convex/monge.hpp

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• Last update: 2023-10-24 13:33:26+09:00
• Include: #include "convex/monge.hpp"
• Link: https://noshi91.github.io/algorithm-encyclopedia/d-edge-shortest-path-monge
• Link: https://topcoder-g-hatena-ne-jp.jag-icpc.org/spaghetti_source/20120915/1347668163.html

Depends on

• convex/larsch.hpp
• convex/smawk.hpp
• other/fibonacci_search.hpp

Verified with

• test/yukicoder/705.test.cpp

Code

#include "convex/larsch.hpp"
#include "convex/smawk.hpp"
#include "other/fibonacci_search.hpp"

// 定義域 [0, N] の範囲で f の monge 性を確認

template <typename T, typename F>
bool check_monge(int N, F f) {
  FOR(l, N + 1) FOR(k, l) FOR(j, k) FOR(i, j) {
    T lhs = f(i, l) + f(j, k);
    T rhs = f(i, k) + f(j, l);
    if (lhs < rhs) {
      print("monge ng");
      print(i, j, k, l, f(i, k), f(i, l), f(j, k), f(j, l), lhs, rhs);
      return false;
    }
  }
  print("monge ok");
  return true;
}

// newdp[j] = min (dp[i] + f(i,j))

template <typename T, typename F>
vc<T> monge_dp_update(int N, vc<T>& dp, F f) {
  assert(len(dp) == N + 1);
  auto select = [&](int i, int j, int k) -> int {
    if (i <= k) return j;
    return (dp[j] + f(j, i) > dp[k] + f(k, i) ? k : j);
  };
  vc<int> I = SMAWK(N + 1, N + 1, select);
  vc<T> newdp(N + 1, infty<T>);
  FOR(j, N + 1) {
    int i = I[j];
    chmin(newdp[j], dp[i] + f(i, j));
  }
  return newdp;
}

// 遷移回数を問わない場合

template <typename T, typename F>
vc<T> monge_shortest_path(int N, F f) {
  vc<T> dp(N + 1, infty<T>);
  dp[0] = 0;
  LARSCH<T> larsch(N, [&](int i, int j) -> T {
    ++i;
    if (i <= j) return infty<T>;
    return dp[j] + f(j, i);
  });
  FOR(r, 1, N + 1) {
    int l = larsch.get_argmin();
    dp[r] = dp[l] + f(l, r);
  }
  return dp;
}

// https://noshi91.github.io/algorithm-encyclopedia/d-edge-shortest-path-monge

// |f| の上限 f_lim も渡す

// ・larsch が結構重いので、自前で dp できるならその方がよい

// ・複数の d で計算するとき：同じ lambda

// に対する計算をメモ化しておくと定数倍高速？ 　・ABC305

template <typename T, typename F>
T monge_shortest_path_d_edge(int N, int d, T f_lim, F f) {
  assert(d <= N);
  auto calc_L = [&](T lambda) -> T {
    auto cost = [&](int frm, int to) -> T { return f(frm, to) + lambda; };
    vc<T> dp = monge_shortest_path<T>(N, cost);
    return dp[N] - lambda * d;
  };

  auto [x, fx] = fibonacci_search<T, false>(calc_L, -3 * f_lim, 3 * f_lim + 1);
  return fx;
}

// https://topcoder-g-hatena-ne-jp.jag-icpc.org/spaghetti_source/20120915/1347668163.html

// Prop 1

// 上三角 monge A, B

// C[i][j] = min_k (A[i][k] + B[k][j])

template <typename T, typename F1, typename F2>
vvc<T> monge_matrix_product(int N, F1 A, F2 B) {
  vv(T, C, N + 1, N + 1, infty<T>);
  vc<int> K(N + 1);
  FOR(i, N + 1) C[i][i] = A(i, i) + B(i, i), K[i] = i;
  FOR(s, 1, N + 1) {
    vc<int> newK(N + 1 - s);
    FOR(i, N + 1 - s) {
      int j = i + s;
      int p = K[i], q = K[i + 1];
      FOR(k, p, q + 1) if (chmin(C[i][j], A(i, k) + B(k, j))) newK[i] = k;
    }
    swap(K, newK);
  }
  return C;
}

#line 2 "convex/larsch.hpp"

