{"id":393,"date":"2025-06-26T22:19:46","date_gmt":"2025-06-26T16:49:46","guid":{"rendered":"https:\/\/codeanddebug.in\/blog\/?p=393"},"modified":"2025-06-26T22:23:53","modified_gmt":"2025-06-26T16:53:53","slug":"number-of-ways-to-arrive-at-destination-python","status":"publish","type":"post","link":"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/","title":{"rendered":"Number of Ways to Arrive at Destination | Leetcode 1976 | Simple Dijkstra + Path-Counting Guide in Python"},"content":{"rendered":"\n<p>Learn how to solve <em>Number of Ways to Arrive at Destination<\/em> with Dijkstra\u2019s algorithm while counting paths. Clear intuition, commented Python code, dry run, and Big-O analysis.<\/p>\n\n\n\n<p>Here&#8217;s the [<strong><a href=\"https:\/\/leetcode.com\/problems\/number-of-ways-to-arrive-at-destination\/description\/\" target=\"_blank\" rel=\"noreferrer noopener\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><span style=\"text-decoration: underline;\">Problem Link<\/span><\/mark><\/a><\/strong>] to begin with.<\/p>\n\n\n<div style=\"max-width: -moz-fit-content; \" class=\"wp-block-ub-table-of-contents-block ub_table-of-contents ub_table-of-contents-collapsed\" id=\"ub_table-of-contents-d1af3a59-396b-46de-94b2-4063074a6cf2\" data-linktodivider=\"false\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" data-enablesmoothscroll=\"true\" data-initiallyhideonmobile=\"false\" data-initiallyshow=\"false\"><div class=\"ub_table-of-contents-header-container\" style=\"\">\n\t\t\t<div class=\"ub_table-of-contents-header\" style=\"text-align: left; \">\n\t\t\t\t<div class=\"ub_table-of-contents-title\">Contents:<\/div>\n\t\t\t\t<div class=\"ub_table-of-contents-header-toggle\">\n\t\t\t<div class=\"ub_table-of-contents-toggle\" style=\"\">\n\t\t\t\u00a0[<a class=\"ub_table-of-contents-toggle-link\" href=\"#\" style=\"\">show<\/a>]\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div><div class=\"ub_table-of-contents-extra-container\" style=\"\">\n\t\t\t<div class=\"ub_table-of-contents-container ub_table-of-contents-1-column ub-hide\">\n\t\t\t\t<ul style=\"\"><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#0-1-what-does-the-problem-ask\" style=\"\">1. What does the problem ask?<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#1-2-example\" style=\"\">2. Example<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#2-3-intuition-amp-approach\" style=\"\">3. Intuition &amp; approach<\/a><ul><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#3-31-dijkstra-finds-the-time\" style=\"\">3.1 Dijkstra finds the time<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#4-32-counting-paths-on-the-fly\" style=\"\">3.2 Counting paths on the fly<\/a><\/li><\/ul><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#5-4%E2%80%82python-code\" style=\"\">4\u2002Python code<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#6-5-step-by-step-explanation\" style=\"\">5. Step-by-step explanation<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#7-6%E2%80%82dry-run-mini-graph\" style=\"\">6\u2002Dry run (mini graph)<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#8-7-complexity\" style=\"\">7. Complexity<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/number-of-ways-to-arrive-at-destination-python\/#9-8-conclusion\" style=\"\">8. Conclusion<\/a><\/li><\/ul>\n\t\t\t<\/div>\n\t\t<\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-1-what-does-the-problem-ask\">1. What does the problem ask?<\/h2>\n\n\n\n<p>You are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>n<\/code> cities numbered <code>0 \u2026 n-1<\/code>,<\/li>\n\n\n\n<li>a list of bidirectional <strong>roads<\/strong> <code>u \u2194 v<\/code> with <strong>travel time<\/strong> <code>w<\/code>.<\/li>\n<\/ul>\n\n\n\n<p>Start at city <code>0<\/code> and finish at city <code>n \u2212 1<\/code>.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Find the <strong>shortest possible travel time<\/strong>.<\/li>\n\n\n\n<li>Return <strong>how many different shortest routes<\/strong> exist, <strong>mod 1 000 000 007<\/strong>.<br>(Two routes are different if the sequence of visited cities differs at any step.)<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1-2-example\">2. Example<\/h2>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"n = 7\nroads = [\n  [0,6,7],[0,1,2],[1,2,3],[1,3,3],\n  [6,3,3],[3,5,1],[6,5,1],[2,5,1],[0,4,5],[4,6,2]\n]\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #D4D4D4\">n = <\/span><span style=\"color: #B5CEA8\">7<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">roads = [<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">  [<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">6<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">7<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">2<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">2<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">3<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">3<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">3<\/span><span