【树上倍增】【割点】 【换根法】3067. 在带权树网络中统计可连接服务器对数目

本文主要是介绍【树上倍增】【割点】 【换根法】3067. 在带权树网络中统计可连接服务器对数目，希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！

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视频算法专题

本文涉及知识点

树上倍增 树 图论 并集查找 换根法 深度优先
割点原理及封装好的割点类(预计2024年3月11号左右发布）

LeetCode3067. 在带权树网络中统计可连接服务器对数目

给你一棵无根带权树，树中总共有 n 个节点，分别表示 n 个服务器，服务器从 0 到 n - 1 编号。同时给你一个数组 edges ，其中 edges[i] = [ai, bi, weighti] 表示节点 ai 和 bi 之间有一条双向边，边的权值为 weighti 。再给你一个整数 signalSpeed 。
如果两个服务器 a ，b 和 c 满足以下条件，那么我们称服务器 a 和 b 是通过服务器 c 可连接的 ：
a < b ，a != c 且 b != c 。
从 c 到 a 的距离是可以被 signalSpeed 整除的。
从 c 到 b 的距离是可以被 signalSpeed 整除的。
从 c 到 b 的路径与从 c 到 a 的路径没有任何公共边。
请你返回一个长度为 n 的整数数组 count ，其中 count[i] 表示通过服务器 i 可连接 的服务器对的 数目 。

示例 1：

输入：edges = [[0,1,1],[1,2,5],[2,3,13],[3,4,9],[4,5,2]], signalSpeed = 1
输出：[0,4,6,6,4,0]
解释：由于 signalSpeed 等于 1 ，count[c] 等于所有从 c 开始且没有公共边的路径对数目。
在输入图中，count[c] 等于服务器 c 左边服务器数目乘以右边服务器数目。
示例 2：

输入：edges = [[0,6,3],[6,5,3],[0,3,1],[3,2,7],[3,1,6],[3,4,2]], signalSpeed = 3
输出：[2,0,0,0,0,0,2]
解释：通过服务器 0 ，有 2 个可连接服务器对(4, 5) 和 (4, 6) 。
通过服务器 6 ，有 2 个可连接服务器对 (4, 5) 和 (0, 5) 。
所有服务器对都必须通过服务器 0 或 6 才可连接，所以其他服务器对应的可连接服务器对数目都为 0 。

提示：

2 <= n <= 1000
edges.length == n - 1
edges[i].length == 3
0 <= ai, bi < n
edges[i] = [ai, bi, weighti]
1 <= weighti <= 10⁶
1 <= signalSpeed <= 10⁶
输入保证 edges 构成一棵合法的树。

树上倍增

本题有三个考点：
一，如何计算树上两个节点x1,x2的距离。
假定这两个节点的最早公共祖先是pub。以任意节点root为根，f(x)表示节点x到root的距离。
x1到x2的距离：f(x1)+f(x2)-2*f(pub)。
二，如何找到最早公共祖先：树上倍增。
记录各节点的1级祖先（父节点）、2级祖先、4级祖先…
三，如果判断ac和bc有公共边。
树是连通无向无环图，因为无环，所以两个节点的路径唯一。
假设公共边是x3x4。则x3到c的路径唯一，假定x3到c的倒数第二个端点是x5，则ab和bc的最后一条边都是$\overrightarrow{x3c}$。断开所以和c相连的边，如果a和b在同一个连通区域，则有公共边。用并查集看是否在同一个连通区域。
时间复杂度： O(nnlogn)。 枚举c，时间复杂度O(n)；枚举ab，时间复杂度O(n)。查公共路径O(logn)。

