @@ -0,0 +1,24 @@ + + + + + +Dijkstra Shortest Path | Graaf lib + + + + +
+

Dijkstra Shortest Path

Dijkstra's algorithm computes shortest paths between nodes in weighted and unweighted graphs. In weighted graphs, +edge weights should be non-negative. Dijkstra's algorithm is implemented with a priority queue and runs +in O(|E|log|V|) for connected graphs, where |E| is the number of edges and |V| the number of vertices in the +graph.

wikipedia

Syntax​

calculates the shortest path between on start_vertex and one end_vertex using Dijkstra's algorithm. Works on both +weighted as well as unweighted graphs. For unweighted graphs, a unit weight is used for each edge.

template <typename V, typename E, graph_type T, typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>
std::optional<graph_path<WEIGHT_T>> 
dijkstra_shortest_path(const graph<V, E, T>& graph, vertex_id_t start_vertex, vertex_id_t end_vertex);
  • graph The graph to extract shortest path from.
  • start_vertex Vertex id where the shortest path should start.
  • end_vertex Vertex id where the shortest path should end.
  • return An optional with the shortest path (list of vertices) if found.

Find the shortest paths from a source vertex to all other vertices in the graph using Dijkstra's algorithm.

template <typename V, typename E, graph_type T, typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>
[[nodiscard]] std::unordered_map<vertex_id_t, graph_path<WEIGHT_T>>
dijkstra_shortest_paths(const graph<V, E, T>& graph, vertex_id_t source_vertex);
  • graph The graph we want to search.
  • source_vertex The source vertex from which to compute shortest paths.
  • return A map containing the shortest paths from the source vertex to all other vertices. The map keys are target +vertex IDs, and the values are instances of graph_path, representing the shortest distance and the path (list of +vertex IDs) from the source to the target. If a vertex is not reachable from the source, its entry will be absent from +the map.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/shortest-path/floyd-warshall.html b/docs/algorithms/shortest-path/floyd-warshall.html new file mode 100644 index 00000000..0100b50f --- /dev/null +++ b/docs/algorithms/shortest-path/floyd-warshall.html @@ -0,0 +1,24 @@ + + + + + +Floyd-Warshall algorithm | Graaf lib + + + + +
+

Floyd-Warshall algorithm

Floyd-Warshall algorithm computes the shortest path between any two vertices in a graph, both directed and undirected. +The algorithm does not work for graphs with negative weight cycles. +The key idea of the algorithm is to relax the weighted shortest path between any two vertices, using any vertex as an +intermediate one. +Advantage of the algorithm is that it processes vertices instead of edges. This advantage can be used when the number of +edges is large enough, aka a dense graph. +Runtime of the algorithm is O(|V3|) and memory consumption is O(|V2|).

wikipedia

Syntax​

Calculates the shortest path between any two vertices.

template <typename V, typename E, graph_type T,
          typename WEIGHT_T = decltype(get_weight(std::declval<E>()))>
std::vector<std::vector<WEIGHT_T>> floyd_warshall_shortest_paths(
    const graph<V, E, T>& graph);
  • graph The graph to extract the shortest path from.
  • return Returns a 2D vector of the shortest path. If a path doesn't exist between two vertices, mark it as +TYPE_MAX.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/strongly-connected-components/tarjan.html b/docs/algorithms/strongly-connected-components/tarjan.html new file mode 100644 index 00000000..fe9fcd88 --- /dev/null +++ b/docs/algorithms/strongly-connected-components/tarjan.html @@ -0,0 +1,17 @@ + + + + + +Tarjan's Strongly Connected Components | Graaf lib + + + + +
+

Tarjan's Strongly Connected Components

Tarjan's algorithm computes the Strongly Connected Components (SCCs) of a directed graph. An SCC is a subset of vertices in the graph for which every vertex is reachable from every other vertex in the subset, i.e. there exists a path between all pairs of vertices for the subset of vertices.