style=\"color: #D4D4D4\">],<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">  [<\/span><span style=\"color: #B5CEA8\">6<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">3<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">3<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">3<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">5<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">6<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">5<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">2<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">5<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">4<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">5<\/span><span style=\"color: #D4D4D4\">],[<\/span><span style=\"color: #B5CEA8\">4<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">6<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #B5CEA8\">2<\/span><span style=\"color: #D4D4D4\">]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">]<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p><em>Shortest time<\/em>: <strong>7<\/strong><br><em>Number of different shortest paths<\/em>: <strong>6<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-3-intuition-amp-approach\">3. Intuition &amp; approach<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"3-31-dijkstra-finds-the-time\">3.1 Dijkstra finds the time<\/h3>\n\n\n\n<p>Classic Dijkstra\u2019s algorithm gives the minimum travel time from city <code>0<\/code> to every other city because all edge weights (<code>w<\/code>) are non-negative.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"4-32-counting-paths-on-the-fly\">3.2 Counting paths on the fly<\/h3>\n\n\n\n<p>While Dijkstra explores the graph we also maintain:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><code>ways[i]<\/code> \u2013 <strong>how many shortest paths<\/strong> reach city <code>i<\/code>.<\/li>\n<\/ul>\n\n\n\n<p>Rules when we process an edge <code>node \u2192 adjNode<\/code> with weight <code>weight<\/code>:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Case<\/th><th>Action<\/th><\/tr><\/thead><tbody><tr><td><code>new_dist &lt; distance[adjNode]<\/code><\/td><td>Found <strong>shorter<\/strong> route \u2192 update <code>distance[adjNode] = new_dist<\/code> and <strong>reset<\/strong> <code>ways[adjNode] = ways[node]<\/code>.<\/td><\/tr><tr><td><code>new_dist == distance[adjNode]<\/code><\/td><td>Found <strong>another<\/strong> route with the <em>same<\/em> shortest time \u2192 increment: <code>ways[adjNode] = (ways[adjNode] + ways[node]) mod MOD<\/code>.<\/td><\/tr><tr><td><code>new_dist &gt; distance[adjNode]<\/code><\/td><td>Route is longer \u2013 ignore.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Because Dijkstra processes nodes by increasing distance, when we pop a node its recorded distance is already the <strong>true shortest<\/strong>, so the counting is safe.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-4%E2%80%82python-code\">4. Python code<\/h2>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"import sys\nimport heapq\nfrom typing import List\n\n\nclass Solution:\n    def countPaths(self, n: int, roads: List[List[int]]) -&gt; int:\n        MOD = 10**9 + 7\n\n        # 1\ufe0f\u20e3 build undirected adjacency list\n        adj_list = [[] for _ in range(n)]\n        for u, v, w in roads:\n            adj_list[u].append([v, w])\n            adj_list[v].append([u, w])\n\n        # 2\ufe0f\u20e3 distance &amp; ways arrays\n        distance = [sys.maxsize for _ in range(n)]\n        ways     = [0 for _ in range(n)]\n        distance[0] = 0        # start city\n        ways[0]     = 1        # one trivial path to itself\n\n        # 3\ufe0f\u20e3 min-heap for Dijkstra (dist, node)\n        priority_queue = [[0, 0]]\n\n        while priority_queue:\n            dist, node = heapq.heappop(priority_queue)\n\n            # Skip stale entries\n            if dist != distance[node]:\n                continue\n\n            # 4\ufe0f\u20e3 relax edges\n            for adjNode, weight in adj_list[node]:\n                new_dist = dist + weight\n\n                if new_dist &lt; distance[adjNode]:\n                    # found a strictly better time\n                    distance[adjNode] = new_dist\n                    heapq.heappush(priority_queue, [new_dist, adjNode])\n                    ways[adjNode] = ways[node]              # reset count\n\n                elif new_dist == distance[adjNode]:\n                    # found another shortest route\n                    ways[adjNode] = (ways[adjNode] + ways[node]) % MOD\n\n        return ways[n - 1] % MOD\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #C586C0\">import<\/span><span style=\"color: #D4D4D4\"> sys<\/span><\/span>\n<span class=\"line\"><span style=\"color: #C586C0\">import<\/span><span style=\"color: #D4D4D4\"> heapq<\/span><\/span>\n<span class=\"line\"><span style=\"color: #C586C0\">from<\/span><span style=\"color: #D4D4D4\"> typing <\/span><span style=\"color: #C586C0\">import<\/span><span style=\"color: #D4D4D4\"> List<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #569CD6\">class<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #4EC9B0\">Solution<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #569CD6\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">countPaths<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">roads<\/span><span style=\"color: #D4D4D4\">: List[List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]]) -&gt; <\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        MOD = <\/span><span style=\"color: #B5CEA8\">10<\/span><span style=\"color: #D4D4D4\">**<\/span><span style=\"color: #B5CEA8\">9<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #B5CEA8\">7<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># 1\ufe0f\u20e3 build undirected adjacency list<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        adj_list = [[] <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(n)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> u, v, w <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> roads:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            adj_list[u].append([v, w])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            adj_list[v].append([u, w])<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># 2\ufe0f\u20e3 distance &amp; ways arrays<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        distance = [sys.maxsize <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(n)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        ways     = [<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> _ <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(n)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        distance[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">] = <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># start city<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        ways[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">]     = <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># one trivial path to itself<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># 3\ufe0f\u20e3 min-heap for Dijkstra (dist, node)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        priority_queue = [[<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">]]<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">while<\/span><span style=\"color: #D4D4D4\"> priority_queue:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dist, node = heapq.heappop(priority_queue)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #6A9955\"># Skip stale entries<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> dist != distance[node]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">continue<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #6A9955\"># 4\ufe0f\u20e3 relax edges<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> adjNode, weight <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> adj_list[node]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                new_dist = dist + weight<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> new_dist &lt; distance[adjNode]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    <\/span><span style=\"color: #6A9955\"># found a strictly better time<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    distance[adjNode] = new_dist<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    heapq.heappush(priority_queue, [new_dist, adjNode])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    ways[adjNode] = ways[node]              <\/span><span style=\"color: #6A9955\"># reset count<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">elif<\/span><span style=\"color: #D4D4D4\"> new_dist == distance[adjNode]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    <\/span><span style=\"color: #6A9955\"># found another shortest route<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    ways[adjNode] = (ways[adjNode] + ways[node]) % MOD<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> ways[n - <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">] % MOD<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-5-step-by-step-explanation\">5. Step-by-step explanation<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Adjacency list<\/strong> \u2013 every road stored twice since travel is two-way.<\/li>\n\n\n\n<li><strong>Arrays<\/strong>\n<ul class=\"wp-block-list\">\n<li><code>distance[i]<\/code> = best travel time seen so far to city <code>i<\/code>.<\/li>\n\n\n\n<li><code>ways[i]<\/code> = number of shortest-time routes to city <code>i<\/code>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Heap pop<\/strong> \u2013 always gives us the city with the smallest current time.<\/li>\n\n\n\n<li><strong>Edge relaxation<\/strong> \u2013 update both <code>distance<\/code> and <code>ways<\/code> according to the table above.<\/li>\n\n\n\n<li><strong>Answer<\/strong> \u2013 after heap empties, <code>ways[n-1]<\/code> is the required count.