cp-library

C++ Library for Competitive Programming

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:heavy_check_mark: 数学/行列/連立一次方程式
(test/math/matrix/linear_equation.test.cpp)

Depends on

Code

/*
 * @title 数学/行列/連立一次方程式
 *
 * verification-helper: PROBLEM https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=2171
 * verification-helper: ERROR 1e-8
 */

#include <iomanip>
#include <iostream>
#include <vector>

#include "emthrm/graph/edge.hpp"
#include "emthrm/graph/shortest_path/dijkstra.hpp"
#include "emthrm/math/matrix/linear_equation.hpp"
#include "emthrm/math/matrix/matrix.hpp"

int main() {
  while (true) {
    int n, s, t;
    std::cin >> n >> s >> t;
    if (n == 0 && s == 0 && t == 0) break;
    --s; --t;
    std::vector<int> q(n);
    for (int i = 0; i < n; ++i) {
      std::cin >> q[i];
    }
    std::vector<std::vector<emthrm::Edge<int>>> graph(n);
    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < n; ++j) {
        int a;
        std::cin >> a;
        if (a > 0) graph[i].emplace_back(i, j, a);
      }
    }
    emthrm::Dijkstra<int> dijkstra(graph);
    const std::vector<int> dist = dijkstra.build(t);
    if (dist[s] == dijkstra.inf) {
      std::cout << "impossible\n";
      continue;
    }
    emthrm::Matrix<int> a(n, n, 0);
    std::vector<int> b(n, 0);
    for (int i = 0; i < n; ++i) {
      if (i == t) {
        a[i][i] = 1;
      } else {
        std::vector<emthrm::Edge<int>> edges;
        if (q[i] == 0) {
          edges = graph[i];
        } else if (q[i] == 1) {
          for (const emthrm::Edge<int>& e : graph[i]) {
            if (dist[e.dst] + e.cost == dist[i]) edges.emplace_back(e);
          }
        }
        a[i][i] = -edges.size();
        for (const emthrm::Edge<int>& e : edges) {
          ++a[i][e.dst];
          b[i] -= e.cost;
        }
      }
    }
    std::cout << std::fixed << std::setprecision(8)
              << emthrm::linear_equation(a, b)[s] << '\n';
  }
  return 0;
}
#line 1 "test/math/matrix/linear_equation.test.cpp"
/*
 * @title 数学/行列/連立一次方程式
 *
 * verification-helper: PROBLEM https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=2171
 * verification-helper: ERROR 1e-8
 */

#include <iomanip>
#include <iostream>
#include <vector>

#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 1 "include/emthrm/graph/shortest_path/dijkstra.hpp"



#include <algorithm>
#include <cassert>
#include <functional>
#include <limits>
#include <queue>
#include <utility>
#line 11 "include/emthrm/graph/shortest_path/dijkstra.hpp"

#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 13 "include/emthrm/graph/shortest_path/dijkstra.hpp"

namespace emthrm {

template <typename CostType>
struct Dijkstra {
  const CostType inf;

  Dijkstra(const std::vector<std::vector<Edge<CostType>>>& graph,
           const CostType inf = std::numeric_limits<CostType>::max())
      : inf(inf), is_built(false), graph(graph) {}

  std::vector<CostType> build(const int s) {
    is_built = true;
    const int n = graph.size();
    std::vector<CostType> dist(n, inf);
    dist[s] = 0;
    prev.assign(n, -1);
    std::priority_queue<std::pair<CostType, int>,
                        std::vector<std::pair<CostType, int>>,
                        std::greater<std::pair<CostType, int>>> que;
    que.emplace(0, s);
    while (!que.empty()) {
      const auto [d, ver] = que.top();
      que.pop();
      if (d > dist[ver]) continue;
      for (const Edge<CostType>& e : graph[ver]) {
        if (dist[ver] + e.cost < dist[e.dst]) {
          dist[e.dst] = dist[ver] + e.cost;
          prev[e.dst] = ver;
          que.emplace(dist[e.dst], e.dst);
        }
      }
    }
    return dist;
  }

  std::vector<int> build_path(int t) const {
    assert(is_built);
    std::vector<int> res;
    for (; t != -1; t = prev[t]) {
      res.emplace_back(t);
    }
    std::reverse(res.begin(), res.end());
    return res;
  }

 private:
  bool is_built;
  std::vector<int> prev;
  std::vector<std::vector<Edge<CostType>>> graph;
};

}  // namespace emthrm


#line 1 "include/emthrm/math/matrix/linear_equation.hpp"



#line 5 "include/emthrm/math/matrix/linear_equation.hpp"
#include <cmath>
#line 7 "include/emthrm/math/matrix/linear_equation.hpp"

#line 1 "include/emthrm/math/matrix/gauss_jordan.hpp"



#line 5 "include/emthrm/math/matrix/gauss_jordan.hpp"

#line 1 "include/emthrm/math/matrix/matrix.hpp"



#line 5 "include/emthrm/math/matrix/matrix.hpp"

namespace emthrm {

template <typename T>
struct Matrix {
  explicit Matrix(const int m, const int n, const T def = 0)
      : data(m, std::vector<T>(n, def)) {}

  int nrow() const { return data.size(); }
  int ncol() const { return data.empty() ? 0 : data.front().size(); }

  Matrix pow(long long exponent) const {
    const int n = nrow();
    Matrix<T> res(n, n, 0), tmp = *this;
    for (int i = 0; i < n; ++i) {
      res[i][i] = 1;
    }
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
    }
    return res;
  }

  inline const std::vector<T>& operator[](const int i) const { return data[i]; }
  inline std::vector<T>& operator[](const int i) { return data[i]; }