// https://noshi91.github.io/Library/algorithm/larsch.cpp.html
template <class T>
class LARSCH {
  struct reduce_row;
  struct reduce_col;

  struct reduce_row {
    int n;
    std::function<T(int, int)> f;
    int cur_row;
    int state;
    std::unique_ptr<reduce_col> rec;

    reduce_row(int n_) : n(n_), f(), cur_row(0), state(0), rec() {
      const int m = n / 2;
      if (m != 0) { rec = std::make_unique<reduce_col>(m); }
    }

    void set_f(std::function<T(int, int)> f_) {
      f = f_;
      if (rec) {
        rec->set_f([&](int i, int j) -> T { return f(2 * i + 1, j); });
      }
    }

    int get_argmin() {
      const int cur_row_ = cur_row;
      cur_row += 1;
      if (cur_row_ % 2 == 0) {
        const int prev_argmin = state;
        const int next_argmin = [&]() {
          if (cur_row_ + 1 == n) {
            return n - 1;
          } else {
            return rec->get_argmin();
          }
        }();
        state = next_argmin;
        int ret = prev_argmin;
        for (int j = prev_argmin + 1; j <= next_argmin; j += 1) {
          if (f(cur_row_, ret) > f(cur_row_, j)) { ret = j; }
        }
        return ret;
      } else {
        if (f(cur_row_, state) <= f(cur_row_, cur_row_)) {
          return state;
        } else {
          return cur_row_;
        }
      }
    }
  };

  struct reduce_col {
    int n;
    std::function<T(int, int)> f;
    int cur_row;
    std::vector<int> cols;
    reduce_row rec;

    reduce_col(int n_) : n(n_), f(), cur_row(0), cols(), rec(n) {}

    void set_f(std::function<T(int, int)> f_) {
      f = f_;
      rec.set_f([&](int i, int j) -> T { return f(i, cols[j]); });
    }

    int get_argmin() {
      const int cur_row_ = cur_row;
      cur_row += 1;
      const auto cs = [&]() -> std::vector<int> {
        if (cur_row_ == 0) {
          return {{0}};
        } else {
          return {{2 * cur_row_ - 1, 2 * cur_row_}};
        }
      }();
      for (const int j: cs) {
        while ([&]() {
          const int size = cols.size();
          return size != cur_row_ && f(size - 1, cols.back()) > f(size - 1, j);
        }()) {
          cols.pop_back();
        }
        if (int(cols.size()) != n) { cols.push_back(j); }
      }
      return cols[rec.get_argmin()];
    }
  };

  std::unique_ptr<reduce_row> base;

public:
  LARSCH(int n, std::function<T(int, int)> f)
      : base(std::make_unique<reduce_row>(n)) {
    base->set_f(f);
  }

  int get_argmin() { return base->get_argmin(); }
};
#line 2 "convex/smawk.hpp"

// select(i,j,k) は (i,j) -> (i,k) を行うかどうか
// 残念ながら monotone minima より高速な場合が存在しない説がある
// https://codeforces.com/contest/1423/problem/M
template <typename F>
vc<int> smawk(int H, int W, F select) {
  auto dfs = [&](auto& dfs, vc<int> X, vc<int> Y) -> vc<int> {
    int N = len(X);
    if (N == 0) return {};
    vc<int> YY;
    for (auto&& y: Y) {
      while (len(YY)) {
        int py = YY.back(), x = X[len(YY) - 1];
        if (!select(x, py, y)) break;
        YY.pop_back();
      }
      if (len(YY) < len(X)) YY.eb(y);
    }
    vc<int> XX;
    FOR(i, 1, len(X), 2) XX.eb(X[i]);
    vc<int> II = dfs(dfs, XX, YY);
    vc<int> I(N);
    FOR(i, len(II)) I[i + i + 1] = II[i];
    int p = 0;
    FOR(i, 0, N, 2) {
      int LIM = (i + 1 == N ? Y.back() : I[i + 1]);
      int best = Y[p];
      while (Y[p] < LIM) {
        ++p;
        if (select(X[i], best, Y[p])) best = Y[p];
      }
      I[i] = best;
    }
    return I;
  };
  vc<int> X(H), Y(W);
  iota(all(X), 0), iota(all(Y), 0);
  return dfs(dfs, X, Y);
}
#line 1 "other/fibonacci_search.hpp"
// [L, R) での極小値をひとつ求める、単峰は不要
template <typename T, bool MINIMIZE, typename F>
pair<ll, T> fibonacci_search(F f, ll L, ll R) {
  assert(L < R);
  --R;
  ll a = L, b = L + 1, c = L + 2, d = L + 3;
  int n = 0;
  while (d < R) { b = c, c = d, d = b + c - a, ++n; }
  auto get = [&](ll x) -> T {
    if (R < x) return infty<T>;
    return (MINIMIZE ? f(x) : -f(x));
  };
  T ya = get(a), yb = get(b), yc = get(c), yd = get(d);
  // この中で極小ならば全体でも極小、を維持する
  FOR(n) {
    if (yb <= yc) {
      d = c, c = b, b = a + d - c;
      yd = yc, yc = yb, yb = get(b);
    } else {
      a = b, b = c, c = a + d - b;
      ya = yb, yb = yc, yc = get(c);
    }
  }
  ll x = a;
  T y = ya;
  if (chmin(y, yb)) x = b;
  if (chmin(y, yc)) x = c;
  if (chmin(y, yd)) x = d;
  if (MINIMIZE) return {x, y};
  return {x, -y};
}
#line 4 "convex/monge.hpp"