Tarjan's algorithm runs in O(|V| + |E|) for directed graphs, where |V| the number of vertices and |E| is the number of edges in the graph. So it runs in linear time.

wikipedia

Syntax​

template <typename V, typename E>
[[nodiscard]] std::vector<std::vector<vertex_id_t>> 
tarjans_strongly_connected_components(const graph<V, E, graph_type::DIRECTED>& graph);
  • graph The graph for which to compute SCCs.
  • return A vector of vectors representing SCCs.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/topological-sort.html b/docs/algorithms/topological-sort.html new file mode 100644 index 00000000..45d8b20e --- /dev/null +++ b/docs/algorithms/topological-sort.html @@ -0,0 +1,19 @@ + + + + + +Topological sort algorithm | Graaf lib + + + + +
+

Topological sort algorithm

Topological sort algorithm processing DAG(directed acyclic graph) using DFS traversal. +Each vertex is visited only after all its dependencies are visited. +The runtime of the algorithm is O(|V|+|E|) and the memory consumption is O(|V|).

wikipedia

Syntax​

template <typename V, typename E>
[[nodiscard]] std::optional<std::vector<vertex_id_t>> topological_sort(
    const graph<V, E, graph_type::DIRECTED>& graph);
  • graph The directed graph to traverse.
  • return Vector of vertices sorted in topological order. If the graph contains cycles, it returns std::nullopt.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/traversal/breadth-first-search.html b/docs/algorithms/traversal/breadth-first-search.html new file mode 100644 index 00000000..eb1570db --- /dev/null +++ b/docs/algorithms/traversal/breadth-first-search.html @@ -0,0 +1,17 @@ + + + + + +Breadth First Search (BFS) | Graaf lib + + + + +
+

Breadth First Search (BFS)

Breadth First Search (BFS) Algorithm​

Breadth First Search (BFS) is a fundamental graph traversal algorithm used to explore and analyze graphs, be they directed or undirected. It operates on the principle of visiting nodes in layers, starting from a given source node and gradually expanding outward to neighboring nodes at increasing distances. BFS ensures that all nodes at a particular distance from the source are visited before moving on to nodes at a greater distance. This process continues until all reachable nodes have been visited, forming a breadth-first exploration of the graph.

The BFS algorithm can be succinctly described using the following steps:

  1. Begin by selecting a source node as the starting point of the traversal and enqueue it in a queue data structure.

  2. While the queue is not empty, repeat the following steps:

    • a. Dequeue a node from the front of the queue.
    • b. Process the dequeued node, which may involve examining its attributes, marking it as visited, or performing other relevant operations.
    • c. Enqueue all unvisited neighbors of the dequeued node into the queue.

  3. Continue this process until the queue becomes empty, indicating that all reachable nodes have been visited.

BFS is particularly useful for:​

  • P2P - Find neighbor nodes:

    • Finds all neighbors, and then all neighbors of these neighbors.

  • Search Engine Crawler:

    • Helps in systematically crawling web pages, exploring links layer by layer.

  • Garbage Collection:

    • Identifies and marks reachable objects, propagating to related objects.

  • Broadcasting in Networks:

    • Efficiently distributes information across nodes in a network.

  • Analyzing the Connectivity of Components:

    • Determines the connected components in a graph.

  • Solving Puzzles like the Sliding Tile Puzzle:

    • Explores possible moves in a puzzle in a systematic manner.

Limitations of BFS:​

  1. Memory Usage: BFS may consume significant memory resources, especially in graphs with many nodes or when searching for paths in deep or complex graphs.

  2. Performance on Dense Graphs: In dense graphs, where the number of edges is close to the maximum possible, BFS may not perform as efficiently as other algorithms designed specifically for dense graphs.

  3. Unweighted Graphs: BFS doesn't incorporate edge weights, which makes it less suitable for finding shortest paths in graphs with weighted edges.

  4. No Negative Weights: BFS is not suited for graphs with negative edge weights, as it assumes that all edges have a non-negative weight. This is because BFS relies on the property that it visits nodes in increasing order of distance from the source, and negative weights can lead to unexpected results.

  5. No Guarantee of Optimality: While BFS can find the shortest path in an unweighted graph, it may not guarantee the shortest path in graphs with weighted edges or other more complex scenarios. Dijkstra's algorithm or the Bellman-Ford algorithm are better suited for such cases.

Complexity and Performance:​

The BFS algorithm is implemented with a priority queue and runs in O(|V| + |E|) time complexity for connected graphs, where |E| is the number of edges and |V| the number of vertices in the graph.

