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[LeetCode] 882. Reachable Nodes In Subdivided Graph · Issue #882 · grandyang/leetcode · GitHub
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Reachable Nodes In Subdivided Graph · Issue #882 · grandyang/leetcode" /><meta name="twitter:description" content="Starting with an undirected graph (the &quot;original graph&quot;) with nodes from 0 to N-1, subdivisions are made to some of the edges. The graph is given as follows: edges[k] is a list of integer..." /> <meta property="og:image" content="https://opengraph.githubassets.com/1af21d27e0b11c4e9b0a0b53198b7cdfa21e344d6a3e93678e6801eab220a910/grandyang/leetcode/issues/882" /><meta property="og:image:alt" content="Starting with an undirected graph (the "original graph") with nodes from 0 to N-1, subdivisions are made to some of the edges. The graph is given as follows: edges[k] is a list of integer pairs (i,..." /><meta property="og:image:width" content="1200" /><meta property="og:image:height" content="600" /><meta property="og:site_name" content="GitHub" /><meta property="og:type" content="object" /><meta property="og:title" content="[LeetCode] 882. Reachable Nodes In Subdivided Graph · Issue #882 · grandyang/leetcode" /><meta property="og:url" content="https://github.com/grandyang/leetcode/issues/882" /><meta property="og:description" content="Starting with an undirected graph (the "original graph") with nodes from 0 to N-1, subdivisions are made to some of the edges. 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data-data-router-enabled="false" > <script type="application/json" data-target="react-app.embeddedData">{"payload":{"preloaded_records":{},"preloadedQueries":[{"queryId":"3dddeeb0bc5f88ccaca7204fa97a09b8","queryName":"IssueViewerViewQuery","variables":{"id":"repository","number":882,"owner":"grandyang","repo":"leetcode"},"result":{"data":{"repository":{"isOwnerEnterpriseManaged":false,"issue":{"id":"MDU6SXNzdWU0NTAzOTMzNzQ=","updatedAt":"2021-04-08T17:35:35Z","title":"[LeetCode] 882. Reachable Nodes In Subdivided Graph","number":882,"repository":{"nameWithOwner":"grandyang/leetcode","id":"MDEwOlJlcG9zaXRvcnkxODk0NDQ0MjM=","name":"leetcode","owner":{"__typename":"User","login":"grandyang","id":"MDQ6VXNlcjg1NTMwMTA=","url":"https://github.com/grandyang"},"isArchived":false,"isPrivate":false,"databaseId":189444423,"slashCommandsEnabled":false,"viewerCanInteract":false,"viewerInteractionLimitReasonHTML":"","planFeatures":{"maximumAssignees":10},"visibility":"PUBLIC","pinnedIssues":{"totalCount":0},"viewerCanPinIssues":false,"issueTypes":null},"titleHTML":"[LeetCode] 882. Reachable Nodes In Subdivided Graph","url":"https://github.com/grandyang/leetcode/issues/882","viewerCanUpdateNext":false,"issueType":null,"state":"OPEN","stateReason":null,"duplicateOf":null,"linkedPullRequests":{"nodes":[]},"subIssuesSummary":{"total":0,"completed":0},"__isLabelable":"Issue","labels":{"edges":[],"pageInfo":{"endCursor":null,"hasNextPage":false}},"__isNode":"Issue","assignees":{"nodes":[]},"milestone":null,"databaseId":450393374,"viewerDidAuthor":false,"locked":false,"author":{"__typename":"User","__isActor":"User","login":"grandyang","id":"MDQ6VXNlcjg1NTMwMTA=","profileUrl":"https://github.com/grandyang","avatarUrl":"https://avatars.githubusercontent.com/u/8553010?u=e77fff8485af7a0dd7160048240bf9272e4be8cf\u0026v=4"},"__isComment":"Issue","body":" \nStarting with an undirected graph (the \"original graph\") with nodes from `0` to `N-1`, subdivisions are made to some of the edges. \n\nThe graph is given as follows: `edges[k]` is a list of integer pairs `(i, j, n)` such that `(i, j)` is an edge of the original graph,\n\nand `n` is the total number of new nodes on that edge.\n\nThen, the edge `(i, j)` is deleted from the original graph, `n` new nodes `(x_1, x_2, ..., x_n)` are added to the original graph,\n\nand `n+1` new edges `(i, x_1), (x_1, x_2), (x_2, x_3), ..., (x_{n-1}, x_n), (x_n, j)` are added to the original graph.\n\nNow, you start at node `0` from the original graph, and in each move, you travel along one edge.\n\nReturn how many nodes you can reach in at most `M`moves.\n\nExample 1:\n \n \n Input: `edges` = [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3\n Output: 13\n Explanation:\n The nodes that are reachable in the final graph after M = 6 moves are indicated below.