// 定義域 [0, N] の範囲で f の monge 性を確認

template <typename T, typename F>
bool check_monge(int N, F f) {
  FOR(l, N + 1) FOR(k, l) FOR(j, k) FOR(i, j) {
    T lhs = f(i, l) + f(j, k);
    T rhs = f(i, k) + f(j, l);
    if (lhs < rhs) {
      print("monge ng");
      print(i, j, k, l, f(i, k), f(i, l), f(j, k), f(j, l), lhs, rhs);
      return false;
    }
  }
  print("monge ok");
  return true;
}

// newdp[j] = min (dp[i] + f(i,j))

template <typename T, typename F>
vc<T> monge_dp_update(int N, vc<T>& dp, F f) {
  assert(len(dp) == N + 1);
  auto select = [&](int i, int j, int k) -> int {
    if (i <= k) return j;
    return (dp[j] + f(j, i) > dp[k] + f(k, i) ? k : j);
  };
  vc<int> I = SMAWK(N + 1, N + 1, select);
  vc<T> newdp(N + 1, infty<T>);
  FOR(j, N + 1) {
    int i = I[j];
    chmin(newdp[j], dp[i] + f(i, j));
  }
  return newdp;
}

// 遷移回数を問わない場合

template <typename T, typename F>
vc<T> monge_shortest_path(int N, F f) {
  vc<T> dp(N + 1, infty<T>);
  dp[0] = 0;
  LARSCH<T> larsch(N, [&](int i, int j) -> T {
    ++i;
    if (i <= j) return infty<T>;
    return dp[j] + f(j, i);
  });
  FOR(r, 1, N + 1) {
    int l = larsch.get_argmin();
    dp[r] = dp[l] + f(l, r);
  }
  return dp;
}

// https://noshi91.github.io/algorithm-encyclopedia/d-edge-shortest-path-monge

// |f| の上限 f_lim も渡す

// ・larsch が結構重いので、自前で dp できるならその方がよい

// ・複数の d で計算するとき：同じ lambda

// に対する計算をメモ化しておくと定数倍高速？ 　・ABC305

template <typename T, typename F>
T monge_shortest_path_d_edge(int N, int d, T f_lim, F f) {
  assert(d <= N);
  auto calc_L = [&](T lambda) -> T {
    auto cost = [&](int frm, int to) -> T { return f(frm, to) + lambda; };
    vc<T> dp = monge_shortest_path<T>(N, cost);
    return dp[N] - lambda * d;
  };

  auto [x, fx] = fibonacci_search<T, false>(calc_L, -3 * f_lim, 3 * f_lim + 1);
  return fx;
}

// https://topcoder-g-hatena-ne-jp.jag-icpc.org/spaghetti_source/20120915/1347668163.html

// Prop 1

// 上三角 monge A, B

// C[i][j] = min_k (A[i][k] + B[k][j])

template <typename T, typename F1, typename F2>
vvc<T> monge_matrix_product(int N, F1 A, F2 B) {
  vv(T, C, N + 1, N + 1, infty<T>);
  vc<int> K(N + 1);
  FOR(i, N + 1) C[i][i] = A(i, i) + B(i, i), K[i] = i;
  FOR(s, 1, N + 1) {
    vc<int> newK(N + 1 - s);
    FOR(i, N + 1 - s) {
      int j = i + s;
      int p = K[i], q = K[i + 1];
      FOR(k, p, q + 1) if (chmin(C[i][j], A(i, k) + B(k, j))) newK[i] = k;
    }
    swap(K, newK);
  }
  return C;
}

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