本文涉及知识点
C++图论 强连通分量 缩点 最长路 拓扑排序
P2656 采蘑菇
题目描述
小胖和 ZYR 要去 ESQMS 森林采蘑菇。
ESQMS 森林间有 N N N 个小树丛， M M M 条小径，每条小径都是单向的，连接两个小树丛，上面都有一定数量的蘑菇。小胖和 ZYR 经过某条小径一次，可以采走这条路上所有的蘑菇。由于 ESQMS 森林是一片神奇的沃土，所以一条路上的蘑菇被采过后，又会长出一些新的蘑菇，数量为原来蘑菇的数量乘上这条路的“恢复系数”，再下取整。
比如，一条路上有 4 4 4 个蘑菇，这条路的“恢复系数”为 0.7 0.7 0.7，则第一~四次经过这条路径所能采到的蘑菇数量分别为 4 , 2 , 1 , 0 4,2,1,0 4,2,1,0。
现在，小胖和 ZYR 从 S S S 号小树丛出发，求他们最多能采到多少蘑菇。
输入格式
第一行两个整数， N N N 和 M M M。
第二行到第 M + 1 M+1 M+1 行，每行四个数，分别表示一条小路的起点，终点，初始蘑菇数，恢复系数。
第 M + 2 M+2 M+2 行，一个整数 S S S。
输出格式
一行一个整数，表示最多能采到多少蘑菇，保证答案不超过 ( 2 31 − 1 ) (2^{31}-1) (231−1)。
输入输出样例 #1
输入 #1
3 3
1 2 4 0.5
1 3 7 0.1
2 3 4 0.6
1
输出 #1
8
说明/提示
对于 30 % 30\% 30% 的数据， N ≤ 7 N\le 7 N≤7， M ≤ 15 M\le15 M≤15
另有 30 % 30\% 30% 的数据，满足所有“恢复系数”为 0 0 0。
对于 100 % 100\% 100% 的数据， 1 ≤ N ≤ 8 × 10 4 1 \le N\le 8\times 10^4 1≤N≤8×104， 1 ≤ M ≤ 2 × 10 5 1\le M\le 2\times 10^5 1≤M≤2×105， 0 ≤ 恢复系数 ≤ 0.8 0\le\text{恢复系数}\le 0.8 0≤恢复系数≤0.8 且最多有一位小数， 1 ≤ S ≤ N 1\le S\le N 1≤S≤N。
强连通分量 缩点 最长路 拓扑排序
建立有向图G，边权是此路能采到的所有蘑菇。
缩点形成有向图G1,点n在G1中对应的点是f(n）。
如果 f ( n 1 ) ≠ f ( n 2 ) , 边不变，边权初始蘑菇。如果相等，取消这条边。 s u m [ f ( n 1 ) ] + = 包括恢复的蘑菇边权 f(n1)\neq f(n2),边不变，边权初始蘑菇。如果相等，取消这条边。sum[f(n1)] += 包括恢复的蘑菇边权 f(n1)=f(n2),边不变，边权初始蘑菇。如果相等，取消这条边。sum[f(n1)]+=包括恢复的蘑菇边权
所有以f(n1)为起点的边，边权增加sum[f(n1)]。
求任意起点的最长路。按拓扑序处理以各节点开始的最长路。
注意一： ans = 点权，避免缩点后只有一个点。
注意二：如果S是缩点后的点，改成缩点后的点。
注意三：计算蘑菇数的时候，需要转成整数再计算，否则误差。
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 CSCCTarjan {
public:CSCCTarjan(vector<vector<int>>& neiBo) :m_neiBo(neiBo) {const int N = neiBo.size();m_vTime.assign(N, -1);m_vBack.assign(N, -1);m_vIsStack.assign(N, false);for (int i = 0; i < N; i++) {DFS(i);}}void InitPtNew() {m_ptNew.resize(m_neiBo.size());iota(m_ptNew.begin(), m_ptNew.end(), 0);for (auto& v : m_sccs) {nth_element(v.begin(), v.begin(), v.end());m_v0.emplace_back(v[0]);for (int i = 1; i < v.size(); i++) {m_ptNew[v[i]] = v[0];}}}vector<vector<int>> GetNewNeiBo() {vector<vector<int>> neiBo(m_neiBo.size());for (int i = 0; i < neiBo.size(); i++) {const int n1 = m_ptNew[i];unordered_set<int> s;for (const auto& next : m_neiBo[i]) {const int n2 = m_ptNew[next];if (n1 == n2) { continue; }//自环s.emplace(n2);}neiBo[n1].insert(neiBo[n1].begin(), s.begin(), s.end());}return neiBo;}vector<vector<int>> m_sccs;vector<int> m_v0, m_ptNew;
protected:void DFS(int cur) {if (-1 != m_vTime[cur]) { return; }m_vTime[cur] = m_vBack[cur] = m_iTimes++;m_vIsStack[cur] = true;m_sta.emplace(cur);for (const auto& next : m_neiBo[cur]) {if (-1 == m_vTime[next]) {DFS(next);m_vBack[cur] = min(m_vBack[cur], m_vBack[next]);}else if (m_vIsStack[next]) {m_vBack[cur] = min(m_vBack[cur], m_vTime[next]);}}if (m_vTime[cur] != m_vBack[cur]) { return; }vector<int> scc;while (m_sta.size()){auto u = m_sta.top(); m_sta.pop();scc.emplace_back(u);m_vIsStack[u] = false;if (cur == u) { break; }}m_sccs.emplace_back(scc);}vector<vector<int>>& m_neiBo;int m_iTimes = 0;vector<int> m_vTime, m_vBack;vector<bool> m_vIsStack;stack<int> m_sta;
};class CDGTopSort
{