\n \n\n\n\nExample 2:\n \n \n Input: `edges` = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4\n Output: 23\n \n\nNote:\n\n 1. `0 \u003c= edges.length \u003c= 10000`\n 2. `0 \u003c= edges[i][0] \u003c edges[i][1] \u003c N`\n 3. There does not exist any `i != j` for which `edges[i][0] == edges[j][0]` and `edges[i][1] == edges[j][1]`.\n 4. The original graph has no parallel edges.\n 5. `0 \u003c= edges[i][2] \u003c= 10000`\n 6. `0 \u003c= M \u003c= 10^9`\n 7. `1 \u003c= N \u003c= 3000`\n 8. A reachable node is a node that can be travelled to using at most M moves starting from node 0.\n\n \n这道题给了我们一个无向图，里面有N个结点，但是每两个结点中间可能有多个不同的结点，假设每到达下一个相邻的结点需要消耗一步，现在我们有M步可以走，问我们在M步内最多可以到达多少个不同的结点。这里虽然有N个有编号的大结点，中间还有若干个没有编号的小结点，但是最后在统计的时候不分大小结点，全都算不同的结点。为了更好的理解这道题，实际上可以把N个有编号的结点当作N个大城市，比如省会城市，每两个省会城市中间有多个小城市，假设我们每次坐飞机只能飞到相邻的下一个城市，现在我们最多能坐M次飞机，问从省会大城市0出发的话，最多能到达多少个城市。由于省会城市是大型中转站，所以只有在这里才能有多个选择去往不同的城市，而在两个省会城市中的每个小城市，只有前后两种选择，所以这道题实际上还是一种图的遍历，只不过不保证每次都能到有编号的结点，只有到达了有编号的结点，才可以继续遍历下去。当到达了有编号的结点时，还要计算此时的剩余步数，就是用前一个有编号结点的剩余步数，减去当前路径上的所有小结点的个数。假如当前的剩余步数不够到达下一个大结点时，此时我们要想办法标记出来我们走过了多少个小结点，不然下次我们通过另一条路径到达相同的下一个大结点时，再往回走就有可能重复统计小结点的个数。由于小结点并没有标号，没法直接标记，只能通过离最近的大结点的个数来标记，所以虽然这道题是一道无向图的题，但是我们需要将其当作有向图来处理，比如两个大结点A和B，中间有10个小结点，此时在A结点时只有6步能走，那么我们走了中间的6个结点，此时就要标记从B出发往A方向的话只有4个小结点能走了。 \n\n再进一步来分析，其实上对于每个结点来说（不论有没有编号），若我们能算出该结点离起始结点的最短距离，且该距离小于等于M的话，那这个结点就一定可以到达。这样来说，其实本质就是求单源点的最短距离，此时就要祭出神器迪杰斯特拉算法 Dijkstra Algorithm 了，LeetCode 中使用了该算法的题目还有 [Network Delay Time](https://www.cnblogs.com/grandyang/p/8278115.html) 和 [The Maze II](http://www.cnblogs.com/grandyang/p/6725380.html)。该算法的一般形式是用一个最小堆来保存到源点的最小距离，这里我们直接统计到源点的最小距离不是很方便，可以使用一个小 trick，即用一个最大堆来统计当前结点所剩的最大步数，因为剩的步数越多，说明距离源点距离越小。由于 Dijkstra 算法是以起点为中心，向外层层扩展，直到扩展到终点为止。根据这特性，用 BFS 来实现时再好不过了，首先来建立邻接链表，这里可以使用一个 NxN 的二维数组 graph，其中 graph[i][j] 表示从大结点i往大结点j方向会经过的小结点个数，建立邻接链表的时候对于每个 edge，要把两个方向都赋值，前面解释过了这里要当作有向图来做。然后使用一个最大堆，里面放剩余步数和结点编号组成的数对儿，把剩余步数放前面就可以默认按步数从大到小排序了，初始化时把 {M,0} 存入最大堆。还需要一个一维数组 visited 来记录某个结点是否访问过。在 while 循环中，首先取出堆顶元素数对儿，分别取出步数 move，和当前结点编号 cur，此时检查若该结点已经访问过了，直接跳过，否则就在 visited 数组中标记为 true。此时结果 res 自增1，因为当前大结点也是新遍历到的，需要累加个数。然后我们需要遍历所有跟 cur 相连的大结点，对于二维数组形式的邻接链表，我们只需要将i从0遍历到N，假如 graph[cur][i] 为 -1，表示结点 cur 和结点i不相连，直接跳过。否则相连的话，两个大结点中小结点的个数为 graph[cur][i]，此时要跟当前 cur 结点时剩余步数 move 比较，假如 move 较大，说明可以到达结点i，将此时到达结点i的剩余步数 move-graph[cur][i]-1（最后的减1是到达结点i需要的额外步数）和i一起组成数对儿，加入最大堆中。由于之前的分析，结点 cur 往结点i走过的所有结点，从结点i就不能再往结点 cur 走了，否则就累加了重复结点，所以 graph[i][cur] 要减去 move 和 graph[cur][i] 中的较小值，同时结果 res 要累加该较小值即可，参见代码如下：\n\n \n解法一： \n \n \n class Solution {\n public:\n int reachableNodes(vector\u003cvector\u003cint\u003e\u003e\u0026 edges, int M, int N) {\n int res = 0;\n vector\u003cvector\u003cint\u003e\u003e graph(N, vector\u003cint\u003e(N, -1));\n vector\u003cbool\u003e visited(N);\n priority_queue\u003cpair\u003cint, int\u003e\u003e pq;\n pq.push({M, 0});\n for (auto \u0026edge : edges) {\n \tgraph[edge[0]][edge[1]] = edge[2];\n \tgraph[edge[1]][edge[0]] = edge[2];\n }\n while (!pq.empty()) {\n \tauto t= pq.top(); pq.pop();\n \tint move = t.first, cur = t.second;\n \tif (visited[cur]) continue;\n \tvisited[cur] = true;\n \t++res;\n \tfor (int i = 0; i \u003c N; ++i) {\n \t\tif (graph[cur][i] == -1) continue;\n \t\tif (move \u003e graph[cur][i] \u0026\u0026 !visited[i]) {\n \t\t\tpq.push({move - graph[cur][i] - 1, i});\n \t\t}\n \t\tgraph[i][cur] -= min(move, graph[cur][i]);\n \t\tres += min(move, graph[cur][i]);\n \t}\n }\n return res;\n }\n };\n \n\n \n我们也可以使用 HashMap 来建立邻接链表，最后的运行速度果然要比二维数组形式的邻接链表要快一些，其他的地方都不变，参见代码如下： \n解法二： \n \n \n class Solution {\n public:\n int reachableNodes(vector\u003cvector\u003cint\u003e\u003e\u0026 edges, int M, int N) {\n int res = 0;\n unordered_map\u003cint, unordered_map\u003cint, int\u003e\u003e graph;\n vector\u003cbool\u003e visited(N);\n priority_queue\u003cpair\u003cint, int\u003e\u003e pq;\n pq.push({M, 0});\n for (auto \u0026edge : edges) {\n \tgraph[edge[0]][edge[1]] = edge[2];\n \tgraph[edge[1]][edge[0]] = edge[2];\n }\n while (!pq.empty()) {\n \tauto t= pq.top(); pq.pop();\n \tint move = t.first, cur = t.second;\n \tif (visited[cur]) continue;\n \tvisited[cur] = true;\n \t++res;\n \tfor (auto \u0026a : graph[cur]) {\n \t\tif (move \u003e a.second \u0026\u0026 !visited[a.first]) {\n \t\t\tpq.push({move - a.second - 1, a.first});\n \t\t}\n \t\tgraph[a.first][cur] -= min(move, a.second);\n \t\tres += min(move, a.second);\n \t}\n }\n return res;\n }\n };\n \n\n \nGithub 同步地址: \n\n\u003chttps://github.com/grandyang/leetcode/issues/882\u003e\n\n \n参考资料： \n\n\u003chttps://leetcode.com/problems/reachable-nodes-in-subdivided-graph/\u003e\n\n\u003chttps://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156777/Java-Dijkstra-Solution\u003e\n\n\u003chttps://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156739/C%2B%2BJavaPython-Dijkstra-%2B-Priority-Queue\u003e\n\n \n[LeetCode All in One 题目讲解汇总(持续更新中...)](https://www.cnblogs.com/grandyang/p/4606334.html) \n","bodyHTML":"\u003cp dir=\"auto\"\u003eStarting with an undirected graph (the \"original graph\") with nodes from \u003ccode class=\"notranslate\"\u003e0\u003c/code\u003e to \u003ccode class=\"notranslate\"\u003eN-1\u003c/code\u003e, subdivisions are made to some of the edges.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe graph is given as follows: \u003ccode class=\"notranslate\"\u003eedges[k]\u003c/code\u003e is a list of integer pairs \u003ccode class=\"notranslate\"\u003e(i, j, n)\u003c/code\u003e such that \u003ccode class=\"notranslate\"\u003e(i, j)\u003c/code\u003e is an edge of the original graph,\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eand \u003ccode class=\"notranslate\"\u003en\u003c/code\u003e is the total number of new nodes on that edge.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThen, the edge \u003ccode class=\"notranslate\"\u003e(i, j)\u003c/code\u003e is deleted from the original graph, \u003ccode class=\"notranslate\"\u003en\u003c/code\u003e new nodes \u003ccode class=\"notranslate\"\u003e(x_1, x_2, ..., x_n)\u003c/code\u003e are added to the original graph,\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eand \u003ccode class=\"notranslate\"\u003en+1\u003c/code\u003e new edges \u003ccode class=\"notranslate\"\u003e(i, x_1), (x_1, x_2), (x_2, x_3), ..., (x_{n-1}, x_n), (x_n, j)\u003c/code\u003e are added to the original graph.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNow, you start at node \u003ccode class=\"notranslate\"\u003e0\u003c/code\u003e from the original graph, and in each move, you travel along one edge.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eReturn how many nodes you can reach in at most \u003ccode class=\"notranslate\"\u003eM\u003c/code\u003emoves.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eExample 1:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"Input: `edges` = [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3\nOutput: 13\nExplanation:\nThe nodes that are reachable in the final graph after M = 6 moves are indicated below.\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode class=\"notranslate\"\u003eInput: `edges` = [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3\nOutput: 13\nExplanation:\nThe nodes that are reachable in the final graph after M = 6 moves are indicated below.\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer nofollow\" href=\"https://camo.githubusercontent.com/697ee12a5477abc0962362ae45a212cb16e5bb11753b428df94f265ee0fe32dd/68747470733a2f2f73332d6c632d75706c6f61642e73332e616d617a6f6e6177732e636f6d2f75706c6f6164732f323031382f30382f30312f6f72696766696e616c2e706e67\"\u003e\u003cimg src=\"https://camo.githubusercontent.com/697ee12a5477abc0962362ae45a212cb16e5bb11753b428df94f265ee0fe32dd/68747470733a2f2f73332d6c632d75706c6f61642e73332e616d617a6f6e6177732e636f6d2f75706c6f6164732f323031382f30382f30312f6f72696766696e616c2e706e67\" alt=\"\" data-canonical-src=\"https://s3-lc-upload.s3.amazonaws.com/uploads/2018/08/01/origfinal.png\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eExample 2:\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"Input: `edges` = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4\nOutput: 23\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode class=\"notranslate\"\u003eInput: `edges` = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4\nOutput: 23\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eNote:\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003e\u003ccode class=\"notranslate\"\u003e0 \u0026lt;= edges.length \u0026lt;= 10000\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode class=\"notranslate\"\u003e0 \u0026lt;= edges[i][0] \u0026lt; edges[i][1] \u0026lt; N\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThere does not exist any \u003ccode class=\"notranslate\"\u003ei != j\u003c/code\u003e for which \u003ccode class=\"notranslate\"\u003eedges[i][0] == edges[j][0]\u003c/code\u003e and \u003ccode class=\"notranslate\"\u003eedges[i][1] == edges[j][1]\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eThe original graph has no parallel edges.