并集查找

class CNeiBo
{
public:	static vector<vector<int>> Two(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0) {vector<vector<int>>  vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase);}}return vNeiBo;}	static vector<vector<std::pair<int, int>>> Three(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0){vector<vector<std::pair<int, int>>> vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase, v[2]);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase, v[2]);}}return vNeiBo;}
};
class CUnionFind
{
public:CUnionFind(int iSize) :m_vNodeToRegion(iSize){for (int i = 0; i < iSize; i++){m_vNodeToRegion[i] = i;}m_iConnetRegionCount = iSize;}	CUnionFind(vector<vector<int>>& vNeiBo):CUnionFind(vNeiBo.size()){for (int i = 0; i < vNeiBo.size(); i++) {for (const auto& n : vNeiBo[i]) {Union(i, n);}}}int GetConnectRegionIndex(int iNode){int& iConnectNO = m_vNodeToRegion[iNode];if (iNode == iConnectNO){return iNode;}return iConnectNO = GetConnectRegionIndex(iConnectNO);}void Union(int iNode1, int iNode2){const int iConnectNO1 = GetConnectRegionIndex(iNode1);const int iConnectNO2 = GetConnectRegionIndex(iNode2);if (iConnectNO1 == iConnectNO2){return;}m_iConnetRegionCount--;if (iConnectNO1 > iConnectNO2){UnionConnect(iConnectNO1, iConnectNO2);}else{UnionConnect(iConnectNO2, iConnectNO1);}}bool IsConnect(int iNode1, int iNode2){return GetConnectRegionIndex(iNode1) == GetConnectRegionIndex(iNode2);}int GetConnetRegionCount()const{return m_iConnetRegionCount;}vector<int> GetNodeCountOfRegion()//各联通区域的节点数量{const int iNodeSize = m_vNodeToRegion.size();vector<int> vRet(iNodeSize);for (int i = 0; i < iNodeSize; i++){vRet[GetConnectRegionIndex(i)]++;}return vRet;}std::unordered_map<int, vector<int>> GetNodeOfRegion(){std::unordered_map<int, vector<int>> ret;const int iNodeSize = m_vNodeToRegion.size();for (int i = 0; i < iNodeSize; i++){ret[GetConnectRegionIndex(i)].emplace_back(i);}return ret;}
private:void UnionConnect(int iFrom, int iTo){m_vNodeToRegion[iFrom] = iTo;}vector<int> m_vNodeToRegion;//各点所在联通区域的索引,本联通区域任意一点的索引，为了增加可理解性，用最小索引int m_iConnetRegionCount;
};class CParents
{
public:CParents(vector<int>& vParent, const vector<int>& vLeve):m_vLeve(vLeve){const int iMaxLeve = *std::max_element(vLeve.begin(), vLeve.end());int iBitNum = 0;for (; (1 << iBitNum) < iMaxLeve; iBitNum++);const int n = vParent.size();m_vParents.assign(iBitNum+1, vector<int>(n, -1));m_vParents[0] = vParent;//树上倍增for (int i = 1; i < m_vParents.size(); i++){for (int j = 0; j < n; j++){const int iPre = m_vParents[i - 1][j];if (-1 != iPre){m_vParents[i][j] = m_vParents[i - 1][iPre];}}}}int GetParent(int iNode, int iLeve)const{int iParent = iNode;for (int iBit = 0; iBit < m_vParents.size(); iBit++){if (-1 == iParent){return iParent;}if (iLeve & (1 << iBit)){iParent = m_vParents[iBit][iParent];}}return iParent;}int GetPublicParent(int iNode1, int iNode2)const{int leve0 = m_vLeve[iNode1];int leve1 = m_vLeve[iNode2];if (leve0 < leve1){iNode2 = GetParent(iNode2, leve1 - leve0);leve1 = leve0;}else{iNode1 = GetParent(iNode1, leve0 - leve1);leve0 = leve1;}//二分查找int left = -1, r = leve0;while (r - left > 1){const auto mid = left + (r - left) / 2;const int iParent0 = GetParent(iNode1, mid);const int iParent1 = GetParent(iNode2, mid);if (iParent0 == iParent1){r = mid;}else{left = mid;}}return GetParent(iNode1, r);}
protected:vector<vector<int>> m_vParents;const vector<int> m_vLeve;
};class Solution {
public:vector<int> countPairsOfConnectableServers(vector<vector<int>>& edges, int signalSpeed) {m_c = edges.size() + 1;m_vDisToRoot.resize(m_c);m_vParent.resize(m_c);m_vLeve.resize(m_c);auto neiBo = CNeiBo::Three(m_c, edges, false, 0);DFS(neiBo, 0, -1, 0,0);	CParents par(m_vParent,m_vLeve);vector<int> vRet(m_c);for (int c = 0; c < m_c; c++){CUnionFind uf(m_c);for (const auto& v : edges){if ((v[0] == c) || (v[1] == c)){continue;}uf.Union(v[0], v[1]);}vector<int> vRegionCnt(m_c);for (int ab = 0; ab < m_c; ab++){if (ab == c ){continue;}const int pub = par.GetPublicParent(ab, c);const int len = m_vDisToRoot[ab] + m_vDisToRoot[c] - 2 * m_vDisToRoot[pub];if (0 != len % signalSpeed){continue;}vRegionCnt[uf.GetConnectRegionIndex(ab)]++;}int&iRet = vRet[c];const int total = std::accumulate(vRegionCnt.begin(), vRegionCnt.end(), 0);for (int c1 = 0; c1 < m_c; c1++){iRet += vRegionCnt[c1] * (total - vRegionCnt[c1]);}	iRet /= 2;}return vRet;}void DFS(vector<vector<std::pair<int, int>>>& neiBo, int cur, int par,int leve,int dis){m_vDisToRoot[cur] =dis;m_vParent[cur] = par;m_vLeve[cur] = leve;for (const auto& [next,len] : neiBo[cur]){if (next == par){continue;}DFS(neiBo, next, cur,leve+1,dis+len);}}vector<int> m_vDisToRoot,m_vParent,m_vLeve;int m_c;
};