  Matrix& operator=(const Matrix& x) = default;

  Matrix& operator+=(const Matrix& x) {
    const int m = nrow(), n = ncol();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        data[i][j] += x[i][j];
      }
    }
    return *this;
  }

  Matrix& operator-=(const Matrix& x) {
    const int m = nrow(), n = ncol();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        data[i][j] -= x[i][j];
      }
    }
    return *this;
  }

  Matrix& operator*=(const Matrix& x) {
    const int m = nrow(), l = ncol(), n = x.ncol();
    std::vector<std::vector<T>> res(m, std::vector<T>(n, 0));
    for (int i = 0; i < m; ++i) {
      for (int k = 0; k < l; ++k) {
        for (int j = 0; j < n; ++j) {
          res[i][j] += data[i][k] * x[k][j];
        }
      }
    }
    data.swap(res);
    return *this;
  }

  Matrix operator+(const Matrix& x) const { return Matrix(*this) += x; }
  Matrix operator-(const Matrix& x) const { return Matrix(*this) -= x; }
  Matrix operator*(const Matrix& x) const { return Matrix(*this) *= x; }

 private:
  std::vector<std::vector<T>> data;
};

}  // namespace emthrm


#line 7 "include/emthrm/math/matrix/gauss_jordan.hpp"

namespace emthrm {

template <bool IS_EXTENDED = false, typename T>
int gauss_jordan(Matrix<T>* a, const T eps = 1e-8) {
  const int m = a->nrow(), n = a->ncol();
  int rank = 0;
  for (int col = 0; col < (IS_EXTENDED ? n - 1 : n); ++col) {
    int pivot = -1;
    T mx = eps;
    for (int row = rank; row < m; ++row) {
      const T abs = ((*a)[row][col] < 0 ? -(*a)[row][col] : (*a)[row][col]);
      if (abs > mx) {
        pivot = row;
        mx = abs;
      }
    }
    if (pivot == -1) continue;
    std::swap((*a)[rank], (*a)[pivot]);
    T tmp = (*a)[rank][col];
    for (int col2 = 0; col2 < n; ++col2) {
      (*a)[rank][col2] /= tmp;
    }
    for (int row = 0; row < m; ++row) {
      if (row != rank &&
          ((*a)[row][col] < 0 ? -(*a)[row][col] : (*a)[row][col]) > eps) {
        tmp = (*a)[row][col];
        for (int col2 = 0; col2 < n; ++col2) {
          (*a)[row][col2] -= (*a)[rank][col2] * tmp;
        }
      }
    }
    ++rank;
  }
  return rank;
}

}  // namespace emthrm


#line 10 "include/emthrm/math/matrix/linear_equation.hpp"

namespace emthrm {

template <typename T, typename U = double>
std::vector<U> linear_equation(const Matrix<T>& a, const std::vector<T>& b,
                               const U eps = 1e-8) {
  const int m = a.nrow(), n = a.ncol();
  Matrix<U> c(m, n + 1);
  for (int i = 0; i < m; ++i) {
    std::copy(a[i].begin(), a[i].end(), c[i].begin());
    c[i].back() = b[i];
  }
  const int rank = gauss_jordan<true>(&c, eps);
  for (int row = rank; row < m; ++row) {
    if ((c[row].back() < 0 ? -c[row].back() : c[row].back()) > eps) {
      return std::vector<U>{};
    }
  }
  std::vector<U> res(n, 0);
  for (int i = 0, j = 0; i < rank; ++i) {
    while ((c[i][j] < 0 ? -c[i][j] : c[i][j]) <= eps) ++j;
    res[j++] = c[i].back();
  }
  return res;
}

}  // namespace emthrm


#line 16 "test/math/matrix/linear_equation.test.cpp"

int main() {
  while (true) {
    int n, s, t;
    std::cin >> n >> s >> t;
    if (n == 0 && s == 0 && t == 0) break;
    --s; --t;
    std::vector<int> q(n);
    for (int i = 0; i < n; ++i) {
      std::cin >> q[i];
    }
    std::vector<std::vector<emthrm::Edge<int>>> graph(n);
    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < n; ++j) {
        int a;
        std::cin >> a;
        if (a > 0) graph[i].emplace_back(i, j, a);
      }
    }
    emthrm::Dijkstra<int> dijkstra(graph);
    const std::vector<int> dist = dijkstra.build(t);
    if (dist[s] == dijkstra.inf) {
      std::cout << "impossible\n";
      continue;
    }
    emthrm::Matrix<int> a(n, n, 0);
    std::vector<int> b(n, 0);
    for (int i = 0; i < n; ++i) {
      if (i == t) {
        a[i][i] = 1;
      } else {
        std::vector<emthrm::Edge<int>> edges;
        if (q[i] == 0) {
          edges = graph[i];
        } else if (q[i] == 1) {
          for (const emthrm::Edge<int>& e : graph[i]) {
            if (dist[e.dst] + e.cost == dist[i]) edges.emplace_back(e);
          }
        }
        a[i][i] = -edges.size();
        for (const emthrm::Edge<int>& e : edges) {
          ++a[i][e.dst];
          b[i] -= e.cost;
        }
      }
    }
    std::cout << std::fixed << std::setprecision(8)
              << emthrm::linear_equation(a, b)[s] << '\n';
  }
  return 0;
}
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