\u003c/li\u003e\n\u003cli\u003e\u003ccode class=\"notranslate\"\u003e0 \u0026lt;= edges[i][2] \u0026lt;= 10000\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode class=\"notranslate\"\u003e0 \u0026lt;= M \u0026lt;= 10^9\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode class=\"notranslate\"\u003e1 \u0026lt;= N \u0026lt;= 3000\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eA reachable node is a node that can be travelled to using at most M moves starting from node 0.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp dir=\"auto\"\u003e这道题给了我们一个无向图，里面有N个结点，但是每两个结点中间可能有多个不同的结点，假设每到达下一个相邻的结点需要消耗一步，现在我们有M步可以走，问我们在M步内最多可以到达多少个不同的结点。这里虽然有N个有编号的大结点，中间还有若干个没有编号的小结点，但是最后在统计的时候不分大小结点，全都算不同的结点。为了更好的理解这道题，实际上可以把N个有编号的结点当作N个大城市，比如省会城市，每两个省会城市中间有多个小城市，假设我们每次坐飞机只能飞到相邻的下一个城市，现在我们最多能坐M次飞机，问从省会大城市0出发的话，最多能到达多少个城市。由于省会城市是大型中转站，所以只有在这里才能有多个选择去往不同的城市，而在两个省会城市中的每个小城市，只有前后两种选择，所以这道题实际上还是一种图的遍历，只不过不保证每次都能到有编号的结点，只有到达了有编号的结点，才可以继续遍历下去。当到达了有编号的结点时，还要计算此时的剩余步数，就是用前一个有编号结点的剩余步数，减去当前路径上的所有小结点的个数。假如当前的剩余步数不够到达下一个大结点时，此时我们要想办法标记出来我们走过了多少个小结点，不然下次我们通过另一条路径到达相同的下一个大结点时，再往回走就有可能重复统计小结点的个数。由于小结点并没有标号，没法直接标记，只能通过离最近的大结点的个数来标记，所以虽然这道题是一道无向图的题，但是我们需要将其当作有向图来处理，比如两个大结点A和B，中间有10个小结点，此时在A结点时只有6步能走，那么我们走了中间的6个结点，此时就要标记从B出发往A方向的话只有4个小结点能走了。\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e再进一步来分析，其实上对于每个结点来说（不论有没有编号），若我们能算出该结点离起始结点的最短距离，且该距离小于等于M的话，那这个结点就一定可以到达。这样来说，其实本质就是求单源点的最短距离，此时就要祭出神器迪杰斯特拉算法 Dijkstra Algorithm 了，LeetCode 中使用了该算法的题目还有 \u003ca href=\"https://www.cnblogs.com/grandyang/p/8278115.html\" rel=\"nofollow\"\u003eNetwork Delay Time\u003c/a\u003e 和 \u003ca href=\"http://www.cnblogs.com/grandyang/p/6725380.html\" rel=\"nofollow\"\u003eThe Maze II\u003c/a\u003e。该算法的一般形式是用一个最小堆来保存到源点的最小距离，这里我们直接统计到源点的最小距离不是很方便，可以使用一个小 trick，即用一个最大堆来统计当前结点所剩的最大步数，因为剩的步数越多，说明距离源点距离越小。由于 Dijkstra 算法是以起点为中心，向外层层扩展，直到扩展到终点为止。根据这特性，用 BFS 来实现时再好不过了，首先来建立邻接链表，这里可以使用一个 NxN 的二维数组 graph，其中 graph[i][j] 表示从大结点i往大结点j方向会经过的小结点个数，建立邻接链表的时候对于每个 edge，要把两个方向都赋值，前面解释过了这里要当作有向图来做。然后使用一个最大堆，里面放剩余步数和结点编号组成的数对儿，把剩余步数放前面就可以默认按步数从大到小排序了，初始化时把 {M,0} 存入最大堆。还需要一个一维数组 visited 来记录某个结点是否访问过。在 while 循环中，首先取出堆顶元素数对儿，分别取出步数 move，和当前结点编号 cur，此时检查若该结点已经访问过了，直接跳过，否则就在 visited 数组中标记为 true。此时结果 res 自增1，因为当前大结点也是新遍历到的，需要累加个数。然后我们需要遍历所有跟 cur 相连的大结点，对于二维数组形式的邻接链表，我们只需要将i从0遍历到N，假如 graph[cur][i] 为 -1，表示结点 cur 和结点i不相连，直接跳过。否则相连的话，两个大结点中小结点的个数为 graph[cur][i]，此时要跟当前 cur 结点时剩余步数 move 比较，假如 move 较大，说明可以到达结点i，将此时到达结点i的剩余步数 move-graph[cur][i]-1（最后的减1是到达结点i需要的额外步数）和i一起组成数对儿，加入最大堆中。由于之前的分析，结点 cur 往结点i走过的所有结点，从结点i就不能再往结点 cur 走了，否则就累加了重复结点，所以 graph[i][cur] 要减去 move 和 graph[cur][i] 中的较小值，同时结果 res 要累加该较小值即可，参见代码如下：\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e解法一：\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"class Solution {\npublic:\n int reachableNodes(vector\u0026lt;vector\u0026lt;int\u0026gt;\u0026gt;\u0026amp; edges, int M, int N) {\n int res = 0;\n vector\u0026lt;vector\u0026lt;int\u0026gt;\u0026gt; graph(N, vector\u0026lt;int\u0026gt;(N, -1));\n vector\u0026lt;bool\u0026gt; visited(N);\n priority_queue\u0026lt;pair\u0026lt;int, int\u0026gt;\u0026gt; pq;\n pq.push({M, 0});\n for (auto \u0026amp;edge : edges) {\n \tgraph[edge[0]][edge[1]] = edge[2];\n \tgraph[edge[1]][edge[0]] = edge[2];\n }\n while (!pq.empty()) {\n \tauto t= pq.top(); pq.pop();\n \tint move = t.first, cur = t.second;\n \tif (visited[cur]) continue;\n \tvisited[cur] = true;\n \t++res;\n \tfor (int i = 0; i \u0026lt; N; ++i) {\n \t\tif (graph[cur][i] == -1) continue;\n \t\tif (move \u0026gt; graph[cur][i] \u0026amp;\u0026amp; !visited[i]) {\n \t\t\tpq.push({move - graph[cur][i] - 1, i});\n \t\t}\n \t\tgraph[i][cur] -= min(move, graph[cur][i]);\n \t\tres += min(move, graph[cur][i]);\n \t}\n }\n return res;\n }\n};\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode class=\"notranslate\"\u003eclass Solution {\npublic:\n int reachableNodes(vector\u0026lt;vector\u0026lt;int\u0026gt;\u0026gt;\u0026amp; edges, int M, int N) {\n int res = 0;\n vector\u0026lt;vector\u0026lt;int\u0026gt;\u0026gt; graph(N, vector\u0026lt;int\u0026gt;(N, -1));\n vector\u0026lt;bool\u0026gt; visited(N);\n priority_queue\u0026lt;pair\u0026lt;int, int\u0026gt;\u0026gt; pq;\n pq.push({M, 0});\n for (auto \u0026amp;edge : edges) {\n \tgraph[edge[0]][edge[1]] = edge[2];\n \tgraph[edge[1]][edge[0]] = edge[2];\n }\n while (!pq.empty()) {\n \tauto t= pq.top(); pq.pop();\n \tint move = t.first, cur = t.second;\n \tif (visited[cur]) continue;\n \tvisited[cur] = true;\n \t++res;\n \tfor (int i = 0; i \u0026lt; N; ++i) {\n \t\tif (graph[cur][i] == -1) continue;\n \t\tif (move \u0026gt; graph[cur][i] \u0026amp;\u0026amp; !visited[i]) {\n \t\t\tpq.push({move - graph[cur][i] - 1, i});\n \t\t}\n \t\tgraph[i][cur] -= min(move, graph[cur][i]);\n \t\tres += min(move, graph[cur][i]);\n \t}\n }\n return res;\n }\n};\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003e我们也可以使用 HashMap 来建立邻接链表，最后的运行速度果然要比二维数组形式的邻接链表要快一些，其他的地方都不变，参见代码如下：\u003cbr\u003e\n解法二：\u003c/p\u003e\n\u003cdiv class=\"snippet-clipboard-content notranslate position-relative overflow-auto\" data-snippet-clipboard-copy-content=\"class Solution {\npublic:\n int reachableNodes(vector\u0026lt;vector\u0026lt;int\u0026gt;\u0026gt;\u0026amp; edges, int M, int N) {\n int res = 0;\n unordered_map\u0026lt;int, unordered_map\u0026lt;int, int\u0026gt;\u0026gt; graph;\n vector\u0026lt;bool\u0026gt; visited(N);\n priority_queue\u0026lt;pair\u0026lt;int, int\u0026gt;\u0026gt; pq;\n pq.push({M, 0});\n for (auto \u0026amp;edge : edges) {\n \tgraph[edge[0]][edge[1]] = edge[2];\n \tgraph[edge[1]][edge[0]] = edge[2];\n }\n while (!pq.empty()) {\n \tauto t= pq.top(); pq.pop();\n \tint move = t.first, cur = t.second;\n \tif (visited[cur]) continue;\n \tvisited[cur] = true;\n \t++res;\n \tfor (auto \u0026amp;a : graph[cur]) {\n \t\tif (move \u0026gt; a.second \u0026amp;\u0026amp; !visited[a.first]) {\n \t\t\tpq.push({move - a.second - 1, a.first});\n \t\t}\n \t\tgraph[a.first][cur] -= min(move, a.second);\n \t\tres += min(move, a.second);\n \t}\n }\n return res;\n }\n};\"\u003e\u003cpre class=\"notranslate\"\u003e\u003ccode class=\"notranslate\"\u003eclass Solution {\npublic:\n int reachableNodes(vector\u0026lt;vector\u0026lt;int\u0026gt;\u0026gt;\u0026amp; edges, int M, int N) {\n int res = 0;\n unordered_map\u0026lt;int, unordered_map\u0026lt;int, int\u0026gt;\u0026gt; graph;\n vector\u0026lt;bool\u0026gt; visited(N);\n priority_queue\u0026lt;pair\u0026lt;int, int\u0026gt;\u0026gt; pq;\n pq.push({M, 0});\n for (auto \u0026amp;edge : edges) {\n \tgraph[edge[0]][edge[1]] = edge[2];\n \tgraph[edge[1]][edge[0]] = edge[2];\n }\n while (!pq.empty()) {\n \tauto t= pq.top(); pq.pop();\n \tint move = t.first, cur = t.second;\n \tif (visited[cur]) continue;\n \tvisited[cur] = true;\n \t++res;\n \tfor (auto \u0026amp;a : graph[cur]) {\n \t\tif (move \u0026gt; a.second \u0026amp;\u0026amp; !visited[a.first]) {\n \t\t\tpq.push({move - a.second - 1, a.first});\n \t\t}\n \t\tgraph[a.first][cur] -= min(move, a.second);\n \t\tres += min(move, a.second);\n \t}\n }\n return res;\n }\n};\n\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eGithub 同步地址:\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca class=\"issue-link js-issue-link\" data-error-text=\"Failed to load title\" data-id=\"450393374\" data-permission-text=\"Title is private\" data-url=\"https://github.com/grandyang/leetcode/issues/882\" data-hovercard-type=\"issue\" data-hovercard-url=\"/grandyang/leetcode/issues/882/hovercard\" href=\"https://github.com/grandyang/leetcode/issues/882\"\u003e#882\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e参考资料：\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/\" rel=\"nofollow\"\u003ehttps://leetcode.com/problems/reachable-nodes-in-subdivided-graph/\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156777/Java-Dijkstra-Solution\" rel=\"nofollow\"\u003ehttps://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156777/Java-Dijkstra-Solution\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156739/C%2B%2BJavaPython-Dijkstra-%2B-Priority-Queue\" rel=\"nofollow\"\u003ehttps://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156739/C%2B%2BJavaPython-Dijkstra-%2B-Priority-Queue\u003c/a\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca