{"id":408,"date":"2025-06-26T22:53:49","date_gmt":"2025-06-26T17:23:49","guid":{"rendered":"https:\/\/codeanddebug.in\/blog\/?p=408"},"modified":"2025-06-26T22:53:51","modified_gmt":"2025-06-26T17:23:51","slug":"find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance","status":"publish","type":"post","link":"https:\/\/codeanddebug.in\/blog\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/","title":{"rendered":"Find the City With the Smallest Number of Neighbors at a Threshold Distance | Floyd Warshall vs. Dijkstra"},"content":{"rendered":"\n<p>Learn to solve \u201cFind the City With the Smallest Number of Neighbors at a Threshold Distance\u201d using Floyd Warshall or a Dijkstra-per-source loop. Simple Python, step-by-step.<\/p>\n\n\n\n<p>Here&#8217;s the [<strong><a href=\"https:\/\/leetcode.com\/problems\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/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-2701668c-a5cf-4e00-aaa5-dd77eb06ba21\" 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\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/#0-1-problem-recap\" style=\"\">1. Problem recap<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/#1-2-intuition\" style=\"\">2. Intuition<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/#2-3-solution-a-%E2%80%93-floyd-warshall-%E2%9C%85\" style=\"\">3. Solution A \u2013 Floyd Warshall \u2705<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/#3-4-solution-b-%E2%80%93-dijkstra-from-every-city\" style=\"\">4. Solution B \u2013 Dijkstra from every city<\/a><\/li><li style=\"\"><a href=\"https:\/\/codeanddebug.in\/blog\/find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\/#4-5-which-one-should-i-use\" style=\"\">5. Which one should I use?<\/a><\/li><\/ul>\n\t\t\t<\/div>\n\t\t<\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-1-problem-recap\">1. Problem recap<\/h2>\n\n\n\n<p>Given<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><code>n<\/code><\/strong> cities numbered <code>0 \u2026 n-1<\/code>,<\/li>\n\n\n\n<li>an undirected weighted edge list <code>edges = [u, v, w]<\/code>,<\/li>\n\n\n\n<li>a <strong>distanceThreshold<\/strong>,<\/li>\n<\/ul>\n\n\n\n<p>compute for every city how many other cities can be reached with total travel distance \u2264 <code>distanceThreshold<\/code>.<br>Return the city that has the <strong>fewest such neighbors<\/strong>; if several tie, return the city with the <strong>largest index<\/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=\"1-2-intuition\">2. Intuition<\/h2>\n\n\n\n<p>This is an <strong>all-pairs shortest-path (APSP)<\/strong> task followed by a simple count:<\/p>\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=\"shortest(i, j)  \u2264  threshold  ?  reachable : not\" 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\">shortest(i, j)  \u2264  threshold  <\/span><span style=\"color: #F44747\">?<\/span><span style=\"color: #D4D4D4\">  reachable : <\/span><span style=\"color: #569CD6\">not<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>Two classic APSP strategies:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Method<\/th><th>Works with<\/th><th>Time<\/th><th>Space<\/th><th>When to prefer<\/th><\/tr><\/thead><tbody><tr><td><strong>Floyd Warshall<\/strong><\/td><td>dense graphs, <code>n<\/code> \u2272 400<\/td><td><code>O(n\u00b3)<\/code><\/td><td><code>O(n\u00b2)<\/code><\/td><td>small\/medium <code>n<\/code><\/td><\/tr><tr><td><strong>Run Dijkstra from every city<\/strong><\/td><td>non-negative weights<\/td><td><code>O(n\u00b7E log n)<\/code><\/td><td><code>O(n+E)<\/code><\/td><td>sparse or large graphs<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Below you\u2019ll find both fully-commented solutions.<\/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-solution-a-%E2%80%93-floyd-warshall-%E2%9C%85\">3. Solution A \u2013 Floyd Warshall<\/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\nfrom typing import List\n\nclass Solution:\n    def findTheCity(\n        self, n: int, edges: List[List[int]], distanceThreshold: int\n    ) -&gt; int:\n        INF = sys.maxsize\n        # build adjacency matrix\n        dist = [[INF] * n for _ in range(n)]\n        for u, v, w in edges:\n            dist[u][v] = w\n            dist[v][u] = w\n        for i in range(n):\n            dist[i][i] = 0\n\n        # Floyd-Warshall triple loop\n        for via in range(n):\n            for i in range(n):\n                for j in range(n):\n                    if dist[i][via] != INF and dist[via][j] != INF:\n                        dist[i][j] = min(dist[i][j], dist[i][via] + dist[via][j])\n\n        # count reachable neighbors\n        min_cnt, city = n + 1, -1\n        for i in range(n):\n            cnt = sum(dist[i][j] &lt;= distanceThreshold for j in range(n))\n            # \u2265 keeps the larger index on ties\n            if cnt &lt;= min_cnt:\n                min_cnt, city = cnt, i\n        return city\" 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\">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 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\">findTheCity<\/span><span