In summary, Breadth First Search is a powerful and versatile algorithm for exploring graphs, but its limitations in handling weighted graphs and negative edge weights should be considered. It provides a straightforward way to explore a graph layer by layer and is particularly useful for unweighted graph scenarios and connectivity analysis.

wikipedia

Syntax​

The bfs_termination_strategy returns true when a certain condition is met, causing to terminate. The bfs_edge_callback is a function that is used as a callback during the BFS traversal to perform some action whenever an edge is traversed.

template <
    typename V, typename E, graph_type T,
    typename EDGE_CALLBACK_T = detail::noop_callback,
    typename SEARCH_TERMINATION_STRATEGY_T = detail::exhaustive_search_strategy>
  requires std::invocable<EDGE_CALLBACK_T &, edge_id_t &> &&
           std::is_invocable_r_v<bool, SEARCH_TERMINATION_STRATEGY_T &,
                                 vertex_id_t>
void breadth_first_traverse(
    const graph<V, E, T> &graph, vertex_id_t start_vertex,
    const EDGE_CALLBACK_T &edge_callback,
    const SEARCH_TERMINATION_STRATEGY_T &search_termination_strategy =
        SEARCH_TERMINATION_STRATEGY_T{});

Explanation of Parameters:​

  • graph: The graph to traverse. This parameter represents the graph data structure on which the traversal will be performed.
  • start_vertex: Vertex id where the traversal should be started. This parameter specifies the initial vertex from which the traversal begins.
  • edge_callback: A callback function that is called for each traversed edge. It should be invocable with an edge_id_t object, representing an edge in the graph.
  • search_termination_strategy: A unary predicate that indicates whether the traversal should continue or not. The traversal continues while this predicate returns false. This parameter is optional and defaults to a predefined search termination strategy, which traverses the graph exhaustively.
  • return: The provided code does not explicitly return a value. The traversal is performed by visiting vertices and edges in the graph based on the specified parameters.
+ + + + \ No newline at end of file diff --git a/docs/algorithms/traversal/depth-first-search.html b/docs/algorithms/traversal/depth-first-search.html new file mode 100644 index 00000000..5facbc18 --- /dev/null +++ b/docs/algorithms/traversal/depth-first-search.html @@ -0,0 +1,18 @@ + + + + + +Depth First Search (DFS) | Graaf lib + + + + +
+

Depth First Search (DFS)

Depth First Search (DFS) Algorithm​

Depth First Search (DFS) is a fundamental graph traversal algorithm used to explore and analyze graphs, whether they are directed or undirected. +DFS traverses deeper into the graph before backtracking to explore other branches.

The DFS algorithm can be succinctly described using the following steps:

  1. Begin by selecting a source node as the starting point of the traversal and push it onto a stack data structure.

  2. While the stack is not empty, repeat the following steps:

    • a. Pop a node from the top of the stack.
    • b. Process the popped node, which may involve examining its attributes, marking it as visited, or performing other relevant operations.
    • c. Push all unvisited neighbors of the popped node onto the stack.

  3. Continue this process until the stack becomes empty, indicating that all reachable nodes have been visited.

The main difference to the BFS is the use of a stack instead of a queue.

DFS is particularly useful for:​

  • Topological Sorting:

    • Finds a linear ordering of nodes that respects the partial order imposed by directed edges in a directed acyclic graph.

  • Pathfinding:

    • Can be used to find paths between nodes, although it may not always find the shortest path.

  • Solving Mazes:

    • Navigates through maze-like structures to find a way from a starting point to an end point.

  • Detecting Cycles:

    • Helps identify cycles in a graph, which is valuable for various applications.

Limitations of DFS:​

  1. Completeness: DFS may not explore all nodes in disconnected graphs unless modifications are made to the algorithm.

  2. Infinite Graphs: DFS can get stuck in an infinite loop if applied to graphs with infinite branches.

  3. Performance on Dense Graphs: In dense graphs, DFS might explore many nodes before reaching a solution, making it less efficient compared to other algorithms.

  4. No Guarantee of Optimality: Like BFS, DFS may not always find the optimal solution, especially in cases where the graph has weighted edges or other complexities.

  5. Memory Usage: DFS on deep graphs may lead to excessive recursion and memory consumption due to the call stack.

  6. Biased Exploration: DFS can lead to biased exploration when some branches are deeper than others, potentially missing relevant solutions.

Complexity and Performance:​

The DFS algorithm is implemented with a stack and runs in O(|V| + |E|) time complexity for connected graphs, where |E| is the number of edges and |V| the number of vertices in the graph.