测试用例

template<class T,class T2>
void Assert(const T& t1, const T2& t2)
{assert(t1 == t2);
}template<class T>
void Assert(const vector<T>& v1, const vector<T>& v2)
{if (v1.size() != v2.size()){assert(false);return;}for (int i = 0; i < v1.size(); i++){Assert(v1[i], v2[i]);}}int main()
{vector<vector<int>> edges;int signalSpeed;{Solution sln;edges = {  }, signalSpeed = 1;auto res = sln.countPairsOfConnectableServers(edges, signalSpeed);Assert({ 0 }, res);}{Solution sln;edges = { {0,1,1} }, signalSpeed = 1;auto res = sln.countPairsOfConnectableServers(edges, signalSpeed);Assert({ 0,0 }, res);}{Solution sln;edges = { {0,1,1},{1,2,1} }, signalSpeed = 1;auto res = sln.countPairsOfConnectableServers(edges, signalSpeed);Assert({ 0,1,0 }, res);}{Solution sln;edges = { {0,1,1},{1,2,5},{2,3,13},{3,4,9},{4,5,2} }, signalSpeed = 1;auto res = sln.countPairsOfConnectableServers(edges, signalSpeed);Assert({ 0,4,6,6,4,0 } , res);}{Solution sln;edges = { {0,6,3},{6,5,3},{0,3,1},{3,2,7},{3,1,6},{3,4,2} }, signalSpeed = 3;auto res = sln.countPairsOfConnectableServers(edges, signalSpeed);Assert({ 2,0,0,0,0,0,2 }, res);}
}

换根法DFS

树中删除一个节点，则各孩子各一个连通区域，除自己及后代外一个区域。如果这个节点是根，则简单得多。各孩子一个连通区域。
DSF(cur) 返回自己及子孙到当前根节点距离是signalSpeed 倍的节点数量。令当前根节点各孩子的返回值是{i1,i2,$\cdots$,im} 。i1*i2+(i1+i2)*i3 $\cdots$ +(I1+i2+$\dots$ + i${}_{m-1}$)*im。这样不必除以二。
a<b ，表示a !=b ，(a,b)和(b,a)只取一个。

class CNeiBo
{
public:	static vector<vector<int>> Two(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0) {vector<vector<int>>  vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase);}}return vNeiBo;}	static vector<vector<std::pair<int, int>>> Three(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0){vector<vector<std::pair<int, int>>> vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase, v[2]);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase, v[2]);}}return vNeiBo;}
};class Solution {
public:vector<int> countPairsOfConnectableServers(vector<vector<int>>& edges, int signalSpeed) {m_c = edges.size() + 1;m_iSignalSpeed = signalSpeed;		auto neiBo = CNeiBo::Three(m_c, edges, false, 0);		vector<int> vRet(m_c);for (int c = 0; c < m_c; c++){int& iRet = vRet[c];int left = 0;for (const auto& [next, len] : neiBo[c]){int cur = DFS(neiBo, next, c, len);iRet += left * cur;left += cur;}}return vRet;}int DFS(vector<vector<std::pair<int, int>>>& neiBo, int cur, int par,int dis){int iRet = (0 ==dis % m_iSignalSpeed);for (const auto& [next,len] : neiBo[cur]){if (next == par){continue;}iRet +=DFS(neiBo, next, cur,dis+len);}return iRet;}int m_iSignalSpeed;int m_c;
};

