Knapsack Limitations(個数制限つきナップサック問題) $O(NW)$ (dp/knapsack-limitations.hpp)

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• Last update: 2024-04-27 01:27:28+09:00
• Include: #include "dp/knapsack-limitations.hpp"

概要

個数制限つきナップサック問題を次に示す.

重さ $w_i$, 価値 $v_i$ であるような $N$ 種類の品物がある. $i$ 番目の品物は $m_i$ 個まで選ぶことができる. 重さの和が $W$ 以下となるように選ぶとき, 価値の最大値を求めよ.

スライド最大値を用いた動的計画法により効率的に計算可能.

• knapsack_limitations(w, m, v, W, NG, comp): W 以下の範囲で, 各重さについて価値の最大値を求める. NG は到達ができない場合の値で, comp は比較演算子.

計算量

• $O(NW)$

Required by

• Knapsack Limitations(個数制限つきナップサック問題) $O(N^2 \max(v_i)^2)$ (dp/knapsack-limitations-2.hpp)

Verified with

• test/verify/aoj-dpl-1-g.test.cpp
• test/verify/aoj-dpl-1-i.test.cpp

Code

template <typename T, typename Compare = greater<T> >
vector<T> knapsack_limitations(const vector<int> &w, const vector<int> &m,
                               const vector<T> &v, const int &W, const T &NG,
                               const Compare &comp = Compare()) {
  const int N = (int)w.size();
  vector<T> dp(W + 1, NG), deqv(W + 1);
  dp[0] = T();
  vector<int> deq(W + 1);
  for (int i = 0; i < N; i++) {
    if (w[i] == 0) {
      for (int j = 0; j <= W; j++) {
        if (dp[j] != NG && comp(dp[j] + v[i] * m[i], dp[j])) {
          dp[j] = dp[j] + v[i] * m[i];
        }
      }
    } else {
      for (int a = 0; a < w[i]; a++) {
        int s = 0, t = 0;
        for (int j = 0; w[i] * j + a <= W; j++) {
          if (dp[w[i] * j + a] != NG) {
            auto val = dp[w[i] * j + a] - j * v[i];
            while (s < t && comp(val, deqv[t - 1])) --t;
            deq[t] = j;
            deqv[t++] = val;
          }
          if (s < t) {
            dp[j * w[i] + a] = deqv[s] + j * v[i];
            if (deq[s] == j - m[i]) ++s;
          }
        }
      }
    }
  }
  return dp;
}

#line 1 "dp/knapsack-limitations.hpp"
template <typename T, typename Compare = greater<T> >
vector<T> knapsack_limitations(const vector<int> &w, const vector<int> &m,
                               const vector<T> &v, const int &W, const T &NG,
                               const Compare &comp = Compare()) {
  const int N = (int)w.size();
  vector<T> dp(W + 1, NG), deqv(W + 1);
  dp[0] = T();
  vector<int> deq(W + 1);
  for (int i = 0; i < N; i++) {
    if (w[i] == 0) {
      for (int j = 0; j <= W; j++) {
        if (dp[j] != NG && comp(dp[j] + v[i] * m[i], dp[j])) {
          dp[j] = dp[j] + v[i] * m[i];
        }
      }
    } else {
      for (int a = 0; a < w[i]; a++) {
        int s = 0, t = 0;
        for (int j = 0; w[i] * j + a <= W; j++) {
          if (dp[w[i] * j + a] != NG) {
            auto val = dp[w[i] * j + a] - j * v[i];
            while (s < t && comp(val, deqv[t - 1])) --t;
            deq[t] = j;
            deqv[t++] = val;
          }
          if (s < t) {
            dp[j * w[i] + a] = deqv[s] + j * v[i];
            if (deq[s] == j - m[i]) ++s;
          }
        }
      }
    }
  }
  return dp;
}

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