In summary, Depth First Search is a powerful and versatile algorithm for exploring graphs, but its limitations in handling weighted graphs and negative edge weights should be considered. It provides a straightforward way to explore a graph layer by layer and is particularly useful for unweighted graph scenarios and connectivity analysis.

wikipedia

Syntax​

The dfs_termination_strategy returns true when a certain condition is met, causing to terminate. The dfs_edge_callback is a function that is used as a callback during the DFS traversal to perform some action whenever an edge is traversed.

template <
    typename V, typename E, graph_type T,
    typename EDGE_CALLBACK_T = detail::noop_callback,
    typename SEARCH_TERMINATION_STRATEGY_T = detail::exhaustive_search_strategy>
  requires std::invocable<EDGE_CALLBACK_T &, edge_id_t &> &&
           std::is_invocable_r_v<bool, SEARCH_TERMINATION_STRATEGY_T &,
                                 vertex_id_t>
void depth_first_traverse(
    const graph<V, E, T> &graph, vertex_id_t start_vertex,
    const EDGE_CALLBACK_T &edge_callback,
    const SEARCH_TERMINATION_STRATEGY_T &search_termination_strategy =
        SEARCH_TERMINATION_STRATEGY_T{});

Explanation of Parameters:​

  • graph: The graph to traverse. This parameter represents the graph data structure on which the traversal will be performed.
  • start_vertex: Vertex id where the traversal should be started. This parameter specifies the initial vertex from which the traversal begins.
  • edge_callback: A callback function that is called for each traversed edge. It should be invocable with an edge_id_t object, representing an edge in the graph.
  • search_termination_strategy: A unary predicate that indicates whether the traversal should continue or not. The traversal continues while this predicate returns false. This parameter is optional and defaults to a predefined search termination strategy, which traverses the graph exhaustively.
  • return: The provided code does not explicitly return a value. The traversal is performed by visiting vertices and edges in the graph based on the specified parameters.
+ + + + \ No newline at end of file diff --git a/docs/category/cycle-detection-algorithms.html b/docs/category/cycle-detection-algorithms.html new file mode 100644 index 00000000..d3c3a07d --- /dev/null +++ b/docs/category/cycle-detection-algorithms.html @@ -0,0 +1,17 @@ + + + + + +Cycle Detection Algorithms | Graaf lib + + + + +
+

Cycle Detection Algorithms

+ + + + \ No newline at end of file diff --git a/docs/category/graph-coloring-algorithms.html b/docs/category/graph-coloring-algorithms.html new file mode 100644 index 00000000..63dcc173 --- /dev/null +++ b/docs/category/graph-coloring-algorithms.html @@ -0,0 +1,17 @@ + + + + + +Graph Coloring Algorithms | Graaf lib + + + + +
+

Graph Coloring Algorithms

+ + + + \ No newline at end of file diff --git a/docs/category/minimum-spanning-tree.html b/docs/category/minimum-spanning-tree.html new file mode 100644 index 00000000..df0dd19a --- /dev/null +++ b/docs/category/minimum-spanning-tree.html @@ -0,0 +1,17 @@ + + + + + +Minimum Spanning Tree | Graaf lib + + + + +
+

Minimum Spanning Tree

+ + + + \ No newline at end of file diff --git a/docs/category/quickstart---basics.html b/docs/category/quickstart---basics.html new file mode 100644 index 00000000..b4be891e --- /dev/null +++ b/docs/category/quickstart---basics.html @@ -0,0 +1,17 @@ + + + + + +Quickstart - Basics | Graaf lib + + + + +
+

Quickstart - Basics

5 minutes to learn the most important Graaf concepts.

+ + + + \ No newline at end of file diff --git a/docs/category/quickstart---installation.html b/docs/category/quickstart---installation.html new file mode 100644 index 00000000..d9d6c3af --- /dev/null +++ b/docs/category/quickstart---installation.html @@ -0,0 +1,17 @@ + + + + + +Quickstart - Installation | Graaf lib + + + + +
+

Quickstart - Installation

5 minutes to learn the most important Graaf concepts.

+ + + + \ No newline at end of file diff --git a/docs/category/shortest-path-algorithms.html b/docs/category/shortest-path-algorithms.html new file mode 100644 index 00000000..7c527006 --- /dev/null +++ b/docs/category/shortest-path-algorithms.html @@ -0,0 +1,17 @@ + + + + + +Shortest Path Algorithms | Graaf lib + + + + +
+

Shortest Path Algorithms

+ + + + \ No newline at end of file diff --git a/docs/category/strongly-connected-component-algorithms.html b/docs/category/strongly-connected-component-algorithms.html new file mode 100644 index 00000000..767aee05 --- /dev/null +++ b/docs/category/strongly-connected-component-algorithms.html @@ -0,0 +1,17 @@ + + + + + +Strongly Connected Component Algorithms | Graaf lib + + + + +
+