割点

本解法过于复杂，除非用了提前封装好的割点扩展类，否则被使用。

class CNeiBo
{
public:	static vector<vector<int>> Two(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0) {vector<vector<int>>  vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase);}}return vNeiBo;}	static vector<vector<std::pair<int, int>>> Three(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0){vector<vector<std::pair<int, int>>> vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase, v[2]);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase, v[2]);}}return vNeiBo;}
};
class CUnionFind
{
public:CUnionFind(int iSize) :m_vNodeToRegion(iSize){for (int i = 0; i < iSize; i++){m_vNodeToRegion[i] = i;}m_iConnetRegionCount = iSize;}	CUnionFind(vector<vector<int>>& vNeiBo):CUnionFind(vNeiBo.size()){for (int i = 0; i < vNeiBo.size(); i++) {for (const auto& n : vNeiBo[i]) {Union(i, n);}}}int GetConnectRegionIndex(int iNode){int& iConnectNO = m_vNodeToRegion[iNode];if (iNode == iConnectNO){return iNode;}return iConnectNO = GetConnectRegionIndex(iConnectNO);}void Union(int iNode1, int iNode2){const int iConnectNO1 = GetConnectRegionIndex(iNode1);const int iConnectNO2 = GetConnectRegionIndex(iNode2);if (iConnectNO1 == iConnectNO2){return;}m_iConnetRegionCount--;if (iConnectNO1 > iConnectNO2){UnionConnect(iConnectNO1, iConnectNO2);}else{UnionConnect(iConnectNO2, iConnectNO1);}}bool IsConnect(int iNode1, int iNode2){return GetConnectRegionIndex(iNode1) == GetConnectRegionIndex(iNode2);}int GetConnetRegionCount()const{return m_iConnetRegionCount;}vector<int> GetNodeCountOfRegion()//各联通区域的节点数量{const int iNodeSize = m_vNodeToRegion.size();vector<int> vRet(iNodeSize);for (int i = 0; i < iNodeSize; i++){vRet[GetConnectRegionIndex(i)]++;}return vRet;}std::unordered_map<int, vector<int>> GetNodeOfRegion(){std::unordered_map<int, vector<int>> ret;const int iNodeSize = m_vNodeToRegion.size();for (int i = 0; i < iNodeSize; i++){ret[GetConnectRegionIndex(i)].emplace_back(i);}return ret;}
private:void UnionConnect(int iFrom, int iTo){m_vNodeToRegion[iFrom] = iTo;}vector<int> m_vNodeToRegion;//各点所在联通区域的索引,本联通区域任意一点的索引，为了增加可理解性，用最小索引int m_iConnetRegionCount;
};class CParents
{
public:CParents(vector<int>& vParent, const int iMaxLeve){	int iBitNum = 0;for (; (1 << iBitNum) < iMaxLeve; iBitNum++);const int n = vParent.size();m_vParents.assign(iBitNum+1, vector<int>(n, -1));m_vParents[0] = vParent;//树上倍增for (int i = 1; i < m_vParents.size(); i++){for (int j = 0; j < n; j++){const int iPre = m_vParents[i - 1][j];if (-1 != iPre){m_vParents[i][j] = m_vParents[i - 1][iPre];}}}}int GetParent(int iNode, int iLeve)const{int iParent = iNode;for (int iBit = 0; iBit < m_vParents.size(); iBit++){if (-1 == iParent){return iParent;}if (iLeve & (1 << iBit)){iParent = m_vParents[iBit][iParent];}}return iParent;}	
protected:vector<vector<int>> m_vParents;
};class C2Parents : CParents
{
