From 6d11fe90f1008ab7aa0a7d6461fa7d451e04862d Mon Sep 17 00:00:00 2001 From: snowan Date: Mon, 26 Aug 2019 01:09:06 -0700 Subject: [PATCH] feat: Add LC #1168 solution analysis (#146) --- README.en.md | 1 + README.md | 1 + ...mize-water-distribution-in-a-village-1.png | Bin 0 -> 697017 bytes ...ter-distribution-in-a-village-example1.png | Bin 0 -> 28851 bytes ...mize-water-distribution-in-a-village-cn.md | 187 +++++++++++++++++ ...mize-water-distribution-in-a-village-en.md | 194 ++++++++++++++++++ 6 files changed, 383 insertions(+) create mode 100644 assets/problems/1168.optimize-water-distribution-in-a-village-1.png create mode 100644 assets/problems/1168.optimize-water-distribution-in-a-village-example1.png create mode 100644 problems/1168.optimize-water-distribution-in-a-village-cn.md create mode 100644 problems/1168.optimize-water-distribution-in-a-village-en.md diff --git a/README.en.md b/README.en.md index 9644c0dcb..c2c133ac4 100644 --- a/README.en.md +++ b/README.en.md @@ -226,6 +226,7 @@ The data structures mainly includes: - [0295.find-median-from-data-stream](./problems/295.find-median-from-data-stream.md) 🆕 - [0301.remove-invalid-parentheses](./problems/301.remove-invalid-parentheses.md) - [0460.lfu-cache](./problems/460.lfu-cache.md) 🆕 +- [1168.optimize-water-distribution-in-a-village](./problems/1168.optimize-water-distribution-in-a-village-en.md) 🆕 diff --git a/README.md b/README.md index b55d729d0..85f080994 100644 --- a/README.md +++ b/README.md @@ -230,6 +230,7 @@ leetcode 题解,记录自己的 leetcode 解题之路。 - [0295.find-median-from-data-stream](./problems/295.find-median-from-data-stream.md) 🆕 - [0301.remove-invalid-parentheses](./problems/301.remove-invalid-parentheses.md) - [0460.lfu-cache](./problems/460.lfu-cache.md) 🆕 +- [1168.optimize-water-distribution-in-a-village](./problems/1168.optimize-water-distribution-in-a-village-cn.md) 🆕 ### 数据结构与算法的总结 diff --git a/assets/problems/1168.optimize-water-distribution-in-a-village-1.png 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z?2MQeB2qAVtlx%e_z`Llg zOiV^w&2GDk>yK@VJ_|>=9|mE(EIVwZcvhn<;xtzh>Myq0ugD zpMC|y9Kk9TY<2Gn?EO~)*w$cPVa|9HEH0l|{yoYLw;;N?(4AA@5Ci`1&+fKeXjAV&Cr5~}VUR%W<>3WCut%8H!TD;L z_a%vTA^S2ibGR#SxeOe|@qeuWA2dk;9-y7HKoTr}E(qpPMD2^ZotYzUvyNE+`f&3s zyR82jj(lhmcmi7J%hB%~<%Zq1d2@{U_&wNdD{uIT52IPj&bOYwdc1eywJyN9C6LRv zp}`_ucJ*2H|7ogqlcrMBNyhaR9~pk+n*PTnV&Vb+s;%TOHvS)a^Dg{=5W@fsP4Khjobj EKitjFzW@LL literal 0 HcmV?d00001 diff --git a/problems/1168.optimize-water-distribution-in-a-village-cn.md b/problems/1168.optimize-water-distribution-in-a-village-cn.md new file mode 100644 index 000000000..f0398a673 --- /dev/null +++ b/problems/1168.optimize-water-distribution-in-a-village-cn.md @@ -0,0 +1,187 @@ +## 题目地址 +https://leetcode.com/problems/optimize-water-distribution-in-a-village/ + +## 题目描述 +``` +There are n houses in a village. We want to supply water for all the houses by building wells and laying pipes. + +For each house i, we can either build a well inside it directly with cost wells[i], or pipe in water from another well to it. The costs to lay pipes between houses are given by the array pipes, where each pipes[i] = [house1, house2, cost] represents the cost to connect house1 and house2 together using a pipe. Connections are bidirectional. + +Find the minimum total cost to supply water to all houses. + +Example 1: + +Input: n = 3, wells = [1,2,2], pipes = [[1,2,1],[2,3,1]] +Output: 3 +Explanation: +The image shows the costs of connecting houses using pipes. +The best strategy is to build a well in the first house with cost 1 and connect the other houses to it with cost 2 so the total cost is 3. + +Constraints: + +1 <= n <= 10000 +wells.length == n +0 <= wells[i] <= 10^5 +1 <= pipes.length <= 10000 +1 <= pipes[i][0], pipes[i][1] <= n +0 <= pipes[i][2] <= 10^5 +pipes[i][0] != pipes[i][1] +``` +example 1 pic: + +![example 1](../assets/problems/1168.optimize-water-distribution-in-a-village-example1.png) + +## 思路 +题意,在每个城市打井需要一定的花费,也可以用其他城市的井水,城市之间建立连接管道需要一定的花费,怎么样安排可以花费最少的前灌溉所有城市。 + +这是一道连通所有点的最短路径/最小生成树问题,把城市看成图中的点,管道连接城市看成是连接两个点之间的边。这里打井的花费是直接在点上,而且并不是所有 +点之间都有边连接,为了方便,我们可以假想一个点`(root)0`,这里自身点的花费可以与 `0` 连接,花费可以是 `0-i` 之间的花费。这样我们就可以构建一个连通图包含所有的点和边。 +那在一个连通图中求最短路径/最小生成树的问题. + +参考延伸阅读中,维基百科针对这类题给出的几种解法。 + +解题步骤: +1. 创建 `POJO EdgeCost(node1, node2, cost) - 节点1 和 节点2 连接边的花费`。 +2. 假想一个`root` 点 `0`,构建图 +3. 连通所有节点和 `0`,`[0,i] - i 是节点 [1,n]`,`0-1` 是节点 `0` 和 `1` 的边,边的值是节点 `i` 上打井的花费 `wells[i]`; +4. 把打井花费和城市连接点转换成图的节点和边。 +5. 对图的边的值排序(从小到大) +6. 遍历图的边,判断两个节点有没有连通 (`Union-Find`), + - 已连通就跳过,继续访问下一条边 + - 没有连通,记录花费,连通节点 +7. 若所有节点已连通,求得的最小路径即为最小花费,返回 +8. 对于每次`union`, 节点数 `n-1`, 如果 `n==0` 说明所有节点都已连通,可以提前退出,不需要继续访问剩余的边。 + +> 这里用加权Union-Find 判断两个节点是否连通,和连通未连通的节点。 + +举例:`n = 5, wells=[1,2,2,3,2], pipes=[[1,2,1],[2,3,1],[4,5,7]]` + +如图: + +![minimum cost](../assets/problems/1168.optimize-water-distribution-in-a-village-1.png) + +从图中可以看到,最后所有的节点都是连通的。 + +#### 复杂度分析 +- *时间复杂度:* `O(ElogE) - E 是图的边的个数` +- *空间复杂度:* `O(E)` + +> 一个图最多有 `n(n-1)/2 - n 是图中节点个数` 条边 (完全连通图) + +## 关键点分析 +1. 构建图,得出所有边 +2. 对所有边排序 +3. 遍历所有的边(从小到大) +4. 对于每条边,检查是否已经连通,若没有连通,加上边上的值,连通两个节点。若已连通,跳过。 + +## 代码 (`Java/Python3`) +*Java code* +```java + class OptimizeWaterDistribution { + public int minCostToSupplyWater(int n, int[] wells, int[][] pipes) { + List costs = new ArrayList<>(); + for (int i = 1; i <= n; i++) { + costs.add(new EdgeCost(0, i, wells[i - 1])); + } + for (int[] p : pipes) { + costs.add(new EdgeCost(p[0], p[1], p[2])); + } + Collections.sort(costs); + int minCosts = 0; + UnionFind uf = new UnionFind(n); + for (EdgeCost edge : costs) { + int rootX = uf.find(edge.node1); + int rootY = uf.find(edge.node2); + if (rootX == rootY) continue; + minCosts += edge.cost; + uf.union(edge.node1, edge.node2); + // for each union, we connnect one node + n--; + // if all nodes already connected, terminate early + if (n == 0) { + return minCosts; + } + } + return minCosts; + } + + class EdgeCost implements Comparable { + int node1; + int node2; + int cost; + public EdgeCost(int node1, int node2, int cost) { + this.node1 = node1; + this.node2 = node2; + this.cost = cost; + } + + @Override + public int compareTo(EdgeCost o) { + return this.cost - o.cost; + } + } + + class UnionFind { + int[] parent; + int[] rank; + public UnionFind(int n) { + parent = new int[n + 1]; + for (int i = 0; i <= n; i++) { + parent[i] = i; + } + rank = new int[n + 1]; + } + public int find(int x) { + return x == parent[x] ? x : find(parent[x]); + } + public void union(int x, int y) { + int px = find(x); + int py = find(y); + if (px == py) return; + if (rank[px] >= rank[py]) { + parent[py] = px; + rank[px] += rank[py]; + } else { + parent[px] = py; + rank[py] += rank[px]; + } + } + } + } +``` +*Pythong3 code* +```python +class Solution: + def minCostToSupplyWater(self, n: int, wells: List[int], pipes: List[List[int]]) -> int: + union_find = {i: i for i in range(n + 1)} + + def find(x): + return x if x == union_find[x] else find(union_find[x]) + + def union(x, y): + px = find(x) + py = find(y) + union_find[px] = py + + graph_wells = [[cost, 0, i] for i, cost in enumerate(wells, 1)] + graph_pipes = [[cost, i, j] for i, j, cost in pipes] + min_costs = 0 + for cost, x, y in sorted(graph_wells + graph_pipes): + if find(x) == find(y): + continue + union(x, y) + min_costs += cost + n -= 1 + if n == 0: + return min_costs +``` + +## 延伸阅读 + +1. [最短路径问题](https://www.wikiwand.com/zh-hans/%E6%9C%80%E7%9F%AD%E8%B7%AF%E9%97%AE%E9%A2%98) +2. [Dijkstra算法](https://www.wikiwand.com/zh-hans/戴克斯特拉算法) +3. [Floyd-Warshall算法](https://www.wikiwand.com/zh-hans/Floyd-Warshall%E7%AE%97%E6%B3%95) +4. [Bellman-Ford算法](https://www.wikiwand.com/zh-hans/%E8%B4%9D%E5%B0%94%E6%9B%BC-%E7%A6%8F%E7%89%B9%E7%AE%97%E6%B3%95) +5. [Kruskal算法](https://www.wikiwand.com/zh-hans/%E5%85%8B%E9%B2%81%E6%96%AF%E5%85%8B%E5%B0%94%E6%BC%94%E7%AE%97%E6%B3%95) +6. [Prim's 算法](https://www.wikiwand.com/zh-hans/%E6%99%AE%E6%9E%97%E5%A7%86%E7%AE%97%E6%B3%95) +7. [最小生成树](https://www.wikiwand.com/zh/%E6%9C%80%E5%B0%8F%E7%94%9F%E6%88%90%E6%A0%91) \ No newline at end of file diff --git a/problems/1168.optimize-water-distribution-in-a-village-en.md b/problems/1168.optimize-water-distribution-in-a-village-en.md new file mode 100644 index 000000000..3eac1ad67 --- /dev/null +++ b/problems/1168.optimize-water-distribution-in-a-village-en.md @@ -0,0 +1,194 @@ +## Problem + + + + +https://leetcode.com/problems/optimize-water-distribution-in-a-village/ +## Problem Description + +## 题目描述 +``` +There are n houses in a village. We want to supply water for all the houses by building wells and laying pipes. + +For each house i, we can either build a well inside it directly with cost wells[i], or pipe in water from another well to it. The costs to lay pipes between houses are given by the array pipes, where each pipes[i] = [house1, house2, cost] represents the cost to connect house1 and house2 together using a pipe. Connections are bidirectional. + +Find the minimum total cost to supply water to all houses. + +Example 1: + +Input: n = 3, wells = [1,2,2], pipes = [[1,2,1],[2,3,1]] +Output: 3 +Explanation: +The image shows the costs of connecting houses using pipes. +The best strategy is to build a well in the first house with cost 1 and connect the other houses to it with cost 2 so the total cost is 3. + +Constraints: + +1 <= n <= 10000 +wells.length == n +0 <= wells[i] <= 10^5 +1 <= pipes.length <= 10000 +1 <= pipes[i][0], pipes[i][1] <= n +0 <= pipes[i][2] <= 10^5 +pipes[i][0] != pipes[i][1] +``` +example 1 pic: + +![example 1](../assets/problems/1168.optimize-water-distribution-in-a-village-example1.png) + + +## Solution + +From example graph, we can see that this is Shortest path problem/Minimum spanning tree problem. In this problem, in a graph, view cities as nodes, pipe connects two cities as edges with cost. +here, wells costs, it is self connected edge, we can add extra node as root node `0`, and connect all `0` and `i` with costs `wells[i]`. So that we can have one graph/tree, +and how to get minimun spanning trees / shortest path problem in a graph. Please see below detailed steps for analysis. + +Analysis Steps: +1. Create `POJO EdgeCost(node1, node2, cost) - node1, node2, and cost of connect node1 and node2` +2. Assume on `root node 0`,build graph with `node 0 and all nodes(cities)` +3. Connect all nodes with`0`,`[0,i] - i is nodes range from [1,n]`,`0-1` meaning `node 0` and `node 1` connect edge ,value is node `i`'s cost `wells[i]`; +4. Turn cities into nodes, wells' costs and pipes' into costs into edges value which connected into two cities. +5. Sort all edges (from min to max) +6. Scan all edges, check whether 2 nodes connected or not:(`Union-Find`), + - if already connected, continue check next edge + - if not yet connected, +costs, connect 2 nodes +7. If all nodes already connected, get minimum costs, return result +8. (#7 Optimization) for each `union`, total nodes number `n-1`, if `n==0`, then meaning all nodes already connected, can terminate early. + +> Here use weighted-Union-find to check whether 2 nodes connceted or not, and union not connected nodes. + +For