Strongly Connected Component Algorithms

+ + + + \ No newline at end of file diff --git a/docs/category/traversal-algorithms.html b/docs/category/traversal-algorithms.html new file mode 100644 index 00000000..c59a069b --- /dev/null +++ b/docs/category/traversal-algorithms.html @@ -0,0 +1,17 @@ + + + + + +Traversal Algorithms | Graaf lib + + + + +
+

Traversal Algorithms

+ + + + \ No newline at end of file diff --git a/docs/examples/Intro.html b/docs/examples/Intro.html new file mode 100644 index 00000000..07d4dec5 --- /dev/null +++ b/docs/examples/Intro.html @@ -0,0 +1,18 @@ + + + + + +Examples | Graaf lib + + + + +
+

Examples

This section contains example usages of the Graaf library. +If there is a usecase you would like to see an example of, please open an issue in our issue tracker.

+ + + + \ No newline at end of file diff --git a/docs/examples/example-basics/dot-serialization.html b/docs/examples/example-basics/dot-serialization.html new file mode 100644 index 00000000..9819ef90 --- /dev/null +++ b/docs/examples/example-basics/dot-serialization.html @@ -0,0 +1,17 @@ + + + + + +Dot Serialization Example | Graaf lib + + + + +
+

Dot Serialization Example

The to_dot function as defined under graaf::io can be used to searialize graphs to the dot format. This can be handy for debugging purposes, as well as for post-processing of your graphs in another tool which supports the format.

Numeric primitive types​

Default vertex and edge writers are provided such that you can serialize graphs with numeric primitive vertices and edges. For instance:

graaf::undirected_graph<int, float> my_graph{};

// ...

graaf::io::to_dot(my_graph, path):

User defined types​

For user defined vertex and edge types, it is necessary to provide your own vertex and edge writers. These writers should take a vertex or edge as a parameter and serialize it to a string. This resulting string is used in the dot attribute list of the respective vertex or edge.

For example, consider the following user defined vertex and edge types:

struct my_vertex {
  int number{};
  std::string name{};
};

enum class edge_priority { LOW, HIGH };

struct my_edge {
  edge_priority priority{edge_priority::LOW};
  float weight{};
};

We define two lambdas to serialize these vertices and edges. Here we can use any of the graphviz attributes. In this example, we use fmtlib to format our strings.

Vertex writer

const auto vertex_writer{[](graaf::vertex_id_t vertex_id,
                              const my_vertex& vertex) -> std::string {
  const auto color{vertex.number <= 25 ? "lightcyan" : "mediumspringgreen"};
  return fmt::format("label=\"{}: {}\", fillcolor={}, style=filled", vertex_id, vertex.name, color);
}};

Edge writer

const auto edge_writer{[](const graaf::vertex_ids_t& /*edge_id*/,
                            const my_edge& edge) -> std::string {
  const auto style{edge.priority == edge_priority::HIGH ? "solid" : "dashed"};
  return fmt::format("label=\"{}\", style={}, color=gray, fontcolor=gray", edge.weight, style);
}};

Now let's create a directed graph and serialize it to dot:

graaf::directed_graph<my_vertex, my_edge> graph{};

const auto vertex_1{graph.add_vertex({10, "some data"})};
const auto vertex_2{graph.add_vertex({20, "some more data"})};
// ...

graph.add_edge(vertex_1, vertex_2, {edge_priority::HIGH, 3.3});
// ...

const std::filesystem::path dof_file_path{"./my_graph.dot"};
graaf::io::to_dot(my_graph, dof_file_path, vertex_writer, edge_writer);

The contents of my_graph.dot can be processed in any tool which supports dot format. For example, you can use the dot command line tool to generate png images:

dot -Tpng ./my_graph.dot -o my_graph.png

Alternatively, you can use graphviz online for easy visualization:

+ + + + \ No newline at end of file diff --git a/docs/examples/example-basics/shortest-path.html b/docs/examples/example-basics/shortest-path.html new file mode 100644 index 00000000..04059df7 --- /dev/null +++ b/docs/examples/example-basics/shortest-path.html @@ -0,0 +1,19 @@ + + + + + +Shortest Path Example | Graaf lib + + + + +
+

Shortest Path Example

The shortest path algorithm implemented in graaf::algorithm::get_shortest_path can be used to compute the shortest +path between any two vertices in a graph.