public:C2Parents(vector<int>& vParent, const vector<int>& vLeve) :m_vLeve(vLeve), CParents(vParent,*std::max_element(vLeve.begin(), vLeve.end())){		}	int GetPublicParent(int iNode1, int iNode2)const{int leve0 = m_vLeve[iNode1];int leve1 = m_vLeve[iNode2];if (leve0 < leve1){iNode2 = GetParent(iNode2, leve1 - leve0);leve1 = leve0;}else{iNode1 = GetParent(iNode1, leve0 - leve1);leve0 = leve1;}//二分查找int left = -1, r = leve0;while (r - left > 1){const auto mid = left + (r - left) / 2;const int iParent0 = GetParent(iNode1, mid);const int iParent1 = GetParent(iNode2, mid);if (iParent0 == iParent1){r = mid;}else{left = mid;}}return GetParent(iNode1, r);}
protected:vector<vector<int>> m_vParents;const vector<int> m_vLeve;
};class CCutPointEx
{
public:CCutPointEx(const vector<vector<int>>& vNeiB) : m_iSize(vNeiB.size()){m_vNodeToTime.assign(m_iSize, -1);m_vChildFirstEnd.resize(m_iSize);m_vNodeToRegion.assign(m_iSize, -1);m_vCut.assign(m_iSize, false);for (int i = 0; i < m_iSize; i++){if (-1 != m_vNodeToTime[i]){continue;}dfs(i, -1, vNeiB);m_iRegionCount++;}m_vTimeToNode.resize(m_iSize);for (int i = 0; i < m_iSize; i++){m_vTimeToNode[m_vNodeToTime[i]] = i;;}}bool Visit(int src, int dest, int iCut)const{if (m_vNodeToRegion[src] != m_vNodeToRegion[dest]){return false;//不在一个连通区域}if (!m_vCut[iCut]){return true;}const int r1 = GetCutRegion(iCut, src);const int r2 = GetCutRegion(iCut, dest);return r1 == r2;}vector<vector<int>> GetSubRegionOfCut(const int iCut)const{//删除iCut及和它相连的边后，iCut所在的区域会分成几个区域：父节点一个区域、各子节点一个区域//父节点所在区域可能为空，如果iCut所在的连通区域只有一个节点，则返回一个没有节点的区域。const auto& v = m_vChildFirstEnd[iCut];vector<vector<int>> vRet(1);			int j = 0;for (int iTime=0;iTime < m_iSize; iTime++ ){const int iNode	= m_vTimeToNode[iTime];if ((j < v.size()) && ( iTime  >= v[j].first )){j++;vRet.emplace_back();}if ((iCut != iNode) && (m_vNodeToRegion[iNode] == m_vNodeToRegion[iCut])){if (v.size()&&(iTime >= v.back().second)){vRet[0].emplace_back(iNode);}else{vRet.back().emplace_back(iNode);}}}return vRet;}
protected:int dfs(int cur, int parent, const vector<vector<int>>& vNeiB){auto& curTime = m_vNodeToTime[cur];m_vNodeToRegion[cur] = m_iRegionCount;curTime = m_iTime++;int iCutChild = 0;int iMinTime = curTime;for (const auto& next : vNeiB[cur]){if (-1 != m_vNodeToTime[next]){iMinTime = min(iMinTime, m_vNodeToTime[next]);continue;}int iChildBeginTime = m_iTime;const int iChildMinTime = dfs(next, cur, vNeiB);iMinTime = min(iMinTime, iChildMinTime);if (iChildMinTime >= curTime){iCutChild++;m_vChildFirstEnd[cur].push_back({ iChildBeginTime,m_iTime });};}m_vCut[cur] = (iCutChild + (-1 != parent)) >= 2;return iMinTime;}int GetCutRegion(int iCut, int iNode)const{const auto& v = m_vChildFirstEnd[iCut];auto it = std::upper_bound(v.begin(), v.end(), m_vNodeToTime[iNode], [](int time, const std::pair<int, int>& pr) {return time < pr.first; });if (v.begin() == it){return v.size();}--it;return (it->second > m_vNodeToTime[iNode]) ? (it - v.begin()) : v.size();}int m_iTime = 0;const int m_iSize;//时间戳int m_iRegionCount = 0;vector<int> m_vNodeToTime;//各节点到达时间，从0开始。 -1表示未处理vector<bool> m_vCut;//是否是割点vector<int> m_vNodeToRegion;//各节点所在区域vector<vector<pair<int, int>>> m_vChildFirstEnd;//左闭右开空间[0,m_vChildFirstEnd[0].first)和[m_vChildFirstEnd.back().second,iSize)是一个区域vector<int> m_vTimeToNode;
};

