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【高精度带权并集查找唯一分解定理】 P4079 [SDOI2016] 齿轮|省选-

news 2025/7/22 5:44:44

本文涉及知识点

C++并集查找
数论：质数、最大公约数、菲蜀定理

P4079 [SDOI2016] 齿轮

题目描述

现有一个传动系统，包含了N个组合齿轮和M个链条。每一个链条连接了两个组合齿轮u和v，并提供了一个传动比x: y。即如果只考虑这两个组合齿轮，编号为u的齿轮转动x圈，编号为v的齿轮会转动y圈。传动比为正表示若编号为u的齿轮顺时针转动，则编号为v的齿轮也顺时针转动。传动比为负表示若编号为u的齿轮顺时针转动，则编号为v的齿轮会逆时针转动。若不同链条的传动比不相容，则有些齿轮无法转动。我们希望知道，系统中的这Ｎ个组合齿轮能否同时转动。

输入格式

有多组数据。

第一行给定整数Ｔ，表示总的数据组数，之后依次给出Ｔ组数据。

每一组数据的第一行给定整数Ｎ和Ｍ，表示齿轮总数和链条总数。

之后有Ｍ行，依次描述了每一个链条，其中每一行给定四个整数u，v，x和y，表示只考虑这一组联动关系的情况下，编号为u的齿轮转动x圈，编号为v的齿轮会转动y圈。请注意，x为正整数，而y为非零整数，但是y有可能为负数。

输出格式

输出T行，对应每一组数据。首先应该输出标识这是第几组数据，参见样例输出。之后输出判定结果，如果N个组合齿轮可以同时正常运行，则输出Yes，否则输出No。

输入输出样例 #1

输入 #1

输出 #1

Case #1: Yes
Case #2: No

说明/提示

对于所有数据，T<=32，N<=1000，M<=10000且x与y的绝对值均不超过100。

高精度 + 并集查找

编号为u的齿轮转动x圈，编号为v的齿轮会转动y圈。令编号为u、v的齿轮半径为r1、r2，则 $2π×r1×x==2π×r2×y2\pi\times r1 \times x == 2\pi \times r2 \times y$ 即 $r1 \times x==r2 \times y$ 。
性质一：某链条为：u，v，x和y，另外一链条为：v,w,x1,y1。则：
v跑 $y×x1y\times x1$ 圈时，u跑 $\times x1$ ,w跑 $\times y1$ ，即 $\times x1,y \times y1$
$u, w,$ 我们把齿轮看成点，链条看成边，则 $\rightarrow v$ 的边权为 $\div y$ ，整个路径的边权为，边权相除。 $\rightarrow u$ 的边权为 $\div x$ ，边反向，边权是倒数。
性质二：如果没有环，边权不会矛盾。按任意顺序将边 $u 、 v$ 加到并集查找中。如果已经连通则不增加，判断带权的并集查找 $\rightarrow g \rightarrow v$ 的边权，和当前边的边权是否相等。如果不等无解。相等则生什么都不做。g是u、v的最近公共祖先。
通过唯一分解定理：将分数分解成质因数的次数。
v 记录100以内的质数。m= v.size()。
分数= $(−1)datamΠi:0m−1(v[i]datai)(-1)^{data_m}\Pi_{i:0}^{m-1}(v[i]^{data_i})$
分数乘法： $data_i$ 相加
取倒数： $datai∗=−1,0≤i<mdata_i *= -1,0 \le i<m$
判断两分数是否相等： $data_i$ 相等， $data_m$ 奇偶性相同。

在这里插入图片描述

u和v连通前，root1和root2的边权都是1。u和v连通后，修改root1或root2边权前，并集查找中 $\rightarrow v$ 的边权为w1。 $\div y$
如果 $\rightarrow root2$ ，则 $w1 \times w[root1] ==w2$ 即 $\div w1$
如果 $\rightarrow root1$ ，则 $w1 \div w[root2] ==w2$ 即 $\div w2$ 图中并集查找 $\rightarrow v$ 的路径为 $\rightarrow root1 \rightarrow root2 \rightarrow v$ ，乘以 $\rightarrow root2$ 的边权。

