Divide And Conquer Optimization (dp/divide-and-conquer-optimization.hpp)

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• Last update: 2024-05-01 23:44:47+09:00
• Include: #include "dp/divide-and-conquer-optimization.hpp"

概要

$dp[i][j] = \min_{0 \leq k \lt j}\{dp[i-1][k]+f(k,j)\}$ の形のDPを高速化するテク.

$f(k,j)$ は $0 \leq k \lt j \leq W$ で定義される $2$ 変数関数.

各行について Monotone-Minima を適用する.

• divide_and_conquer_optimization(H, W, INF, f): dp 配列を返す.

計算量

• $O(HW \log W)$

Depends on

• Monotone Minima (dp/monotone-minima.hpp)

Verified with

• test/verify/aoj-2603.test.cpp

Code

#include "monotone-minima.hpp"

template <typename T, typename Compare = less<T> >
vector<vector<T> > divide_and_conquer_optimization(
    int H, int W, T INF, const function<T(int, int)> &f,
    const Compare &comp = Compare()) {
  vector<vector<T> > dp(H + 1, vector<T>(W + 1, INF));
  dp[0][0] = 0;
  for (int i = 1; i <= H; i++) {
    function<T(int, int)> get_cost = [&](int y, int x) {
      if (x >= y) return INF;
      return dp[i - 1][x] + f(x, y);
    };
    auto ret = monotone_minima(W + 1, W + 1, get_cost, comp);
    for (int j = 0; j <= W; j++) dp[i][j] = ret[j].second;
  }
  return dp;
}

#line 1 "dp/monotone-minima.hpp"
template <typename T, typename Compare = less<T> >
vector<pair<int, T> > monotone_minima(int H, int W,
                                      const function<T(int, int)> &f,
                                      const Compare &comp = Compare()) {
  vector<pair<int, T> > dp(H);
  function<void(int, int, int, int)> dfs = [&](int top, int bottom, int left,
                                               int right) {
    if (top > bottom) return;
    int line = (top + bottom) / 2;
    T ma;
    int mi = -1;
    for (int i = left; i <= right; i++) {
      T cst = f(line, i);
      if (mi == -1 || comp(cst, ma)) {
        ma = cst;
        mi = i;
      }
    }
    dp[line] = make_pair(mi, ma);
    dfs(top, line - 1, left, mi);
    dfs(line + 1, bottom, mi, right);
  };
  dfs(0, H - 1, 0, W - 1);
  return dp;
}
#line 2 "dp/divide-and-conquer-optimization.hpp"

template <typename T, typename Compare = less<T> >
vector<vector<T> > divide_and_conquer_optimization(
    int H, int W, T INF, const function<T(int, int)> &f,
    const Compare &comp = Compare()) {
  vector<vector<T> > dp(H + 1, vector<T>(W + 1, INF));
  dp[0][0] = 0;
  for (int i = 1; i <= H; i++) {
    function<T(int, int)> get_cost = [&](int y, int x) {
      if (x >= y) return INF;
      return dp[i - 1][x] + f(x, y);
    };
    auto ret = monotone_minima(W + 1, W + 1, get_cost, comp);
    for (int j = 0; j <= W; j++) dp[i][j] = ret[j].second;
  }
  return dp;
}

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