2024年3月9号 新封装

class CNeiBo
{
public:	static vector<vector<int>> Two(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0) {vector<vector<int>>  vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase);}}return vNeiBo;}	static vector<vector<std::pair<int, int>>> Three(int n, vector<vector<int>>& edges, bool bDirect, int iBase = 0){vector<vector<std::pair<int, int>>> vNeiBo(n);for (const auto& v : edges){vNeiBo[v[0] - iBase].emplace_back(v[1] - iBase, v[2]);if (!bDirect){vNeiBo[v[1] - iBase].emplace_back(v[0] - iBase, v[2]);}}return vNeiBo;}
};
class CUnionFind
{
public:CUnionFind(int iSize) :m_vNodeToRegion(iSize){for (int i = 0; i < iSize; i++){m_vNodeToRegion[i] = i;}m_iConnetRegionCount = iSize;}	CUnionFind(vector<vector<int>>& vNeiBo):CUnionFind(vNeiBo.size()){for (int i = 0; i < vNeiBo.size(); i++) {for (const auto& n : vNeiBo[i]) {Union(i, n);}}}int GetConnectRegionIndex(int iNode){int& iConnectNO = m_vNodeToRegion[iNode];if (iNode == iConnectNO){return iNode;}return iConnectNO = GetConnectRegionIndex(iConnectNO);}void Union(int iNode1, int iNode2){const int iConnectNO1 = GetConnectRegionIndex(iNode1);const int iConnectNO2 = GetConnectRegionIndex(iNode2);if (iConnectNO1 == iConnectNO2){return;}m_iConnetRegionCount--;if (iConnectNO1 > iConnectNO2){UnionConnect(iConnectNO1, iConnectNO2);}else{UnionConnect(iConnectNO2, iConnectNO1);}}bool IsConnect(int iNode1, int iNode2){return GetConnectRegionIndex(iNode1) == GetConnectRegionIndex(iNode2);}int GetConnetRegionCount()const{return m_iConnetRegionCount;}vector<int> GetNodeCountOfRegion()//各联通区域的节点数量{const int iNodeSize = m_vNodeToRegion.size();vector<int> vRet(iNodeSize);for (int i = 0; i < iNodeSize; i++){vRet[GetConnectRegionIndex(i)]++;}return vRet;}std::unordered_map<int, vector<int>> GetNodeOfRegion(){std::unordered_map<int, vector<int>> ret;const int iNodeSize = m_vNodeToRegion.size();for (int i = 0; i < iNodeSize; i++){ret[GetConnectRegionIndex(i)].emplace_back(i);}return ret;}
private:void UnionConnect(int iFrom, int iTo){m_vNodeToRegion[iFrom] = iTo;}vector<int> m_vNodeToRegion;//各点所在联通区域的索引,本联通区域任意一点的索引，为了增加可理解性，用最小索引int m_iConnetRegionCount;
};class CParents
{
public:CParents(vector<int>& vParent, const int iMaxLeve){	int iBitNum = 0;for (; (1 << iBitNum) < iMaxLeve; iBitNum++);const int n = vParent.size();m_vParents.assign(iBitNum+1, vector<int>(n, -1));m_vParents[0] = vParent;//树上倍增for (int i = 1; i < m_vParents.size(); i++){for (int j = 0; j < n; j++){const int iPre = m_vParents[i - 1][j];if (-1 != iPre){m_vParents[i][j] = m_vParents[i - 1][iPre];}}}}int GetParent(int iNode, int iLeve)const{int iParent = iNode;for (int iBit = 0; iBit < m_vParents.size(); iBit++){if (-1 == iParent){return iParent;}if (iLeve & (1 << iBit)){iParent = m_vParents[iBit][iParent];}}return iParent;}	
protected:vector<vector<int>> m_vParents;
};class C2Parents : CParents
{