href=\"https://www.cnblogs.com/grandyang/p/4606334.html\" rel=\"nofollow\"\u003eLeetCode All in One 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ActivityHeader-module__edits--LwHTE"></div><div class="Box-sc-g0xbh4-0 kpoUe"><div class="Box-sc-g0xbh4-0 ezcJRX"></div><div class="Box-sc-g0xbh4-0 dnyPuu"><button data-component="IconButton" type="button" aria-haspopup="true" aria-expanded="false" tabindex="0" class="prc-Button-ButtonBase-c50BI prc-Button-IconButton-szpyj" data-loading="false" data-no-visuals="true" data-size="small" data-variant="invisible" aria-describedby=":R15bd5db:-loading-announcement" aria-labelledby=":Rv5bd5db:" id=":R15bd5db:"><svg aria-hidden="true" focusable="false" class="octicon octicon-kebab-horizontal" viewBox="0 0 16 16" width="16" height="16" fill="currentColor" display="inline-block" overflow="visible" style="vertical-align:text-bottom"><path d="M8 9a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3ZM1.5 9a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Zm13 0a1.5 1.5 0 1 0 0-3 1.5 1.5 0 0 0 0 3Z"></path></svg></button><span class="prc-TooltipV2-Tooltip-cYMVY" data-direction="s" aria-hidden="true" id=":Rv5bd5db:">Issue body actions</span></div></div></div></div></div><div class="IssueBodyViewer-module__IssueBody--MXyFt"><div data-testid="markdown-body" data-team-hovercards-enabled="true" class="markdown-body" data-turbolinks="false"><div class="Box-sc-g0xbh4-0 markdown-body NewMarkdownViewer-module__safe-html-box--cRsz0"><p dir="auto">Starting with an undirected graph (the "original graph") with nodes from <code class="notranslate">0</code> to <code class="notranslate">N-1</code>, subdivisions are made to some of the edges.</p> <p dir="auto">The graph is given as follows: <code class="notranslate">edges[k]</code> is a list of integer pairs <code class="notranslate">(i, j, n)</code> such that <code class="notranslate">(i, j)</code> is an edge of the original graph,</p> <p dir="auto">and <code class="notranslate">n</code> is the total number of new nodes on that edge.</p> <p dir="auto">Then, the edge <code class="notranslate">(i, j)</code> is deleted from the original graph, <code class="notranslate">n</code> new nodes <code class="notranslate">(x_1, x_2, ..., x_n)</code> are added to the original graph,</p> <p dir="auto">and <code class="notranslate">n+1</code> new edges <code class="notranslate">(i, x_1), (x_1, x_2), (x_2, x_3), ..., (x_{n-1}, x_n), (x_n, j)</code> are added to the original graph.</p> <p dir="auto">Now, you start at node <code class="notranslate">0</code> from the original graph, and in each move, you travel along one edge.</p> <p dir="auto">Return how many nodes you can reach in at most <code class="notranslate">M</code>moves.</p> <p dir="auto">Example 1:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="Input: `edges` = [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3 Output: 13 Explanation: The nodes that are reachable in the final graph after M = 6 moves are indicated below."><pre class="notranslate"><code class="notranslate">Input: `edges` = [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3 Output: 13 Explanation: The nodes that are reachable in the final graph after M = 6 moves are indicated below. </code></pre></div> <p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://camo.githubusercontent.com/697ee12a5477abc0962362ae45a212cb16e5bb11753b428df94f265ee0fe32dd/68747470733a2f2f73332d6c632d75706c6f61642e73332e616d617a6f6e6177732e636f6d2f75706c6f6164732f323031382f30382f30312f6f72696766696e616c2e706e67"><img src="https://camo.githubusercontent.com/697ee12a5477abc0962362ae45a212cb16e5bb11753b428df94f265ee0fe32dd/68747470733a2f2f73332d6c632d75706c6f61642e73332e616d617a6f6e6177732e636f6d2f75706c6f6164732f323031382f30382f30312f6f72696766696e616c2e706e67" alt="" data-canonical-src="https://s3-lc-upload.s3.amazonaws.com/uploads/2018/08/01/origfinal.png" style="max-width: 100%;"></a></p> <p dir="auto">Example 2:</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="Input: `edges` = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4 Output: 23"><pre class="notranslate"><code class="notranslate">Input: `edges` = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4 Output: 23 </code></pre></div> <p dir="auto">Note:</p> <ol dir="auto"> <li><code class="notranslate">0 <= edges.length <= 10000</code></li> <li><code class="notranslate">0 <= edges[i][0] < edges[i][1] < N</code></li> <li>There does not exist any <code class="notranslate">i != j</code> for which <code class="notranslate">edges[i][0] == edges[j][0]</code> and <code class="notranslate">edges[i][1] == edges[j][1]</code>.