Consider the following graph:

In order to compute the shortest path between vertex 0 and vertex 2, we call:

const auto maybe_shortest_path{bfs_shortest_path(graph, start, target)};

// Assert that we found a path at all
assert(maybe_shortest_path.has_value());
auto shortest_path{maybe_shortest_path.value()};

Visualizing the shortest path​

If we want to visualize the shortest path on the graph, we can create our own vertex and edge writers. These writers +then determine the vertex and edge attributes based on whether the vertex or edge is contained in the shortest path.

First, we create a datastructure of all edges on the shortest path such that we can query it in the edge writer:

// We use a set here for O(1) time contains checks
std::unordered_set<graaf::vertex_ids_t, graaf::vertex_ids_hash> edges_on_shortest_path{};

// Convert the list of vertices on the shortest path to edges
graaf::vertex_id_t prev{shortest_path.vertices.front()};
shortest_path.vertices.pop_front();
for (const auto current : shortest_path.vertices) {
  edges_on_shortest_path.insert(std::make_pair(prev, current));
  prev = current;
}

Now we can specify our custom writers:

const auto vertex_writer{
  [start, target](graaf::vertex_id_t vertex_id, int vertex) -> std::string {
    if (vertex_id == start) {
      return "label=start";
    } else if (vertex_id == target) {
      return "label=target";
    }
    return "label=\"\"";
}};

const auto edge_writer{
[&edges_on_shortest_path](const graaf::vertex_ids_t& edge_id, int edge) -> std::string {
  if (edges_on_shortest_path.contains(edge_id)) {
    return "label=\"\", color=red";
  }
  return "label=\"\", color=gray, style=dashed";
}};

This yields us the following visualization:

+ + + + \ No newline at end of file diff --git a/docs/examples/example-basics/transport-example.html b/docs/examples/example-basics/transport-example.html new file mode 100644 index 00000000..ee666268 --- /dev/null +++ b/docs/examples/example-basics/transport-example.html @@ -0,0 +1,21 @@ + + + + + +Network Example | Graaf lib + + + + +
+

Network Example

This example showcases graph traversal and shortest path algorithms in an undirected graph network. As such, it demonstrates the usage of the following classes and algorithms:

  • The undirected_graph implemented in graaf::undirected_graph
  • The shortest path algorithm implemented in graaf::algorithm::get_shortest_path
  • The graph traversal implemented in graaf::algorithm::graph_traversal

Using the following graph:

Custom vertex and edge. In order to use Dijkstra, we should provide the get_weight() function for the edge.

struct station {
  std::string name{};
};

struct railroad : public graaf::weighted_edge<double> {
  double kilometers{};
  [[nodiscard]] double get_weight() const noexcept override {
    return kilometers;
  }
  railroad(double distance) : kilometers(distance) {}
  ~railroad() {}
};

Initializing graph, start and end vertecies

First, we create data structure and initializing graph with vertices and edges

struct graph_with_start_and_target {
    graaf::undirected_graph<station, road> graph{};
    graaf::vertex_id_t start{};
    graaf::vertex_id_t target{};
};

graph_with_start_and_target create_graph_with_start_and_target() {
...
}

Visualizing graph traversal result​

For shortest path, colouring edges with red to indicate the shortest path for both weighted and unweighted graph +We need to specify the start and end vertices in order to find the shortest path between the start and end vertices.

Result of unweighted shortest path, chosen edges are coloured red

Result of weighted shortest path, chosen edges are coloured red

void print_shortest_path(const graaf::undirected_graph<station, road>& graph,
                const std::optional<graaf::algorithm::graph_path<int>>& path, const std::string & filepath) {
...
}

void print_visited_vertices(const graaf::undirected_graph<station, road>& graph,
                            seen_vertices_t& seen,
                            const std::string& filepath) {
...
}

Creating an edge callback structure that will be passed as an argument in the graph traverse function +The function is needed in order to be called inside the traverse function; see graph.tpp for context.

using seen_edges_t = std::unordered_set<graaf::edge_id_t, graaf::edge_id_hash>;
struct record_edges_callback {
  seen_edges_t& seen_edges;

  record_edges_callback(seen_edges_t& seen_edges)
      : seen_edges{seen_edges} {}

  void operator()(const graaf::edge_id_t& edge) const {
    seen_edges.insert(edge);
  }
};