style=\"color: #D4D4D4\">(<\/span><\/span>\n<span class=\"line\"><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\">edges<\/span><span style=\"color: #D4D4D4\">: List[List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]], <\/span><span style=\"color: #9CDCFE\">distanceThreshold<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #4EC9B0\">int<\/span><\/span>\n<span class=\"line\"><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\">        INF = sys.maxsize<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># build adjacency matrix<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        dist = [[INF] * n <\/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\"> edges:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dist[u][v] = w<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dist[v][u] = w<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> i <\/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\">            dist[i][i] = <\/span><span style=\"color: #B5CEA8\">0<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># Floyd-Warshall triple loop<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> via <\/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\"> i <\/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\"> j <\/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\">if<\/span><span style=\"color: #D4D4D4\"> dist[i][via] != INF <\/span><span style=\"color: #569CD6\">and<\/span><span style=\"color: #D4D4D4\"> dist[via][j] != INF:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                        dist[i][j] = <\/span><span style=\"color: #DCDCAA\">min<\/span><span style=\"color: #D4D4D4\">(dist[i][j], dist[i][via] + dist[via][j])<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #6A9955\"># count reachable neighbors<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        min_cnt, city = n + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, -<\/span><span style=\"color: #B5CEA8\">1<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> i <\/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\">            cnt = <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">(dist[i][j] &lt;= distanceThreshold <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> j <\/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: #6A9955\"># \u2265 keeps the larger index on ties<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> cnt &lt;= min_cnt:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                min_cnt, city = cnt, i<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> city<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p><em>Complexity<\/em>\u2003<code>O(n\u00b3)<\/code> time | <code>O(n\u00b2)<\/code> memory.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-4-solution-b-%E2%80%93-dijkstra-from-every-city\">4. Solution B \u2013 Dijkstra from every city<\/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, heapq\nfrom typing import List\n\nclass Solution:\n    def findTheCity(self, n: int, edges: List[List[int]], distanceThreshold: int) -&gt; int:\n        # adjacency list\n        adj = [[] for _ in range(n)]\n        for u, v, w in edges:\n            adj[u].append((v, w))\n            adj[v].append((u, w))\n\n        INF = sys.maxsize\n        def dijkstra(src: int) -&gt; List[int]:\n            dist = [INF] * n\n            dist[src] = 0\n            pq = [(0, src)]\n            while pq:\n                d, u = heapq.heappop(pq)\n                if d &gt; dist[u]:\n                    continue\n                for v, w in adj[u]:\n                    nd = d + w\n                    if nd &lt; dist[v]:\n                        dist[v] = nd\n                        heapq.heappush(pq, (nd, v))\n            return dist\n\n        min_cnt, city = n + 1, -1\n        for i in range(n):\n            row = dijkstra(i)\n            cnt = sum(d &lt;= distanceThreshold for d in row)\n            if cnt &lt;= min_cnt:            # \u201c&lt;=\u201d keeps larger index on tie\n                min_cnt, city = cnt, i\n        return city\" 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, 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 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\">findTheCity<\/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\">edges<\/span><span style=\"color: #D4D4D4\">: List[List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]], <\/span><span style=\"color: #9CDCFE\">distanceThreshold<\/span><span style=\"color: #D4D4D4\">: <\/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\">        <\/span><span style=\"color: #6A9955\"># adjacency list<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        adj = [[] <\/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\"> edges:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            adj[u].append((v, w))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            adj[v].append((u, w))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        INF = sys.maxsize<\/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\">dijkstra<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">src<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">) -&gt; List[<\/span><span style=\"color: #4EC9B0\">int<\/span><span style=\"color: #D4D4D4\">]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dist = [INF] * n<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            dist[src] = <\/span><span style=\"color: #B5CEA8\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            pq = [(<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, src)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">while<\/span><span style=\"color: #D4D4D4\"> pq:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                d, u = heapq.heappop(pq)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> d &gt; dist[u]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    <\/span><span style=\"color: #C586C0\">continue<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> v, w <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> adj[u]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    nd = d + w<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                    <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> nd &lt; dist[v]:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                        dist[v] = nd<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                        heapq.heappush(pq, (nd, v))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> dist<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        min_cnt, city = n + <\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, -<\/span><span style=\"color: #B5CEA8\">1<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> i <\/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\">            row = dijkstra(i)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            cnt = <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">(d &lt;= distanceThreshold <\/span><span style=\"color: #C586C0\">for<\/span><span style=\"color: #D4D4D4\"> d <\/span><span style=\"color: #C586C0\">in<\/span><span style=\"color: #D4D4D4\"> row)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> cnt &lt;= min_cnt:            <\/span><span style=\"color: #6A9955\"># \u201c&lt;=\u201d keeps larger index on tie<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">                min_cnt, city = cnt, i<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> city<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p><em>Complexity<\/em>\u2003<code>O(n \u00b7 E log n)<\/code> time | <code>O(n + E)<\/code> memory.<br>For sparse graphs (<code>E \u226a n\u00b2<\/code>) this is usually faster than Floyd Warshall.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-5-which-one-should-i-use\">5. Which one should I use?<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Small <code>n<\/code>, dense input, or you need negative-edge support?<\/strong><br>Go with <strong>Floyd Warshall<\/strong> \u2013 tiny code, deterministic <code>O(n\u00b3)<\/code>.<\/li>\n\n\n\n<li><strong>Large <code>n<\/code> (\u2248 1 000) and sparse roads (like LeetCode\u2019s limits)?<\/strong><br>Prefer <strong>multi-source Dijkstra<\/strong> \u2013 scales roughly <code>O(n\u00b2 log n)<\/code> in practice.<\/li>\n<\/ul>\n\n\n\n<p>Either way you\u2019ll correctly \u201c<strong>Find the City With the Smallest Number of Neighbors at a Threshold Distance<\/strong>\u201d and ace the interview.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Learn to solve \u201cFind the City With the Smallest Number of Neighbors at a Threshold Distance\u201d using Floyd Warshall or a Dijkstra-per-source loop. Simple Python, step-by-step. Here&#8217;s the [Problem Link] to begin with. 1. Problem recap Given compute for every city how many other cities can be reached with total travel distance \u2264 distanceThreshold.Return the<\/p>\n","protected":false},"author":1,"featured_media":409,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,5],"tags":[24,25,17,18],"class_list":{"0":"post-408","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-floyd-warshall","11":"tag-graph","12":"tag-hard"},"featured_image_src":"https:\/\/codeanddebug.in\/blog\/wp-content\/uploads\/2025\/06\/find-the-city-with-the-smallest-number-of-neighbors-at-a-thresholddistance-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\/408","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=408"}],"version-history":[{"count":1,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/408\/revisions"}],"predecessor-version":[{"id":410,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/posts\/408\/revisions\/410"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/media\/409"}],"wp:attachment":[{"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/media?parent=408"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/categories?post=408"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/codeanddebug.in\/blog\/wp-json\/wp\/v2\/tags?post=408"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}