public:C2Parents(vector<int>& vParent, const vector<int>& vLeve) :m_vLeve(vLeve), CParents(vParent,*std::max_element(vLeve.begin(), vLeve.end())){		}	int GetPublicParent(int iNode1, int iNode2)const{int leve0 = m_vLeve[iNode1];int leve1 = m_vLeve[iNode2];if (leve0 < leve1){iNode2 = GetParent(iNode2, leve1 - leve0);leve1 = leve0;}else{iNode1 = GetParent(iNode1, leve0 - leve1);leve0 = leve1;}//二分查找int left = -1, r = leve0;while (r - left > 1){const auto mid = left + (r - left) / 2;const int iParent0 = GetParent(iNode1, mid);const int iParent1 = GetParent(iNode2, mid);if (iParent0 == iParent1){r = mid;}else{left = mid;}}return GetParent(iNode1, r);}
protected:vector<vector<int>> m_vParents;const vector<int> m_vLeve;
};//割点
class CCutPoint
{
public:CCutPoint(const vector<vector<int>>& vNeiB) : m_iSize(vNeiB.size()){m_vNodeToTime.assign(m_iSize, -1);m_vCutNewRegion.resize(m_iSize);for (int i = 0; i < m_iSize; i++){if (-1 == m_vNodeToTime[i]){m_vRegionFirstTime.emplace_back(m_iTime);dfs(vNeiB, i, -1);}}	}int dfs(const vector<vector<int>>& vNeiB,const int cur, const int parent){int iMinTime = m_vNodeToTime[cur] = m_iTime++;int iRegionCount = (-1 != parent);//根连通区域数量for (const auto& next : vNeiB[cur])		{if (-1  != m_vNodeToTime[next])			{iMinTime = min(iMinTime, m_vNodeToTime[next]);continue;}const int childMinTime = dfs(vNeiB, next, cur);iMinTime = min(iMinTime, childMinTime);if (childMinTime >= m_vNodeToTime[cur])			{iRegionCount++;m_vCutNewRegion[cur].emplace_back(m_vNodeToTime[next], m_iTime);}}if (iRegionCount < 2){m_vCutNewRegion[cur].clear();}return iMinTime;}const int m_iSize;const vector<int>& Time()const { return m_vNodeToTime; }//各节点的时间戳const vector<int>& RegionFirstTime()const { return m_vRegionFirstTime; }//各连通区域的最小时间戳vector<bool> Cut()const { vector<bool> ret;for (int i = 0; i < m_iSize; i++){ret.emplace_back(m_vCutNewRegion[i].size());}return ret; }//const vector < vector<pair<int, int>>>& NewRegion()const { return m_vCutNewRegion; };
protected:vector<int> m_vNodeToTime;vector<int> m_vRegionFirstTime;vector < vector<pair<int, int>>> m_vCutNewRegion; //m_vCutNewRegion[c]如果存在[left,r) 表示割掉c后，时间戳[left,r)的节点会形成新区域int m_iTime = 0;
};class CConnectAfterCutPoint 
{
public:CConnectAfterCutPoint(const vector<vector<int>>& vNeiB) :m_ct(vNeiB){m_vTimeToNode.resize(m_ct.m_iSize);m_vNodeToRegion.resize(m_ct.m_iSize);for (int iNode = 0; iNode < m_ct.m_iSize; iNode++){m_vTimeToNode[m_ct.Time()[iNode]] = iNode;}for (int iTime = 0,iRegion= 0; iTime < m_ct.m_iSize; iTime++){if ((iRegion < m_ct.RegionFirstTime().size()) && (m_ct.RegionFirstTime()[iRegion] == iTime)){iRegion++;}m_vNodeToRegion[m_vTimeToNode[iTime]] = (iRegion - 1);}}bool Connect(int src, int dest, int iCut)const{if (m_vNodeToRegion[src] != m_vNodeToRegion[dest]){return false;//不在一个连通区域}if (0 == m_ct.NewRegion()[iCut].size()){//不是割点return true;}const int r1 = GetCutRegion(iCut, src);const int r2 = GetCutRegion(iCut, dest);return r1 == r2;}vector<vector<int>> GetSubRegionOfCut(const int iCut)const{//删除iCut及和它相连的边后，iCut所在的区域会分成几个区域：父节点一个区域、各子节点		一个区域//父节点所在区域可能为空，如果iCut所在的连通区域只有一个节点，则返回一个没有节点的			区域。