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 <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 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();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 CCreatePrime {
public:CCreatePrime(int iMax) :m_isPrime(iMax + 1, true){m_isPrime[0] = m_isPrime[1] = false;for (int i = 2; i <= iMax; i++){if (m_isPrime[i]){m_vPrime.emplace_back(i);}for (const auto& n : m_vPrime){if ((long long)n * i > iMax) { break; }m_isPrime[n * i] = false;if (0 == i % n) { break; }}}}vector<int> m_vPrime;vector<bool> m_isPrime;
};
struct CBigNumMulStatic {vector<int> m_data;CBigNumMulStatic operator*(const CBigNumMulStatic& o)const {CBigNumMulStatic ret = *this;for (int i = 0; i < ret.m_data.size(); i++) {ret.m_data[i] += o.m_data[i];}return ret;}CBigNumMulStatic operator/(const CBigNumMulStatic& o)const {CBigNumMulStatic ret = *this;for (int i = 0; i < ret.m_data.size(); i++) {ret.m_data[i] -= o.m_data[i];}return ret;}bool operator==(const CBigNumMulStatic& o)const {if ((m_data.back() + o.m_data.back()) % 2) {return false;}for (int i = 0; i + 1 < m_data.size(); i++) {if (m_data[i] != o.m_data[i]) {return false;}}return true;};
};
class CBigNumMulStaticFactory {
public:CBigNumMulStaticFactory() {CCreatePrime cp(100);m_prime.swap(cp.m_vPrime);}CBigNumMulStatic Create(int x, int y) {CBigNumMulStatic b;b.m_data.resize(m_prime.size() + 1);if (x * y < 0) { b.m_data.back() = 1; }for (int i = 0; i < m_prime.size(); i++) {while (0 == x % m_prime[i]) {b.m_data[i]++;x /= m_prime[i];}while (0 == y % m_prime[i]) {b.m_data[i]--;y /= m_prime[i];}}return b;}CBigNumMulStatic Create(int x) {CBigNumMulStatic b;b.m_data.resize(m_prime.size() + 1);if (x < 0) { b.m_data.back() = 1; }for (int i = 0; i < m_prime.size(); i++) {while (0 == x % m_prime[i]) {b.m_data[i]++;x /= m_prime[i];}}return b;}
private:vector<int> m_prime;
};
class CUnionFundWSingle
{
public:CUnionFundWSingle(int N) :m_par(N) {for (int i = 0; i < N; i++) { m_par[i] = i; }}pair<int, int> GetRootLeve(int node) const {int leve = 0;int root = node;while (root != m_par[root]) { root = m_par[root]; leve++; }return { root,leve };}pair<int, bool> Union(int node1, int node2) {const auto& [root1, le1] = GetRootLeve(node1);const auto& [root2, le2] = GetRootLeve(node2);if (root1 == root2) { return { -1,false }; }if (le2 < le1) {m_par[root2] = root1;return { root2,false };}m_par[root1] = root2;return { root1,true };}tuple<int, vector<int>, vector<int>> LCA(int node1, int node2) {//node1和node2相等或不在一个连通区域为空，否则路径的所有的边auto [root1, le1] = GetRootLeve(node1);auto [root2, le2] = GetRootLeve(node2);vector<int> ans1, ans2;if (root1 != root2) {return make_tuple(-1, ans1, ans2);}for (; le2 > le1; le2--) {ans2.emplace_back(node2);node2 = m_par[node2];}for (; le1 > le2; le1--) {ans1.emplace_back(node1);node1 = m_par[node1];}for (; node1 != node2; node1 = m_par[node1], node2 = m_par[node2]) {ans1.emplace_back(node1);ans2.emplace_back(node2);}return make_tuple(node1, ans1, ans2);}bool IsConnet(int node1, int node2) {auto [root1, le1] = GetRootLeve(node1);auto [root2, le2] = GetRootLeve(node2);return root1 == root2;}vector<int> m_par;};
class Solution {
public:bool Ans(const int N, vector<tuple<int, int, int, int>>& edge) {static CBigNumMulStaticFactory bf;CUnionFundWSingle uf(N);vector<CBigNumMulStatic> ws(N, bf.Create(1, 1));auto GetW = [&](int u, int v) {CBigNumMulStatic bg = bf.Create(1, 1);auto [g, ans1, ans2] = uf.LCA(u, v);for (auto& i : ans1) {bg = bg * ws[i];}for (auto& i : ans2) {bg = bg / ws[i];}return bg;};for (auto [u, v, x, y] : edge) {u--, v--;const auto [root1, bLeftConnectRight] = uf.Union(u, v);auto bgOld = GetW(u, v);const auto bgNew = bf.Create(x, y);if (-1 != root1) {ws[root1] = bLeftConnectRight ? (bgNew / bgOld) : (bgOld / bgNew);}else {if (!(bgOld == bgNew)) { return false; }}}return true;}
};int main() {
#ifdef _DEBUGfreopen("a.in", "r", stdin);
#endif // DEBUGios::sync_with_stdio(0); cin.tie(nullptr); cout.tie(nullptr);int T,N;cin >> T;for(int i=1;i <= T;i++){cin >> N;auto edge = Read<tuple<int, int, int, int>>();
#ifdef _DEBUGprintf("N=%d", N);Out(edge, ",edge=");//Out(st, ",st=");
#endif // DEBUGauto res = Solution().Ans(N, edge);		cout << "Case #" <<i << ": " << (res ? "Yes" : "No") << "\n";}return 0;
}