public:template <class T = vector<int> >CDGTopSort(const vector<T>& vNeiBo) :m_vDeg(vNeiBo.size()), m_neiBo(vNeiBo) {const int N = vNeiBo.size();m_backNeiBo.resize(N);for (int cur = 0; cur < N; cur++){m_vDeg[cur] = vNeiBo[cur].size();for (const auto& next : vNeiBo[cur]){m_backNeiBo[next].emplace_back(cur);}}}void Init() {auto Add = [&](int i) {if (0 != m_vDeg[i]) { return; }m_que.emplace(i);};for (int i = 0; i < m_vDeg.size(); i++){Add(i);}while (m_que.size()){const int cur = m_que.front(); m_que.pop();if (!OnDo(cur)) { continue; }for (const auto& next : m_backNeiBo[cur]){m_vDeg[next]--;Add(next);}};}queue<int> m_que;vector<int> m_vDeg;vector<int> m_vSort;
protected:const vector<vector<int>>& m_neiBo;vector<vector<int>> m_backNeiBo;virtual bool OnDo(int cur) {m_vSort.emplace_back(cur);return true;};
};template<class T = int, T iDef = INT_MAX / 2>
class CDisNegativeRing //贝尔曼-福特算法
{
public:bool Dis(int N, vector<tuple<int, int, int>> edgeFromToW, int start) {vector<T> pre(N, iDef);pre[start] = 0;for (int t = 0; t < N; t++) {auto cur = pre;for (const auto& [u, v, w] : edgeFromToW) {cur[v] = min(cur[v], pre[u] + w);}if (t + 1 == N) {for (int i = 0; i < N; i++) {if (pre[i] != cur[i]) { return false; }}}pre.swap(cur);}m_vDis = pre;return true;}vector<T> m_vDis;
};class Solution {
public:int Ans(const int N, vector<tuple<int, int, int, double>>& edge, int S) {S--;vector<vector<int>> neiBo(N);for (auto& [u, v, w, d] : edge) {u--, v--;neiBo[u].emplace_back(v);}CSCCTarjan scc(neiBo);scc.InitPtNew();vector<int> pw(N);vector < vector<pair<int, int>>> neiBo1(N);for (auto& [u, v, w, d] : edge) {const int i = scc.m_ptNew[u];const int j = scc.m_ptNew[v];if (i == j) {pw[i] += F(w, d);}else {neiBo1[i].emplace_back(j, w);}}CDGTopSort topSort(scc.GetNewNeiBo());topSort.Init();vector<int> ans = pw;for (const auto& cur : topSort.m_vSort) {for (const auto& [child, w] : neiBo1[cur]) {ans[cur] = max(ans[cur], ans[child] + w + pw[cur]);}}return ans[scc.m_ptNew[S]];}int F(long long iInit, double d) {long long ret = 0;int p = d * 10;while (iInit) {ret += iInit;iInit = iInit * p / 10;}return ret;}
};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,S;cin >> N ;auto edge = Read<tuple<int, int, int, double>>();cin >> S;
#ifdef _DEBUG //printf("N=%d,S=%d",N,S);//Out(ws, ",ws=");//Out(edge, ",edge=");
#endif // DEBUG auto res = Solution().Ans(N,edge,S);cout << res << "\n";return 0;
};
单元测试
int N,S;vector<tuple<int, int, int, double>> edge;TEST_METHOD(TestMethod01){N = 3, S = 1, edge = { {1,2,4,0.5},{1,3,7,0.1},{2,3,4,0.6} };auto res = Solution().Ans(N, edge,S);AssertEx(8, res);}TEST_METHOD(TestMethod02){N = 3, S = 1, edge = { {1,2,4,0.5},{2,1,4,0.5},{2,3,4,0.6} };auto res = Solution().Ans(N, edge, S);AssertEx(18, res);}TEST_METHOD(TestMethod03){N = 3, S = 1, edge = { {1,2,4,0.5},{2,1,4,0.5} };auto res = Solution().Ans(N, edge, S);AssertEx(14, res);}TEST_METHOD(TestMethod04){N = 3, S = 2, edge = { {1,2,4,0.5},{2,1,4,0.5} };auto res = Solution().Ans(N, edge, S);AssertEx(14, res);}
扩展阅读
| 我想对大家说的话 |
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| 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》。 |
| 学习算法：按章节学习《喜缺全书算法册》，大量的题目和测试用例，打包下载。重视操作 |
| 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。
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