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C++算法：前缀和、前缀乘积、前缀异或的原理、源码及测试用例 包括课程视频
C++DFS
树上倍增 LCA

P10391 [蓝桥杯 2024 省 A] 零食采购

题目描述

小蓝准备去星际旅行，出发前想在本星系采购一些零食，星系内有 $n$ 颗星球，由 $n-1$ 条航路连接为连通图，第 $i$ 颗星球卖第 $c_i$ 种零食特产。小蓝想出了 $q$ 个采购方案，第 $i$ 个方案的起点为星球 $s_i$ ，终点为星球 $t_i$ ，对于每种采购方案，小蓝将从起点走最短的航路到终点，并且可以购买所有经过的星球上的零食（包括起点终点），请计算每种采购方案最多能买多少种不同的零食。

输入格式

输入的第一行包含两个正整数 $n$，$q$，用一个空格分隔。
第二行包含 $n$ 个整数 $c_1,c_2,\cdots ,c_n$，相邻整数之间使用一个空格分隔。
接下来 $n-1$ 行，第 $i$ 行包含两个整数 $u_i,v_i$，用一个空格分隔，表示一条
航路将星球 $u_i$ 与 $v_i$ 相连。
接下来 $q$ 行，第 $i$ 行包含两个整数 $s_i
, t_i $，用一个空格分隔，表示一个采购方案。

输出格式

输出 $q$ 行，每行包含一个整数，依次表示每个采购方案的答案。

输入输出样例 #1

输入 #1

4 2
1 2 3 1
1 2
1 3
2 4
4 3
1 4

输出 #1

3
2

说明/提示

第一个方案路线为 {4,2,1,3}\{4, 2, 1, 3\}{4,2,1,3}，可以买到第 $1,2,3$ 种零食；
第二个方案路线为 {1,2,4}\{1, 2, 4\}{1,2,4}，可以买到第 $1,2$ 种零食。

对于 20% 的评测用例，$1 ≤ n, q ≤ 5000 $；
对于所有评测用例，$1\le n,q\le 10^5，1\le c_i\le 20，1\le u_i,v_i\le n，1\le s_i,t_i\le n$。

DFS 树上前缀和 LCA

以1(0)为根，cnt[i][j]记录第j个星球是否有货物i。preSum[i][j]，节点j到根节点整个路径包括货物i的星球数量。初始化只需要一次DFS。时间复杂度：O(20n)
每次查询，令u和v的最近公共祖先g，如果preSum[i][u]+preSum[i][v]-2preSum[i][g]+cnt[i][g] > 0，则可以买到货物i。
是否复杂度：O(20qlogn)