Result of shortest path BFS, visited edges are coloured red

Graph example usage​

First code block: traversing a weighted graph for the shortest path (Dijkstra) and printing the result to .dot file. +Second code block: traversing an unweighted graph for the shortest path and printing the result to .dot file. +The last one is traversing the graph from a given vertex and printing the result to *.dot file.

  const auto [graph, start, target]{create_graph_with_start_and_target()};

  const auto weighted_shortest_path{
      graaf::algorithm::dijkstra_shortest_path(graph, start, target)};
  print_shortest_path(graph, weighted_shortest_path,
                      "example_weighted_graph.dot");

  const auto unweighted_shortest_path{
      graaf::algorithm::bfs_shortest_path(graph, start, target)};
  print_shortest_path(graph, unweighted_shortest_path,
                      "example_unwieghted_graph.dot");

  seen_edges_t seen_edges{};
  graaf::algorithm::breadth_first_traverse(
      graph, start, record_edges_callback{seen_edges});
  print_visited_vertices(graph, seen_edges,
                         "example_traverse_BFS_graph.dot");
+ + + + \ No newline at end of file diff --git a/docs/quickstart/basics/architecture.html b/docs/quickstart/basics/architecture.html new file mode 100644 index 00000000..c44db7ba --- /dev/null +++ b/docs/quickstart/basics/architecture.html @@ -0,0 +1,17 @@ + + + + + +Graaf Architecture | Graaf lib + + + + +
+

Graaf Architecture

From a very high level, the project is structured in two parts:

  • The graph classes and core data structures
  • Algorithms and additional functionality

Graph classes and core data structures​

The main class of the library is the abstract graph class:

enum class edge_type { WEIGHTED, UNWEIGHTED };
enum class graph_spec { DIRECTED, UNDIRECTED };

template <typename VERTEX_T, typename EDGE_T, edge_type EDGE_TYPE_V, graph_spec GRAPH_SPEC_V>
class graph {...};

An instance of a graph can have user provided types for the vertices and edges. Internally, it stores the graph in an adjacency list, and has separate containers for the vertex and edge instances:

// N.B. These types are a bit more abstracted in the codebase behind using
// declarations, but for clarity I have left this out.

// Adjacency information is stored in a set for fast existence checks and fast removal
std::unordered_map<vertex_id_t, std::unordered_set<vertex_id_t>> adjacency_list_{};

// Storing these in a separate container has the advantage that
// vertices and edges are only in memory once
std::unordered_map<vertex_id_t, VERTEX_T> vertices_{};
std::unordered_map<std::pair<vertex_id_t, vertex_id_t>, edge_t, edge_id_hash> edges_{};

The graph class is abstract as it contains pure virtual private methods related to the handling of edges (do_has_edge, do_get_edge, do_add_edge, and do_remove_edge).

Directed and undirected graphs​

There are two classes which publicly derive from graph:

  • directed_graph
  • undirected_graph
template <typename VERTEX_T, typename EDGE_T, edge_type EDGE_TYPE_V = edge_type::UNWEIGHTED>
class directed_graph final
    : public graph<VERTEX_T, EDGE_T, EDGE_TYPE_V, graph_spec::DIRECTED> 
    {...};

template <typename VERTEX_T, typename EDGE_T, edge_type EDGE_TYPE_V = edge_type::UNWEIGHTED>
class undirected_graph final
    : public graph<VERTEX_T, EDGE_T, EDGE_TYPE_V, graph_spec::UNDIRECTED> 
    {...};

These are the classes which the user instantiates.

They provide implementations for the pure virtual methods related to handling edges. The unweighted_graph first sorts the pair of vertex ids related to an edge before interacting with the internal edges_ data structure. This ensures that an edge a->b is the same as an edge from b->a.

Weighted graphs​

Certain algorithms (such as A*) operate on weighted graphs. A graph is automatically weighted if a primitive numeric type is passed as a template parameter to EDGE_T. Alternatively, user provided edge classes can publicly derive from weighted_edge.

The weighted_edge class provides a default implementation for the get_weight method, but this can be overridden in the derived class:

template <typename WEIGHT_T = int>
class weighted_edge {
 public:
  using weight_t = WEIGHT_T;
  /**
   * By default an edge has a unit weight.
   */
  [[nodiscard]] virtual WEIGHT_T get_weight() const noexcept { return 1; };
};

To create an unweighted graph, simply do not derive from weighted_edge in your edge class.

Algorithms and additional functionality​

The idea here is to keep the graph classes as general-purpose as possible, and to not include use case specific logic (such as dot serialization) as member functions. Therefore, each algorithm/utility function is implemented as a free function.

