説明

$dp[i][j] = \min_{0 \leq k \lt j}\{dp[i-1][k]+f(k,j)\}$

各行について Monotone-Minima

うく

説明を書く

計算量

  • $O(HW \log W)$

実装例

依存ライブラリ Monotone-Minima

  • divide_and_conquer_optimization($H$, $W$, $INF$, $f$): dp 配列を返す。$f(i, j)$ は区間 $[i, j)$ のコスト。
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;
}

検証

Codeforces Codeforces Round #438 F - Yet Another Minimization Problem

int main() {
  int N, K;
  cin >> N >> K;
  vector< int > A(N);
  for(int i = 0; i < N; i++) {
    cin >> A[i];
    --A[i];
  }
  constexpr int64_t INF = 1LL << 58;

  int64 L = 0, R = 0, sum = 0;
  vector< int > appear(100000);
  auto add = [&](int idx) { sum += appear[A[idx]]++; };
  auto del = [&](int idx) { sum -= --appear[A[idx]]; };
  function< int64_t(int l, int r) > f = [&](int l, int r) {
    while(L > l) add(--L);
    while(R < r) add(R++);
    while(L < l) del(L++);
    while(R > r) del(--R);
    return sum;
  };
  cout << divide_and_conquer_optimization(K, N, INF, f).back().back() << endl;
}