代码

核心代码

#include <iostream>
#include <sstream>
#include <vector>
#include<map>
#include<unordered_map>
#include<set>
#include<unordered_set>
#include<string>
#include<algorithm>
#include<functional>
#include<queue>
#include <stack>
#include<iomanip>
#include<numeric>
#include <math.h>
#include <climits>
#include<assert.h>
#include<cstring>
#include<list>
#include<array>#include <bitset>
using namespace std;template<class T1, class T2>
std::istream& operator >> (std::istream& in, pair<T1, T2>& pr) {in >> pr.first >> pr.second;return in;
}template<class T1, class T2, class T3 >
std::istream& operator >> (std::istream& in, tuple<T1, T2, T3>& t) {in >> get<0>(t) >> get<1>(t) >> get<2>(t);return in;
}template<class T1, class T2, class T3, class T4 >
std::istream& operator >> (std::istream& in, tuple<T1, T2, T3, T4>& t) {in >> get<0>(t) >> get<1>(t) >> get<2>(t) >> get<3>(t);return in;
}template<class T1, class T2, class T3, class T4, class T5, class T6, class T7 >
std::istream& operator >> (std::istream& in, tuple<T1, T2, T3, T4,T5,T6,T7>& t) {in >> get<0>(t) >> get<1>(t) >> get<2>(t) >> get<3>(t) >> get<4>(t) >> get<5>(t) >> get<6>(t);return in;
}template<class T = int>
vector<T> Read() {int n;cin >> n;vector<T> ret(n);for (int i = 0; i < n; i++) {cin >> ret[i];}return ret;
}
template<class T = int>
vector<T> ReadNotNum() {vector<T> ret;T tmp;while (cin >> tmp) {ret.emplace_back(tmp);if ('\n' == cin.get()) { break; }}return ret;
}template<class T = int>
vector<T> Read(int n) {vector<T> ret(n);for (int i = 0; i < n; i++) {cin >> ret[i];}return ret;
}template<int N = 1'000'000>
class COutBuff
{
public:COutBuff() {m_p = puffer;}template<class T>void write(T x) {int num[28], sp = 0;if (x < 0)*m_p++ = '-', x = -x;if (!x)*m_p++ = 48;while (x)num[++sp] = x % 10, x /= 10;while (sp)*m_p++ = num[sp--] + 48;AuotToFile();}void writestr(const char* sz) {strcpy(m_p, sz);m_p += strlen(sz);AuotToFile();}inline void write(char ch){*m_p++ = ch;AuotToFile();}inline void ToFile() {fwrite(puffer, 1, m_p - puffer, stdout);m_p = puffer;}~COutBuff() {ToFile();}
private:inline void AuotToFile() {if (m_p - puffer > N - 100) {ToFile();}}char  puffer[N], * m_p;
};template<int N = 1'000'000>
class CInBuff
{
public:inline CInBuff() {}inline CInBuff<N>& operator>>(char& ch) {FileToBuf();while (('\r' == *S) || ('\n' == *S) || (' ' == *S)) { S++; }//忽略空格和回车ch = *S++;return *this;}inline CInBuff<N>& operator>>(int& val) {FileToBuf();int x(0), f(0);while (!isdigit(*S))f |= (*S++ == '-');while (isdigit(*S))x = (x << 1) + (x << 3) + (*S++ ^ 48);val = f ? -x : x; S++;//忽略空格换行		return *this;}inline CInBuff& operator>>(long long& val) {FileToBuf();long long x(0); int f(0);while (!isdigit(*S))f |= (*S++ == '-');while (isdigit(*S))x = (x << 1) + (x << 3) + (*S++ ^ 48);val = f ? -x : x; S++;//忽略空格换行return *this;}template<class T1, class T2>inline CInBuff& operator>>(pair<T1, T2>& val) {*this >> val.first >> val.second;return *this;}template<class T1, class T2, class T3>inline CInBuff& operator>>(tuple<T1, T2, T3>& val) {*this >> get<0>(val) >> get<1>(val) >> get<2>(val);return *this;}template<class T1, class T2, class T3, class T4>inline CInBuff& operator>>(tuple<T1, T2, T3, T4>& val) {*this >> get<0>(val) >> get<1>(val) >> get<2>(val) >> get<3>(val);return *this;}template<class T = int>inline CInBuff& operator>>(vector<T>& val) {int n;*this >> n;val.resize(n);for (int i = 0; i < n; i++) {*this >> val[i];}return *this;}template<class T = int>vector<T> Read(int n) {vector<T> ret(n);for (int i = 0; i < n; i++) {*this >> ret[i];}return ret;}template<class T = int>vector<T> Read() {vector<T> ret;*this >> ret;return ret;}