example:`n = 5, wells=[1,2,2,3,2], pipes=[[1,2,1],[2,3,1],[4,5,7]]` + +As below pic: + +![minimum cost](../assets/problems/1168.optimize-water-distribution-in-a-village-1.png) + +From pictures, we can see that all nodes already connected with minimum costs. + +#### Complexity Analysis +- *Time Complexity:* `O(ElogE) - E number of edge in graph` +- *Space Complexity:* `O(E)` + + + +> A graph at most have `n(n-1)/2 - n number of nodes in graph` edges ([Complete Graph](https://www.wikiwand.com/en/Complete_graph)) + +## Key Points +1. Build graph with all possible edges. +2. Sort edges by value (costs) +3. Iterate all edges (from min value to max value) +4. For each edges, check whether two nodes already connected (union-find), + - if already connected, then skip + - if not connected, then union two nodes, add costs to result + +## Code (`Java/Python3`) +*Java code* +```java + class OptimizeWaterDistribution { + public int minCostToSupplyWater(int n, int[] wells, int[][] pipes) { + List costs = new ArrayList<>(); + for (int i = 1; i <= n; i++) { + costs.add(new EdgeCost(0, i, wells[i - 1])); + } + for (int[] p : pipes) { + costs.add(new EdgeCost(p[0], p[1], p[2])); + } + Collections.sort(costs); + int minCosts = 0; + UnionFind uf = new UnionFind(n); + for (EdgeCost edge : costs) { + int rootX = uf.find(edge.node1); + int rootY = uf.find(edge.node2); + if (rootX == rootY) continue; + minCosts += edge.cost; + uf.union(edge.node1, edge.node2); + // for each union, we connnect one node + n--; + // if all nodes already connected, terminate early + if (n == 0) { + return minCosts; + } + } + return minCosts; + } + + class EdgeCost implements Comparable { + int node1; + int node2; + int cost; + public EdgeCost(int node1, int node2, int cost) { + this.node1 = node1; + this.node2 = node2; + this.cost = cost; + } + + @Override + public int compareTo(EdgeCost o) { + return this.cost - o.cost; + } + } + + class UnionFind { + int[] parent; + int[] rank; + public UnionFind(int n) { + parent = new int[n + 1]; + for (int i = 0; i <= n; i++) { + parent[i] = i; + } + rank = new int[n + 1]; + } + public int find(int x) { + return x == parent[x] ? x : find(parent[x]); + } + public void union(int x, int y) { + int px = find(x); + int py = find(y); + if (px == py) return; + if (rank[px] >= rank[py]) { + parent[py] = px; + rank[px] += rank[py]; + } else { + parent[px] = py; + rank[py] += rank[px]; + } + } + } + } +``` +*Pythong3 code* +```python +class Solution: + def minCostToSupplyWater(self, n: int, wells: List[int], pipes: List[List[int]]) -> int: + union_find = {i: i for i in range(n + 1)} + + def find(x): + return x if x == union_find[x] else find(union_find[x]) + + def union(x, y): + px = find(x) + py = find(y) + union_find[px] = py + + graph_wells = [[cost, 0, i] for i, cost in enumerate(wells, 1)] + graph_pipes = [[cost, i, j] for i, j, cost in pipes] + min_costs = 0 + for cost, x, y in sorted(graph_wells + graph_pipes): + if find(x) == find(y): + continue + union(x, y) + min_costs += cost + n -= 1 + if n == 0: + return min_costs +``` + +## External Resources + +1. [Shortest path problem](https://www.wikiwand.com/en/Shortest_path_problem) +2. [Dijkstra's algorithm](https://www.wikiwand.com/en/Dijkstra%27s_algorithm) +3. [Floyd–Warshall algorithm](https://www.wikiwand.com/en/Floyd%E2%80%93Warshall_algorithm) +4. [Bellman–Ford algorithm](https://www.wikiwand.com/en/Bellman%E2%80%93Ford_algorithm) +5. [Kruskal's algorithm](https://www.wikiwand.com/en/Kruskal%27s_algorithm) +6. [Prim's algorithm](https://www.wikiwand.com/en/Prim%27s_algorithm) +7. [Minimum spanning tree](https://www.wikiwand.com/en/Minimum_spanning_tree) \ No newline at end of file