</li> <li>The original graph has no parallel edges.</li> <li><code class="notranslate">0 <= edges[i][2] <= 10000</code></li> <li><code class="notranslate">0 <= M <= 10^9</code></li> <li><code class="notranslate">1 <= N <= 3000</code></li> <li>A reachable node is a node that can be travelled to using at most M moves starting from node 0.</li> </ol> <p dir="auto">这道题给了我们一个无向图，里面有N个结点，但是每两个结点中间可能有多个不同的结点，假设每到达下一个相邻的结点需要消耗一步，现在我们有M步可以走，问我们在M步内最多可以到达多少个不同的结点。这里虽然有N个有编号的大结点，中间还有若干个没有编号的小结点，但是最后在统计的时候不分大小结点，全都算不同的结点。为了更好的理解这道题，实际上可以把N个有编号的结点当作N个大城市，比如省会城市，每两个省会城市中间有多个小城市，假设我们每次坐飞机只能飞到相邻的下一个城市，现在我们最多能坐M次飞机，问从省会大城市0出发的话，最多能到达多少个城市。由于省会城市是大型中转站，所以只有在这里才能有多个选择去往不同的城市，而在两个省会城市中的每个小城市，只有前后两种选择，所以这道题实际上还是一种图的遍历，只不过不保证每次都能到有编号的结点，只有到达了有编号的结点，才可以继续遍历下去。当到达了有编号的结点时，还要计算此时的剩余步数，就是用前一个有编号结点的剩余步数，减去当前路径上的所有小结点的个数。假如当前的剩余步数不够到达下一个大结点时，此时我们要想办法标记出来我们走过了多少个小结点，不然下次我们通过另一条路径到达相同的下一个大结点时，再往回走就有可能重复统计小结点的个数。由于小结点并没有标号，没法直接标记，只能通过离最近的大结点的个数来标记，所以虽然这道题是一道无向图的题，但是我们需要将其当作有向图来处理，比如两个大结点A和B，中间有10个小结点，此时在A结点时只有6步能走，那么我们走了中间的6个结点，此时就要标记从B出发往A方向的话只有4个小结点能走了。</p> <p dir="auto">再进一步来分析，其实上对于每个结点来说（不论有没有编号），若我们能算出该结点离起始结点的最短距离，且该距离小于等于M的话，那这个结点就一定可以到达。这样来说，其实本质就是求单源点的最短距离，此时就要祭出神器迪杰斯特拉算法 Dijkstra Algorithm 了，LeetCode 中使用了该算法的题目还有 <a href="https://www.cnblogs.com/grandyang/p/8278115.html" rel="nofollow">Network Delay Time</a> 和 <a href="http://www.cnblogs.com/grandyang/p/6725380.html" rel="nofollow">The Maze II</a>。该算法的一般形式是用一个最小堆来保存到源点的最小距离，这里我们直接统计到源点的最小距离不是很方便，可以使用一个小 trick，即用一个最大堆来统计当前结点所剩的最大步数，因为剩的步数越多，说明距离源点距离越小。由于 Dijkstra 算法是以起点为中心，向外层层扩展，直到扩展到终点为止。根据这特性，用 BFS 来实现时再好不过了，首先来建立邻接链表，这里可以使用一个 NxN 的二维数组 graph，其中 graph[i][j] 表示从大结点i往大结点j方向会经过的小结点个数，建立邻接链表的时候对于每个 edge，要把两个方向都赋值，前面解释过了这里要当作有向图来做。然后使用一个最大堆，里面放剩余步数和结点编号组成的数对儿，把剩余步数放前面就可以默认按步数从大到小排序了，初始化时把 {M,0} 存入最大堆。还需要一个一维数组 visited 来记录某个结点是否访问过。在 while 循环中，首先取出堆顶元素数对儿，分别取出步数 move，和当前结点编号 cur，此时检查若该结点已经访问过了，直接跳过，否则就在 visited 数组中标记为 true。此时结果 res 自增1，因为当前大结点也是新遍历到的，需要累加个数。然后我们需要遍历所有跟 cur 相连的大结点，对于二维数组形式的邻接链表，我们只需要将i从0遍历到N，假如 graph[cur][i] 为 -1，表示结点 cur 和结点i不相连，直接跳过。否则相连的话，两个大结点中小结点的个数为 graph[cur][i]，此时要跟当前 cur 结点时剩余步数 move 比较，假如 move 较大，说明可以到达结点i，将此时到达结点i的剩余步数 move-graph[cur][i]-1（最后的减1是到达结点i需要的额外步数）和i一起组成数对儿，加入最大堆中。由于之前的分析，结点 cur 往结点i走过的所有结点，从结点i就不能再往结点 cur 走了，否则就累加了重复结点，所以 graph[i][cur] 要减去 move 和 graph[cur][i] 中的较小值，同时结果 res 要累加该较小值即可，参见代码如下：</p> <p dir="auto">解法一：</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="class Solution { public: int reachableNodes(vector<vector<int>>& edges, int M, int N) { int res = 0; vector<vector<int>> graph(N, vector<int>(N, -1)); vector<bool> visited(N); priority_queue<pair<int, int>> pq; pq.push({M, 0}); for (auto &edge : edges) { graph[edge[0]][edge[1]] = edge[2]; graph[edge[1]][edge[0]] = edge[2]; } while (!pq.empty()) { auto t= pq.top(); pq.pop(); int move = t.first, cur = t.second; if (visited[cur]) continue; visited[cur] = true; ++res; for (int i = 0; i < N; ++i) { if (graph[cur][i] == -1) continue; if (move > graph[cur][i] && !visited[i]) { pq.push({move - graph[cur][i] - 1, i}); } graph[i][cur] -= min(move, graph[cur][i]); res += min(move, graph[cur][i]); } } return res; } };"><pre class="notranslate"><code class="notranslate">class Solution { public: int reachableNodes(vector<vector<int>>& edges, int M, int N) { int res = 0; vector<vector<int>> graph(N, vector<int>(N, -1)); vector<bool> visited(N); priority_queue<pair<int, int>> pq; pq.push({M, 0}); for (auto &edge : edges) { graph[edge[0]][edge[1]] = edge[2]; graph[edge[1]][edge[0]] = edge[2]; } while (!pq.empty()) { auto t= pq.top(); pq.pop(); int move = t.first, cur = t.second; if (visited[cur]) continue; visited[cur] = true; ++res; for (int i = 0; i < N; ++i) { if (graph[cur][i] == -1) continue; if (move > graph[cur][i] && !visited[i]) { pq.push({move - graph[cur][i] - 1, i}); } graph[i][cur] -= min(move, graph[cur][i]); res += min(move, graph[cur][i]); } } return res; } }; </code></pre></div> <p dir="auto">我们也可以使用 HashMap 来建立邻接链表，最后的运行速度果然要比二维数组形式的邻接链表要快一些，其他的地方都不变，参见代码如下：<br> 解法二：</p> <div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="class Solution { public: int reachableNodes(vector<vector<int>>& edges, int M, int N) { int res = 0; unordered_map<int, unordered_map<int, int>> graph; vector<bool> visited(N); priority_queue<pair<int, int>> pq; pq.push({M, 0}); for (auto &edge : edges) { graph[edge[0]][edge[1]] = edge[2]; graph[edge[1]][edge[0]] = edge[2]; } while (!pq.empty()) { auto t= pq.top(); pq.pop(); int move = t.first, cur = t.second; if (visited[cur]) continue; visited[cur] = true; ++res; for (auto &a : graph[cur]) { if (move > a.second && !visited[a.first]) { pq.push({move - a.second - 1, a.first}); } graph[a.first][cur] -= min(move, a.second); res += min(move, a.second); } } return res; } };"><pre class="notranslate"><code class="notranslate">class Solution { public: int reachableNodes(vector<vector<int>>& edges, int M, int N) { int res = 0; unordered_map<int, unordered_map<int, int>> graph; vector<bool> visited(N); priority_queue<pair<int, int>> pq; pq.push({M, 0}); for (auto &edge : edges) { graph[edge[0]][edge[1]] = edge[2]; graph[edge[1]][edge[0]] = edge[2]; } while (!pq.empty()) { auto t= pq.top(); pq.pop(); int move = t.first, cur = t.second; if (visited[cur]) continue; visited[cur] = true; ++res; for (auto &a : graph[cur]) { if (move > a.second && !visited[a.first]) { pq.push({move - a.second - 1, a.first}); } graph[a.first][cur] -= min(move, a.second); res += min(move, a.second); } } return res; } }; </code></pre></div> <p dir="auto">Github 同步地址:</p> <p dir="auto"><a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="450393374" data-permission-text="Title is private" data-url="https://github.com/grandyang/leetcode/issues/882" data-hovercard-type="issue" data-hovercard-url="/grandyang/leetcode/issues/882/hovercard" href="https://github.com/grandyang/leetcode/issues/882">#882</a></p> <p dir="auto">参考资料：</p> <p dir="auto"><a href="https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/" rel="nofollow">https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/</a></p> <p dir="auto"><a href="https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156777/Java-Dijkstra-Solution" rel="nofollow">https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156777/Java-Dijkstra-Solution</a></p> <p dir="auto"><a href="https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156739/C%2B%2BJavaPython-Dijkstra-%2B-Priority-Queue" rel="nofollow">https://leetcode.com/problems/reachable-nodes-in-subdivided-graph/discuss/156739/C%2B%2BJavaPython-Dijkstra-%2B-Priority-Queue</a></p> <p dir="auto"><a href="https://www.cnblogs.com/grandyang/p/4606334.html" rel="nofollow">LeetCode All in One 题目讲解汇总(持续更新中...)</a></p></div></div><div class="IssueBodyViewer-module__IssueBodyTaskList--r4XEH"><!--$--><div role="toolbar" aria-label="Reactions" class="d-flex gap-1 flex-wrap"></div><!--/$--></div></div></div></div></div></div></div><div data-testid="issue-viewer-comments-container" class="react-comments-container"><div class="Box-sc-g0xbh4-0 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aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-check js-clipboard-check-icon color-fg-success d-none m-2"> <path d="M13.78 4.22a.75.75 0 0 1 0 1.06l-7.25 7.25a.75.75 0 0 1-1.06 0L2.22 9.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018L6 10.94l6.72-6.72a.75.75 0 0 1 1.06 0Z"></path> </svg> </clipboard-copy> </div> </template> <template id="snippet-clipboard-copy-button-unpositioned"> <div class="zeroclipboard-container"> <clipboard-copy aria-label="Copy" class="ClipboardButton btn btn-invisible js-clipboard-copy m-2 p-0 d-flex flex-justify-center flex-items-center" data-copy-feedback="Copied!" data-tooltip-direction="w"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-copy js-clipboard-copy-icon"> <path d="M0 6.75C0 5.784.784 5 1.75 5h1.5a.75.75 0 0 1 0 1.5h-1.5a.25.25 0 0 0-.25.25v7.5c0 .138.112.25.25.25h7.5a.25.25 0 0 0 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