private:inline void FileToBuf() {const int canRead = m_iWritePos - (S - buffer);if (canRead >= 100) { return; }if (m_bFinish) { return; }for (int i = 0; i < canRead; i++){buffer[i] = S[i];//memcpy出错			}m_iWritePos = canRead;buffer[m_iWritePos] = 0;S = buffer;int readCnt = fread(buffer + m_iWritePos, 1, N - m_iWritePos, stdin);if (readCnt <= 0) { m_bFinish = true; return; }m_iWritePos += readCnt;buffer[m_iWritePos] = 0;S = buffer;}int m_iWritePos = 0; bool m_bFinish = false;char buffer[N + 10], * S = buffer;
};class CNeiBo
{
public:static vector<vector<int>> Two(int n, const vector<pair<int, int>>& edges, bool bDirect, int iBase = 0){vector<vector<int>>  vNeiBo(n);for (const auto& [i1, i2] : edges){vNeiBo[i1 - iBase].emplace_back(i2 - iBase);if (!bDirect){vNeiBo[i2 - iBase].emplace_back(i1 - iBase);}}return vNeiBo;}static vector<vector<int>> Two(int n, const 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;}static vector<vector<std::pair<int, int>>> Three(int n, const vector<tuple<int, int, int>>& edges, bool bDirect, int iBase = 0){vector<vector<std::pair<int, int>>> vNeiBo(n);for (const auto& [u, v, w] : edges){vNeiBo[u - iBase].emplace_back(v - iBase, w);if (!bDirect){vNeiBo[v - iBase].emplace_back(u - iBase, w);}}return vNeiBo;}static vector<vector<int>> Mat(vector<vector<int>>& neiBoMat){vector<vector<int>> neiBo(neiBoMat.size());for (int i = 0; i < neiBoMat.size(); i++){for (int j = i + 1; j < neiBoMat.size(); j++){if (neiBoMat[i][j]){neiBo[i].emplace_back(j);neiBo[j].emplace_back(i);}}}return neiBo;}
};class CBFSLeve {
public:static vector<int> Leve(const vector<vector<int>>& neiBo, vector<int> start) {vector<int> leves(neiBo.size(), -1);for (const auto& s : start) {leves[s] = 0;}for (int i = 0; i < start.size(); i++) {for (const auto& next : neiBo[start[i]]) {if (-1 != leves[next]) { continue; }leves[next] = leves[start[i]] + 1;start.emplace_back(next);}}return leves;}template<class NextFun>static vector<int> Leve(int N, NextFun nextFun, vector<int> start) {vector<int> leves(N, -1);for (const auto& s : start) {leves[s] = 0;}for (int i = 0; i < start.size(); i++) {auto nexts = nextFun(start[i]);for (const auto& next : nexts) {if (-1 != leves[next]) { continue; }leves[next] = leves[start[i]] + 1;start.emplace_back(next);}}return leves;}static vector<vector<int>> LeveNodes(const vector<int>& leves) {const int iMaxLeve = *max_element(leves.begin(), leves.end());vector<vector<int>> ret(iMaxLeve + 1);for (int i = 0; i < leves.size(); i++) {ret[leves[i]].emplace_back(i);}return ret;};static vector<int> LeveSort(const vector<int>& leves) {const int iMaxLeve = *max_element(leves.begin(), leves.end());vector<vector<int>> leveNodes(iMaxLeve + 1);for (int i = 0; i < leves.size(); i++) {leveNodes[leves[i]].emplace_back(i);}vector<int> ret;for (const auto& v : leveNodes) {ret.insert(ret.end(), v.begin(), v.end());}return ret;};
};
class CParents
{
public:CParents(vector<int>& vParent, long long iMaxDepth){int iBitNum = 0;for (; iMaxDepth; iBitNum++) {const auto mask = 1LL << iBitNum;if (mask & iMaxDepth) { iMaxDepth = iMaxDepth ^ mask; }}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 iDepth)const{int iParent = iNode;for (int iBit = 0; iBit < m_vParents.size(); iBit++){if (-1 == iParent){return iParent;}if (iDepth & (1 << iBit)){iParent = m_vParents[iBit][iParent];}}return iParent;}inline int GetBitCnt()const { return m_vParents.size(); };inline const int& GetPow2Parent(int iNode, int n)const {return m_vParents[n][iNode];}//在leftNodeExclude的1到rightLeve级祖先中查找符合pr的最近祖先template<class _Pr>int FindFirst(int leftNodeExclude, int rightLeve, _Pr pr) {for (int iBit = GetBitCnt() - 1; iBit >= 0; iBit--) {const int iMask = 1 << iBit;if (!(iMask & rightLeve)) { continue; }if (pr(m_vParents[iBit][leftNodeExclude])) {return BFindFirst(leftNodeExclude, iBit, pr);}leftNodeExclude = m_vParents[iBit][leftNodeExclude];}return -1;}//在node的0到rightLeve级祖先中查找符合pr的最远祖先比node高多少层次,这些层次必须存在template<class _Pr>int FindEnd(int node, int rightLeve, _Pr pr) {int leve = 0;for (int iBit = GetBitCnt() - 1; iBit >= 0; iBit--) {const int iMask = 1 << iBit;if (!(iMask & rightLeve)) { continue; }if (!pr(m_vParents[iBit][node])) {return leve + BFindEnd(node, iBit, pr);}node = m_vParents[iBit][node];leve = leve ^ iMask;}return leve;}
