[pull] master from wisdompeak:master

Binary_Search/3585.Find-Weighted-Median-Node-in-Tree/3585.Find-Weighted-Median-Node-in-Tree.cpp

@@ -0,0 +1,110 @@
+using ll = long long;
+const int MAXN = 100000;
+const int LOGN = 17;
+
+class Solution {
+public:
+    vector<pair<int,int>> adj[MAXN];
+    int up[MAXN][LOGN+1];
+    int depth[MAXN];
+    ll distRoot[MAXN];
+
+    void dfs(int cur, int parent)
+    {
+        up[cur][0] = parent;
+        for(auto &[v,w]: adj[cur])
+        {
+            if(v == parent) continue;
+            depth[v] = depth[cur] + 1;
+            distRoot[v] = distRoot[cur] + w;
+            dfs(v, cur);
+        }
+    }
+
+    int lca(int a, int b)
+    {
+        if(depth[a] < depth[b]) swap(a,b);
+        int diff = depth[a] - depth[b];
+        for(int k = 0; k <= LOGN; k++){
+            if(diff & (1<<k)) a = up[a][k];
+        }
+        if(a == b) return a;
+        for(int k = LOGN; k >= 0; k--){
+            if(up[a][k] != up[b][k]){
+                a = up[a][k];
+                b = up[b][k];
+            }
+        }
+        return up[a][0];
+    }
+
+    int stepUp(int u, int k) {        
+        for (int i=16; i>=0; i--) {
+            if ((k>>i)&1) {
+                u = up[u][i];
+            }
+        }
+        return u;
+    }
+
+    ll dist(int a, int b)
+    {
+        int c = lca(a,b);
+        return distRoot[a] + distRoot[b] - 2*distRoot[c];
+    }
+
+    vector<int> findMedian(int n, vector<vector<int>>& edges, vector<vector<int>>& queries) {
+        for (int i = 0; i < n-1; i++)
+        {
+            int u = edges[i][0], v = edges[i][1], w = edges[i][2];
+            adj[u].push_back({v,w});
+            adj[v].push_back({u,w});
+        }
+
+        depth[0] = 0;
+        distRoot[0] = 0;
+        dfs(0, 0);
+
+        for(int k = 1; k <= LOGN; k++) {
+            for(int v = 0; v < n; v++) {
+                up[v][k] = up[up[v][k-1]][k-1];
+            }
+        }
+
+        vector<int>rets;
+        for (auto& q: queries)
+        {                    
+            int u = q[0], v = q[1];
+            int c = lca(u,v);
+            ll total = dist(u,c)+dist(c,v);
+
+            int step1 = depth[u]-depth[c];
+            int step2 = depth[v]-depth[c];
+
+            int low = 0, high = step1+step2;
+            int k;
+            while (low < high) {
+                int mid = low + (high-low)/2;
+                ll d;
+                if (mid <= step1) {
+                    k = stepUp(u, mid);
+                    d = distRoot[u] - distRoot[k];
+                } else {
+                    k = stepUp(v, step2 - (mid-step1));
+                    d = total - (distRoot[v] - distRoot[k]);
+                }
+                if (d >= total*0.5) 
+                    high = mid;
+                else
+                    low = mid+1;
+            }
+            int step = low;
+            if (step<=step1)
+                rets.push_back(stepUp(u, step));
+            else
+                rets.push_back(stepUp(v, step2-(step-step1)));
+        }
+
+        return rets;
+    }
+};

Binary_Search/3585.Find-Weighted-Median-Node-in-Tree/Readme.md

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+### 3585.Find-Weighted-Median-Node-in-Tree
+
+对于任何一个query，我们只需要找到u到v路径（途中经过LCA的点记作c），假设路径的总步长是d，路径的总权重和是total。我们只需要在[0,d]之间进行二分搜索一个合适的步数k：即从u走k步，恰好走过的路径长度超过total的一半。
+
+注意，我们在二分搜索对k进行判定的时候，需要分类讨论k是否在u到c的路径上，还是c到v的路径上。即看是否`dist(u,c) >= total * 0.5`. 如果k是在u到c的路径上，那么经过的路径长度就是dist(u,k)。如果k是在c到v的路径上，那么经过的路径长度就是dist(u,c)+dist(c,k)。
+
+根据binary lifting的算法，树里任意两个节点之间的距离都可以用log(n)的时间求解。

Readme.md

@@ -108,6 +108,7 @@
 [2846.Minimum-Edge-Weight-Equilibrium-Queries-in-a-Tree](https://github.com/wisdompeak/LeetCode/tree/master/Binary_Search/2846.Minimum-Edge-Weight-Equilibrium-Queries-in-a-Tree) (H)      
 [2851.String-Transformation](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/2851.String-Transformation) (H+)      
 [3553.Minimum-Weighted-Subgraph-With-the-Required-Paths-II](https://github.com/wisdompeak/LeetCode/tree/master/Binary_Search/3553.Minimum-Weighted-Subgraph-With-the-Required-Paths-II) (H)      
+[3585.Find-Weighted-Median-Node-in-Tree](https://github.com/wisdompeak/LeetCode/tree/master/Binary_Search/3585.Find-Weighted-Median-Node-in-Tree) (H)      
 * ``Binary Search by Value``  
 [410.Split-Array-Largest-Sum](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/410.Split-Array-Largest-Sum) (H-)  
 [774.Minimize-Max-Distance-to-Gas-Station](https://github.com/wisdompeak/LeetCode/tree/master/Priority_Queue/774.Minimize-Max-Distance-to-Gas-Station) (H)