单元测试

#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 <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 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();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 CCreatePrime {
public:CCreatePrime(int iMax) :m_isPrime(iMax + 1, true){m_isPrime[0] = m_isPrime[1] = false;for (int i = 2; i <= iMax; i++){if (m_isPrime[i]){m_vPrime.emplace_back(i);}for (const auto& n : m_vPrime){if ((long long)n * i > iMax) { break; }m_isPrime[n * i] = false;if (0 == i % n) { break; }}}}vector<int> m_vPrime;vector<bool> m_isPrime;
};
struct CBigNumMulStatic {vector<int> m_data;CBigNumMulStatic operator*(const CBigNumMulStatic& o)const {CBigNumMulStatic ret = *this;for (int i = 0; i < ret.m_data.size(); i++) {ret.m_data[i] += o.m_data[i];}return ret;}CBigNumMulStatic operator/(const CBigNumMulStatic& o)const {CBigNumMulStatic ret = *this;for (int i = 0; i < ret.m_data.size(); i++) {ret.m_data[i] -= o.m_data[i];}return ret;}bool operator==(const CBigNumMulStatic& o)const {if ((m_data.back() + o.m_data.back()) % 2) {return false;}for (int i = 0; i + 1 < m_data.size(); i++) {if (m_data[i] != o.m_data[i]) {return false;}}return true;};
};
class CBigNumMulStaticFactory {
public:CBigNumMulStaticFactory() {CCreatePrime cp(100);m_prime.swap(cp.m_vPrime);}CBigNumMulStatic Create(int x, int y) {CBigNumMulStatic b;b.m_data.resize(m_prime.size() + 1);if (x * y < 0) { b.m_data.back() = 1; }for (int i = 0; i < m_prime.size(); i++) {while (0 == x % m_prime[i]) {b.m_data[i]++;x /= m_prime[i];}while (0 == y % m_prime[i]) {b.m_data[i]--;y /= m_prime[i];}}return b;}CBigNumMulStatic Create(int x) {CBigNumMulStatic b;b.m_data.resize(m_prime.size() + 1);if (x < 0) { b.m_data.back() = 1; }for (int i = 0; i < m_prime.size(); i++) {while (0 == x % m_prime[i]) {b.m_data[i]++;x /= m_prime[i];}}return b;}
private:vector<int> m_prime;
};
class CUnionFundWSingle
{
public:CUnionFundWSingle(int N) :m_par(N) {for (int i = 0; i < N; i++) { m_par[i] = i; }}pair<int, int> GetRootLeve(int node) const {int leve = 0;int root = node;while (root != m_par[root]) { root = m_par[root]; leve++; }return { root,leve };}pair<int, bool> Union(int node1, int node2) {const auto& [root1, le1] = GetRootLeve(node1);const auto& [root2, le2] = GetRootLeve(node2);if (root1 == root2) { return { -1,false }; }if (le2 < le1) {m_par[root2] = root1;return { root2,false };}m_par[root1] = root2;return { root1,true };}tuple<int, vector<int>, vector<int>> LCA(int node1, int node2) {//node1和node2相等或不在一个连通区域为空，否则路径的所有的边auto [root1, le1] = GetRootLeve(node1);auto [root2, le2] = GetRootLeve(node2);vector<int> ans1, ans2;if (root1 != root2) {return make_tuple(-1, ans1, ans2);}for (; le2 > le1; le2--) {ans2.emplace_back(node2);node2 = m_par[node2];}for (; le1 > le2; le1--) {ans1.emplace_back(node1);node1 = m_par[node1];}for (; node1 != node2; node1 = m_par[node1], node2 = m_par[node2]) {ans1.emplace_back(node1);ans2.emplace_back(node2);}return make_tuple(node1, ans1, ans2);}bool IsConnet(int node1, int node2) {auto [root1, le1] = GetRootLeve(node1);auto [root2, le2] = GetRootLeve(node2);return root1 == root2;}vector<int> m_par;};
class Solution {
public:bool Ans(const int N, vector<tuple<int, int, int, int>>& edge) {static CBigNumMulStaticFactory bf;CUnionFundWSingle uf(N);vector<CBigNumMulStatic> ws(N, bf.Create(1, 1));auto GetW = [&](int u, int v) {CBigNumMulStatic bg = bf.Create(1, 1);auto [g, ans1, ans2] = uf.LCA(u, v);for (auto& i : ans1) {bg = bg * ws[i];}for (auto& i : ans2) {bg = bg / ws[i];}return bg;};for (auto [u, v, x, y] : edge) {u--, v--;const auto [root1, bLeftConnectRight] = uf.Union(u, v);auto bgOld = GetW(u, v);const auto bgNew = bf.Create(x, y);if (-1 != root1) {ws[root1] = bLeftConnectRight ? (bgNew / bgOld) : (bgOld / bgNew);}else {if (!(bgOld == bgNew)) { return false; }}}return true;}
};int main() {
#ifdef _DEBUGfreopen("a.in", "r", stdin);
#endif // DEBUGios::sync_with_stdio(0); cin.tie(nullptr); cout.tie(nullptr);int T,N;cin >> T;for(int i=1;i <= T;i++){cin >> N;auto edge = Read<tuple<int, int, int, int>>();
#ifdef _DEBUGprintf("N=%d", N);Out(edge, ",edge=");//Out(st, ",st=");
#endif // DEBUGauto res = Solution().Ans(N, edge);		cout << "Case #" <<i << ": " << (res ? "Yes" : "No") << "\n";}return 0;
}

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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》。
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