protected://在leftNodeExclude的1到2^pow^级祖先中查找符合pr的最近祖先template<class _Pr>int BFindFirst(int leftNodeExclude, int pow, _Pr pr) {while (pow > 0) {const int& mid = m_vParents[pow - 1][leftNodeExclude];if (pr(mid)) {}else {leftNodeExclude = mid;}pow--;}return m_vParents[0][leftNodeExclude];}//在node的[0,2^pow^-1]级祖先中寻找符合的最后一个template<class _Pr>int BFindEnd(int node, int pow, _Pr pr) {int leve = 0;while (pow > 0) {pow--;const int& mid = m_vParents[pow][node];if (pr(mid)) {node = mid;leve = leve ^ (1 << pow);}else {}}return leve;}vector<vector<int>> m_vParents;
};class C2Parents : public CParents
{
public:C2Parents(vector<int>& vParent, const vector<int>& vDepth) :m_vDepth(vDepth), CParents(vParent, *std::max_element(vDepth.begin(), vDepth.end())){}int GetPublicParent(int iNode1, int iNode2)const{int leve0 = m_vDepth[iNode1];int leve1 = m_vDepth[iNode2];if (leve0 < leve1){iNode2 = GetParent(iNode2, leve1 - leve0);leve1 = leve0;}else{iNode1 = GetParent(iNode1, leve0 - leve1);leve0 = leve1;}if (iNode1 == iNode2) { return iNode1; }for (int iBit = GetBitCnt() - 1; iBit >= 0; iBit--) {const int iMask = 1 << iBit;if (iMask & leve0) {const int i1 = GetPow2Parent(iNode1, iBit);const int i2 = GetPow2Parent(iNode2, iBit);if (i1 == i2) {while (iBit > 0) {const int i3 = GetPow2Parent(iNode1, iBit - 1);const int i4 = GetPow2Parent(iNode2, iBit - 1);if (i3 != i4) {iNode1 = i3; iNode2 = i4;}iBit--;}return GetPow2Parent(iNode1, 0);}else {iNode1 = i1; iNode2 = i2; leve0 -= iMask;}}}return iNode1;}
protected:vector<vector<int>> m_vParents;const vector<int> m_vDepth;
};
class Solution {
public:vector<int> Ans(const int N, vector<int>& c, vector<pair<int, int>>& edge, vector<pair<int, int>>& que) {auto neiBo = CNeiBo::Two(N, edge, false, 1);vector<vector<int>> cnt(20, vector<int>(N)), preSum(20, vector<int>(N));vector<int> vpar(N, -1);function<void(int, int)> DFS = [&](int cur, int par) {vpar[cur] = par;for (int i = 0; i < 20; i++){cnt[i][cur] = (i == c[cur] - 1);preSum[i][cur] = cnt[i][cur];if (-1 != par) {preSum[i][cur] += preSum[i][par];}}for (const auto& next : neiBo[cur]) {if (next == par) { continue; }DFS(next, cur);}};DFS(0, -1);auto leves = CBFSLeve::Leve(neiBo, { 0 });C2Parents p2(vpar, leves);vector<int> ans;for (auto [u, v] : que) {u--, v--;const int g = p2.GetPublicParent(u, v);int cur = 0;for (int i = 0; i < 20; i++) {cur += (preSum[i][u] + preSum[i][v] - 2 * preSum[i][g] + cnt[i][g] > 0);}ans.emplace_back(cur);}return ans;}
};int main() {
#ifdef _DEBUGfreopen("a.in", "r", stdin);
#endif // DEBUG	ios::sync_with_stdio(0); cin.tie(nullptr);//CInBuff<> in; COutBuff<10'000'000> ob;int N, Q;cin >> N >> Q;auto c = Read<int>(N);auto edge = Read<pair<int, int>>(N - 1);auto que = Read<pair<int, int>>(Q);
#ifdef _DEBUG	printf("N=%d", N);Out(c, ",c=");Out(que, ",que=");Out(edge, ",edge=");//Out(edge2, ",edge2=");//Out(rr, ",rr=");//Out(ab, ",ab=");//Out(par, "par=");//Out(que, "que=");//Out(B, "B=");
#endif // DEBUG	Solution slu;auto res = slu.Ans(N,c,edge,que);for (const auto& i : res){cout << i << "\n";}return 0;
};

单元测试

int N;vector<int> c;vector<pair<int, int>> edge, que;TEST_METHOD(TestMethod01){N = 4, c = { 1,2,3,1 }, que = { {4,3},{1,4} }, edge = { {1,2},{1,3},{2,4} };auto res = Solution().Ans(N, c, edge, que);AssertEx({ 3,2 }, res);}

扩展阅读

我想对大家说的话
工作中遇到的问题，可以按类别查阅鄙人的算法文章，请点击《算法与数据汇总》。
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有效学习：明确的目标 及时的反馈 拉伸区（难度合适） 专注
闻缺陷则喜(喜缺)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
失败+反思=成功 成功+反思=成功

视频课程

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
如何你想快速形成战斗了，为老板分忧，请学习C#入职培训、C++入职培训等课程
https://edu.csdn.net/lecturer/6176

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。