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-6%E2%80%82dry-run-mini-graph\">6. Dry run (mini graph)<\/h2>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"0 --1--&gt; 1\n0 --1--&gt; 2\n1 --1--&gt; 3\n2 --1--&gt; 3\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #D4D4D4\">&gt; <\/span><span style=\"color: #B5CEA8\">1<\/span><\/span>\n<span class=\"line\"><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #D4D4D4\">&gt; <\/span><span style=\"color: #B5CEA8\">2<\/span><\/span>\n<span class=\"line\"><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #D4D4D4\">&gt; <\/span><span style=\"color: #B5CEA8\">3<\/span><\/span>\n<span class=\"line\"><span style=\"color: #B5CEA8\">2<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #F44747\">--<\/span><span style=\"color: #D4D4D4\">&gt; <\/span><span style=\"color: #B5CEA8\">3<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Pop <code>(dist,node)<\/code><\/th><th>Update<\/th><th><code>distance<\/code><\/th><th><code>ways<\/code><\/th><\/tr><\/thead><tbody><tr><td>(0,0)<\/td><td>0\u21921: dist=1, ways=1<br>0\u21922: dist=1, ways=1<\/td><td>[0,1,1,\u221e]<\/td><td>[1,1,1,0]<\/td><\/tr><tr><td>(1,1)<\/td><td>1\u21923: dist=2, ways=1<\/td><td>[0,1,1,2]<\/td><td>[1,1,1,1]<\/td><\/tr><tr><td>(1,2)<\/td><td>2\u21923: new_dist=2 == 2 \u2192 ways[3]+=ways[2]=1 \u2192 ways[3]=2<\/td><td>\u2013<\/td><td>[1,1,1,2]<\/td><\/tr><tr><td>(2,3)<\/td><td>goal popped<\/td><td>\u2013<\/td><td>ways[3]=2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Two different shortest routes: <code>0-1-3<\/code> and <code>0-2-3<\/code>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-7-complexity\">7. Complexity<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Metric<\/th><th>Big-O<\/th><th>Why<\/th><\/tr><\/thead><tbody><tr><td><strong>Time<\/strong><\/td><td><code>O((V + E) log V)<\/code><\/td><td>Standard heap-based Dijkstra.<\/td><\/tr><tr><td><strong>Space<\/strong><\/td><td><code>O(V + E)<\/code><\/td><td>Adjacency list + arrays + heap.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><code>V = n<\/code>, <code>E = len(roads)<\/code>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-8-conclusion\">8. Conclusion<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Using Dijkstra while <strong>carrying a <code>ways<\/code> counter<\/strong> lets us solve <strong>Number of Ways to Arrive at Destination<\/strong> in one pass.<\/li>\n\n\n\n<li>Update rules: <strong>reset<\/strong> on a better distance, <strong>add<\/strong> on an equal distance.<\/li>\n\n\n\n<li>Always take results <strong>mod 1 000 000 007<\/strong> to prevent overflow.<\/li>\n<\/ul>\n\n\n\n<p>With these steps you can confidently compute both the fastest time and the exact count of fastest routes between two cities.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-container-core-buttons-is-layout-16018d1d wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"https:\/\/codeanddebug.in\/course\/zero-to-hero-python-dsa\" target=\"_blank\" rel=\"noreferrer noopener\">Join our Advance DSA COURSE<\/a><\/div>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><em>For any changes to the article, kindly email at <a href=\"mailto:code@codeanddebug.in\" target=\"_blank\" rel=\"noreferrer noopener\">code@codeanddebug.in<\/a> or contact us at <a href=\"tel:+91-9712928220\" target=\"_blank\" rel=\"noreferrer noopener\">+91-9712928220<\/a>.<\/em><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Learn how to solve Number of Ways to Arrive at Destination with Dijkstra\u2019s algorithm while counting paths. Clear intuition, commented Python code, dry run, and Big-O analysis. Here&#8217;s the [Problem Link] to begin with. 1. What does the problem ask? You are given: Start at city 0 and finish at city n \u2212 1. 2.<\/p>\n","protected":false},"author":1,"featured_media":394,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,5],"tags":[24,17,18],"class_list":{"0":"post-393","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-structures-and-algorithm","8":"category-expert","9":"tag-dijkstra-algorithm","10":"tag-graph","11":"tag-hard"},"featured_image_src":"https:\/\/codeanddebug.in\/blog\/wp-content\/uploads\/2025\/06\/number-of-ways-to-arrive-at-destination-featured-image.png","author_info":{"display_name":"codeanddebug","author_link":"https:\/\/codeanddebug.in\/blog\/author\/codeanddebug\/"},"_links":{"self":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/393","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/comments?post=393"}],"version-history":[{"count":4,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/393\/revisions"}],"predecessor-version":[{"id":401,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/393\/revisions\/401"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/media\/394"}],"wp:attachment":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/media?parent=393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/categories?post=393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/tags?post=393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}