+ + + + \ No newline at end of file diff --git a/docs/quickstart/basics/creating-your-first-graph.html b/docs/quickstart/basics/creating-your-first-graph.html new file mode 100644 index 00000000..eead2925 --- /dev/null +++ b/docs/quickstart/basics/creating-your-first-graph.html @@ -0,0 +1,17 @@ + + + + + +Creating Your First Graph | Graaf lib + + + + +
+
+ + + + \ No newline at end of file diff --git a/docs/quickstart/basics/using-algorithms.html b/docs/quickstart/basics/using-algorithms.html new file mode 100644 index 00000000..934895ba --- /dev/null +++ b/docs/quickstart/basics/using-algorithms.html @@ -0,0 +1,17 @@ + + + + + +Using Algorithms | Graaf lib + + + + +
+
+ + + + \ No newline at end of file diff --git a/docs/quickstart/installation.html b/docs/quickstart/installation.html new file mode 100644 index 00000000..d00dc944 --- /dev/null +++ b/docs/quickstart/installation.html @@ -0,0 +1,17 @@ + + + + + +Installation | Graaf lib + + + + +
+

Installation

Graaf Header-Only Installation​

Installing Graaf on your project is easy! Simply copy the graaflib directory to your project and add it to your include path.

  1. Copy graaflip to your project.

  2. Before compiling, add the directory to your include path.

    # For C compiler
    export C_INCLUDE_PATH="/full/path/to/include/graaflib/:$C_INCLUDE_PATH"
    # For Cpp compiler
    export CPLUS_INCLUDE_PATH="/full/path/to/include/graaflib/:$CPLUS_INCLUDE_PATH"

    Or in CMake:

    include_directories("graaf/src/graaflib")

  3. Include the graaf header in your sources.

    #include <graaflib/directed_graph.h>

CMake FetchContent​

Alternatively, this project can be pulled in using CMake's FetchContent:

include(FetchContent)
FetchContent_Declare(
  graaflib
  GIT_REPOSITORY https://github.com/bobluppes/graaf.git
  GIT_TAG        main
)
FetchContent_MakeAvailable(graaflib)

Now you can link your target against Graaf_lib:

target_link_libraries(${PROJECT_NAME} PRIVATE Graaf_lib)
+ + + + \ No newline at end of file diff --git a/docs/quickstart/installation/alternative-methods.html b/docs/quickstart/installation/alternative-methods.html new file mode 100644 index 00000000..2d4035f4 --- /dev/null +++ b/docs/quickstart/installation/alternative-methods.html @@ -0,0 +1,17 @@ + + + + + +Alternative Installation Methods | Graaf lib + + + + +
+
+ + + + \ No newline at end of file diff --git a/docs/quickstart/intro.html b/docs/quickstart/intro.html new file mode 100644 index 00000000..6ed1aeab --- /dev/null +++ b/docs/quickstart/intro.html @@ -0,0 +1,17 @@ + + + + + +Quickstart Intro | Graaf lib + + + + +
+

Quickstart Intro

Let's install Graaf in your project...

Getting Started​

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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/img/undraw_docusaurus_react.svg b/img/undraw_docusaurus_react.svg new file mode 100644 index 00000000..94b5cf08 --- /dev/null +++ b/img/undraw_docusaurus_react.svg @@ -0,0 +1,170 @@ + + Powered by React + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/img/undraw_docusaurus_tree.svg b/img/undraw_docusaurus_tree.svg new file mode 100644 index 00000000..d9161d33 --- /dev/null +++ b/img/undraw_docusaurus_tree.svg @@ -0,0 +1,40 @@ + + Focus on What Matters + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/index.html b/index.html new file mode 100644 index 00000000..3f875ca9 --- /dev/null +++ b/index.html @@ -0,0 +1,17 @@ + + + + + +Hello from Graaf lib | Graaf lib + + + + +

+

Graaf lib

A general-purpose lightweight graph library implemented in C++

Graaf Quickstart - 5min ⏱️
Easy to Use

Easy to Use

Graaf is designed as a lightweight alternative for Boost Graph. The library is created to be easy to use right from the start.

Focus on What Matters

General-Purpose

Graphs can wrap arbitrary types, i.e. graaf::directed_graph<MyVertexClass, MyEdgeClass>

Powered by React

Lightning Fast

Graaf is written in C++ with performance in mind. This allows users to efficiently perform complex algorithms on large graphs.

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