const auto& v = m_ct.NewRegion()[iCut];vector<int> vParen;const int iRegion = m_vNodeToRegion[iCut];const int iEndTime = (iRegion + 1 == m_ct.RegionFirstTime().size()) ? m_ct.m_iSize : m_ct.RegionFirstTime()[iRegion+1];vector<vector<int>> vRet;	for (int iTime = m_ct.RegionFirstTime()[iRegion],j=-1; iTime < iEndTime; iTime++){if (iCut == m_vTimeToNode[iTime]){continue;}if ((j + 1 < v.size()) && (v[j + 1].first == iTime)){j++;vRet.emplace_back();}const int iNode = m_vTimeToNode[iTime];if ((-1 != j ) && (iTime >= v[j].first) && (iTime < v[j].second)){vRet.back().emplace_back(iNode);}else{vParen.emplace_back(iNode);}			}vRet.emplace_back();vRet.back().swap(vParen);return vRet;}	
protected:int GetCutRegion(int iCut, int iNode)const{const auto& v = m_ct.NewRegion()[iCut];auto it = std::upper_bound(v.begin(), v.end(), m_ct.Time()[iNode], [](int time, const std::pair<int, int>& pr) {return  time < pr.first; });if (v.begin() == it){return v.size();}--it;return (it->second > m_ct.Time()[iNode]) ? (it - v.begin()) : v.size();}vector<int> m_vTimeToNode;vector<int> m_vNodeToRegion;//各节点所在区域const CCutPoint m_ct;
};class Solution {
public:vector<int> countPairsOfConnectableServers(vector<vector<int>>& edges, int signalSpeed) {m_c = edges.size() + 1;m_vDisToRoot.resize(m_c);m_vParent.resize(m_c);m_vLeve.resize(m_c);auto neiBo = CNeiBo::Three(m_c, edges, false, 0);DFS(neiBo, 0, -1, 0, 0);C2Parents par(m_vParent, m_vLeve);auto neiBo2 = CNeiBo::Two(m_c, edges, false, 0);CConnectAfterCutPoint cut(neiBo2);vector<int> vRet(m_c);for (int c = 0; c < m_c; c++){auto regs = cut.GetSubRegionOfCut(c);int left = 0;int& iRet = vRet[c];for (const auto& subRegion : regs){int cur = 0;for (const auto& ab : subRegion){const int pub = par.GetPublicParent(ab, c);const int len = m_vDisToRoot[ab] + m_vDisToRoot[c] - 2 * m_vDisToRoot[pub];if (0 != len % signalSpeed){continue;}cur++;}iRet += left * cur;left += cur;}}return vRet;}void DFS(vector<vector<std::pair<int, int>>>& neiBo, int cur, int par, int leve, int dis){m_vDisToRoot[cur] = dis;m_vParent[cur] = par;m_vLeve[cur] = leve;for (const auto& [next, len] : neiBo[cur]){if (next == par){continue;}DFS(neiBo, next, cur, leve + 1, dis + len);}}vector<int> m_vDisToRoot, m_vParent, m_vLeve;int m_c;
};

DFS代替树上倍增

时间复杂度的瓶颈在 树上倍增。可以直接DFS 最近公共祖先。

扩展阅读

视频课程

有效学习：明确的目标 及时的反馈 拉伸区（难度合适），可以先学简单的课程，请移步CSDN学院，听白银讲师（也就是鄙人）的讲解。
https://edu.csdn.net/course/detail/38771

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https://edu.csdn.net/lecturer/6176

相关

下载

想高屋建瓴的学习算法，请下载《喜缺全书算法册》doc版
https://download.csdn.net/download/he_zhidan/88348653

我想对大家说的话
闻缺陷则喜是一个美好的愿望，早发现问题，早修改问题，给老板节约钱。
子墨子言之：事无终始，无务多业。也就是我们常说的专业的人做专业的事。
如果程序是一条龙，那算法就是他的是睛

测试环境

操作系统：win7 开发环境： VS2019 C++17
或者 操作系统：win10 开发环境： VS2022 **C+

+17**
如无特殊说明，本算法用**C++**实现。

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