{"id":3405,"date":"2021-07-06T15:43:57","date_gmt":"2021-07-06T10:13:57","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=3405"},"modified":"2024-07-30T11:27:37","modified_gmt":"2024-07-30T05:57:37","slug":"what-is-single-source-shortest-path","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-single-source-shortest-path\/","title":{"rendered":"What is Single Source Shortest Path?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Features of Single Source Shortest Path<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Single Source Shortest Path is Weighted graph directed.<\/li>\n\n\n\n<li>Edges may have adverse costs.<\/li>\n\n\n\n<li>No loop whose price is &lt; 0.0.<\/li>\n\n\n\n<li>Find the shortest path to each of the n vertices of the digraph from a given source vertex.<\/li>\n\n\n\n<li>Where there are negative-cost edges, Dijkstra&#8217;s O(n2) single-source greedy algorithm does not work.<\/li>\n\n\n\n<li>The Bellman-Ford algorithm finds the bottom-up gap. It first finds certain distances in the route that have only one edge. Increase the length of the route after that to find all possible solutions.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Single Source <strong>Shortest Path Problem<\/strong><\/h2>\n\n\n\n<p>Given a non-negative linked directed graph G with a non-negative graph Edge weights and root vertex r, find a directed path P(x) from r to x for each vertex x, such that the sum of the edge weights in path P(x) is as small as possible.&nbsp;<\/p>\n\n\n\n<p>In 1959, by the Dutch computer scientist Edsger Dijkstra.<\/p>\n\n\n\n<p>Solves a graph with non-negative edge weights for the single-source shortest path problem.<\/p>\n\n\n\n<p>In routing, this algorithm is also used.<\/p>\n\n\n\n<p>E.g.: The algorithm of Dijkstra is generally the working theory behind the link-state.&nbsp; Protocols of Routing<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"dijkstra&#039;s shortest path algorithm\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/pN2IjiXEQeY?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Bellman-ford Algorithm<\/strong> <\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>All-destinations of single-source shortest paths in<\/li>\n\n\n\n<li>Digraphs with cost-negative edges.<\/li>\n\n\n\n<li>Dynamic programming is used.<\/li>\n\n\n\n<li>Runs when adjacency matrices are used in O(n3) time.<\/li>\n\n\n\n<li>Runs in O(ne) time while using adjacency lists.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Algorithm<\/strong><\/h2>\n\n\n\n<p>bellmanFord(dist, pred, source)<\/p>\n\n\n\n<p>Input \u2212 Distance list, predecessor list and the source vertex.<\/p>\n\n\n\n<p>Output<strong> <\/strong>\u2212 True, when a negative cycle is found.<\/p>\n\n\n\n<p>Begin<\/p>\n\n\n\n<p>iCount := 1<\/p>\n\n\n\n<p>\u00a0maxEdge := n * (n &#8211; 1) \/ 2 \/\/n is the number of vertices<\/p>\n\n\n\n<p>for all vertices v of the graph, do<\/p>\n\n\n\n<p>dist[v] := \u221e<\/p>\n\n\n\n<p>pred[v] := \u03d5<\/p>\n\n\n\n<p>done<\/p>\n\n\n\n<p>dist[source] := 0<\/p>\n\n\n\n<p>\u00a0eCount := number of edges present in the graph<\/p>\n\n\n\n<p>create edge list named edgeList<\/p>\n\n\n\n<p>while iCount &lt; n, do<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;for i := 0 to eCount, do<\/p>\n\n\n\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0if dist[edgeList[i].v] > dist[edgeList[i].u] + (cost[u,v] for edge i)<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;dist[edgeList[i].v] &gt; dist[edgeList[i].u] + (cost[u,v] for edge i)<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;pred[edgeList[i].v] := edgeList[i].u<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;done<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;done<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;iCount := iCount + 1<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;for all vertices i in the graph, do<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if dist[edgeList[i].v] &gt; dist[edgeList[i].u] + (cost[u,v] for edge i), then<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;return true<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;done<\/p>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;return false<\/p>\n\n\n\n<p>End<\/p>\n\n\n\n<p>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<br><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-machine-learning\" target=\"_blank\" rel=\"noreferrer noopener\">What is machine learning?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-data-structure\" target=\"_blank\" rel=\"noreferrer noopener\">Define data structure?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-recursion\" target=\"_blank\" rel=\"noreferrer noopener\">What is recursion?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/index.php\/2021\/05\/31\/what-is-sql\/\" target=\"_blank\" rel=\"noreferrer noopener\">Explain SQL?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/index.php\/2021\/05\/31\/what-is-a-compiler\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is compilers?<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Single Source Shortest Path is the shortest path to each of the n vertices of the digraph from a given source vertex<\/p>\n","protected":false},"author":3,"featured_media":2857,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[36],"tags":[],"class_list":["post-3405","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-computer-science-software-engineering-blog"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is Single Source Shortest Path? 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