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【图论强连通分量】P1653 [USACO04DEC] Cow Ski Area G|普及+

news 2025/6/7 8:32:23

本文涉及知识点

C++图论强连通分量

P1653 [USACO04DEC] Cow Ski Area G

题目描述

约翰的表哥罗恩生活在科罗拉多州。他近来打算教他的奶牛们滑雪，但是奶牛们非常害羞，不敢在游人组织的度假胜地滑雪。没办法，他只好自己建滑雪场了。罗恩的雪场可以划分为 $W$ 列 $L$ 行 $(1\le W\le 500, 1\le L\le 500)$ ，每个方格有一个特定的高度 $H(0\le H\le 9999)$ 。奶牛可以在相邻方格间滑雪，而且不能由低到高滑。

为了保证任意方格可以互通，罗恩打算造一些直达缆车。缆车很强大，可以连接任意两个方格，而且是双向的。而且同一个方格也可以造多台缆车。但是缆车的建造费用贵得吓人，所以他希望造尽量少的缆车。那最少需要造多少台呢？

输入格式

第 $1$ 行： $W$ ， $L$ 。

接下来输入宽 $W$ 高 $L$ 的矩阵地图。

输出格式

输出最小需要的缆车数。

输入输出样例 #1

输入 #1

9 3
1 1 1 2 2 2 1 1 1
1 2 1 2 3 2 1 2 1
1 1 1 2 2 2 1 1 1

输出 #1

说明/提示

$1\le W,L\le 500$ ， $0\le H\le 9999$ 。

P1653 [USACO04DEC] Cow Ski Area G

强连通分量缩点
建立有向图，缩点。如果只剩下一个点，则不需要缆车。
in0记录入度为0的，out0记录出度为0的点，答案是max(in0, out0)

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];for (const auto& next : m_neiBo[i]) {const int n2 = m_ptNew[next];if (n1 == n2) { continue; }//自环neiBo[n1].emplace_back(n2);}}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 Solution {
public:int Ans(vector<vector<int>>& grid) {const int R = grid.size(), C = grid[0].size();const int RC = R * C;vector<vector<int>> neiBo(RC);auto Add = [&](int r, int c, int r1, int c1) {if (grid[r][c] >= grid[r1][c1]) {neiBo[r * C + c].emplace_back(r1 * C + c1);}if (grid[r1][c1] >= grid[r][c]) {neiBo[r1 * C + c1].emplace_back(r * C + c);}};for (int r = 0; r < R; r++) {for (int c = 0; c < C; c++) {if (r + 1 < R) {Add(r, c, r + 1, c);}if (c + 1 < C) {Add(r, c, r, c + 1);}}}CSCCTarjan scc(neiBo);if (scc.m_sccs.size() == 1) { return 0; }scc.InitPtNew();auto neiBo1 = scc.GetNewNeiBo();vector<int> in(RC), out(RC);for (int i = 0; i < RC; i++) {for (const auto& j : neiBo1[i]) {out[i] ++; in[j]++;}}int in0 = 0, out0 = 0;for (const auto& i : scc.m_v0) {in0 += (0 == in[i]);out0 += (0 == out[i]);}return max(in0, out0);}
};
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 R, C;cin >> C >> R;vector<vector<int>> grid(R);for (int r = 0; r < R; r++) {grid[r] = Read<int>(C);}
#ifdef _DEBUG	//printf("N=%d",n);Out(grid, ",grid=");
#endif // DEBUG	auto res = Solution().Ans(grid);cout << res << "\n";return 0;
};

单元测试

vector<vector<int>> grid;TEST_METHOD(TestMethod01){grid = { {1,1,1,2,2,2,1,1,1},{1,2,1,2,3,2,1,2,1},{1,1,1,2,2,2,1,1,1} };auto res = Solution().Ans(grid);AssertEx(3, res);}

扩展阅读

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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