接続行列の連立方程式 (src/Graph/incidence_matrix_equation.hpp)

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• Last update: 2024-09-20 12:04:43+09:00
• Include: #include "src/Graph/incidence_matrix_equation.hpp"

関数名	概要	計算量
incidence_matrix_equation<T>(g, b)	有向グラフ $g$ の接続行列 $A$ と頂点数 $n$ の次元を持つベクトル $\boldsymbol{b}$ に対して $\displaystyle A\boldsymbol{x}=\boldsymbol{b}$ を満たす辺数 $m$ の次元を持つベクトル $\boldsymbol{x}$ を一つ返す. 引数は Graphクラス と vector<T>. 戻り値は vector<T> . 解なしなら空集合を返す. また T=bool の場合は有限体 $\mathbb{F}_2$ だとみなす.	$O(n+m)$

Verify

• AtCoder Beginner Contest 155 F - Perils in Parallel (sp judge)
  
• AtCoder Grand Contest 035 B - Even Degrees (sp judge)

Depends on

• グラフ (src/Graph/Graph.hpp)
• CSR 表現を用いた二次元配列 他 (src/Internal/ListRange.hpp)

Verified with

• test/aoj/3142.test.cpp
• test/aoj/3142.UFPU.test.cpp
• test/atcoder/arc106_b.test.cpp
• test/atcoder/cf17_final_e.test.cpp
• test/sample_test/abc155_f.test.cpp
• test/sample_test/agc035_b.test.cpp

Code

#pragma once
#include <cassert>
#include "src/Graph/Graph.hpp"
template <class T> std::vector<T> incidence_matrix_equation(const Graph &g, std::vector<T> b) {
 const int n= g.vertex_size();
 assert((int)b.size() == n);
 std::vector<T> x(g.edge_size());
 auto adje= g.adjacency_edge(0);
 std::vector<int> pre(n, -2), ei(adje.p.begin(), adje.p.begin() + n);
 for (int s= 0, p, e; s < n; ++s)
  if (pre[s] == -2)
   for (pre[p= s]= -1;;) {
    if (ei[p] == adje.p[p + 1]) {
     if (e= pre[p]; e < 0) {
      if (b[p] != T()) return {};  // no solution
      break;
     }
     T tmp= b[p];
     p= g[e].to(p);
     if constexpr (std::is_same_v<T, bool>) x[e]= tmp, b[p]= tmp ^ b[p];
     else x[e]= g[e].second == p ? -tmp : tmp, b[p]+= tmp;
    } else if (int q= g[e= adje.dat[ei[p]++]].to(p); pre[q] == -2) pre[p= q]= e;
   }
 return x;
}

#line 2 "src/Graph/incidence_matrix_equation.hpp"
#include <cassert>
#line 2 "src/Internal/ListRange.hpp"
#include <vector>
#include <iostream>
#include <iterator>
#include <type_traits>
#define _LR(name, IT, CT) \
 template <class T> struct name { \
  using Iterator= typename std::vector<T>::IT; \
  Iterator bg, ed; \
  Iterator begin() const { return bg; } \
  Iterator end() const { return ed; } \
  size_t size() const { return std::distance(bg, ed); } \
  CT &operator[](int i) const { return bg[i]; } \
 }
_LR(ListRange, iterator, T);
_LR(ConstListRange, const_iterator, const T);
#undef _LR
template <class T> struct CSRArray {
 std::vector<T> dat;
 std::vector<int> p;
 size_t size() const { return p.size() - 1; }
 ListRange<T> operator[](int i) { return {dat.begin() + p[i], dat.begin() + p[i + 1]}; }
 ConstListRange<T> operator[](int i) const { return {dat.cbegin() + p[i], dat.cbegin() + p[i + 1]}; }
};
template <template <class> class F, class T> std::enable_if_t<std::disjunction_v<std::is_same<F<T>, ListRange<T>>, std::is_same<F<T>, ConstListRange<T>>, std::is_same<F<T>, CSRArray<T>>>, std::ostream &> operator<<(std::ostream &os, const F<T> &r) {
 os << '[';
 for (int _= 0, __= r.size(); _ < __; ++_) os << (_ ? ", " : "") << r[_];
 return os << ']';
}
#line 3 "src/Graph/Graph.hpp"
struct Edge: std::pair<int, int> {
 using std::pair<int, int>::pair;
 Edge &operator--() { return --first, --second, *this; }
 int to(int v) const { return first ^ second ^ v; }
 friend std::istream &operator>>(std::istream &is, Edge &e) { return is >> e.first >> e.second, is; }
};
struct Graph: std::vector<Edge> {
 size_t n;
 Graph(size_t n= 0, size_t m= 0): vector(m), n(n) {}
 size_t vertex_size() const { return n; }
 size_t edge_size() const { return size(); }
 size_t add_vertex() { return n++; }
 size_t add_edge(int s, int d) { return emplace_back(s, d), size() - 1; }
 size_t add_edge(Edge e) { return emplace_back(e), size() - 1; }
#define _ADJ_FOR(a, b) \
 for (auto [u, v]: *this) a; \
 for (size_t i= 0; i < n; ++i) p[i + 1]+= p[i]; \
 for (int i= size(); i--;) { \
  auto [u, v]= (*this)[i]; \
  b; \
 }
#define _ADJ(a, b) \
 vector<int> p(n + 1), c(size() << !dir); \
 if (!dir) { \
  _ADJ_FOR((++p[u], ++p[v]), (c[--p[u]]= a, c[--p[v]]= b)) \
 } else if (dir > 0) { \
  _ADJ_FOR(++p[u], c[--p[u]]= a) \
 } else { \
  _ADJ_FOR(++p[v], c[--p[v]]= b) \
 } \
 return {c, p}
 CSRArray<int> adjacency_vertex(int dir) const { _ADJ(v, u); }
 CSRArray<int> adjacency_edge(int dir) const { _ADJ(i, i); }
#undef _ADJ
#undef _ADJ_FOR
};
#line 4 "src/Graph/incidence_matrix_equation.hpp"
template <class T> std::vector<T> incidence_matrix_equation(const Graph &g, std::vector<T> b) {
 const int n= g.vertex_size();
 assert((int)b.size() == n);
 std::vector<T> x(g.edge_size());
 auto adje= g.adjacency_edge(0);
 std::vector<int> pre(n, -2), ei(adje.p.begin(), adje.p.begin() + n);
 for (int s= 0, p, e; s < n; ++s)
  if (pre[s] == -2)
   for (pre[p= s]= -1;;) {
    if (ei[p] == adje.p[p + 1]) {
     if (e= pre[p]; e < 0) {
      if (b[p] != T()) return {};  // no solution
      break;
     }
     T tmp= b[p];
     p= g[e].to(p);
     if constexpr (std::is_same_v<T, bool>) x[e]= tmp, b[p]= tmp ^ b[p];
     else x[e]= g[e].second == p ? -tmp : tmp, b[p]+= tmp;
    } else if (int q= g[e= adje.dat[ei[p]++]].to(p); pre[q] == -2) pre[p= q]= e;
   }
 return x;
}

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