Submission #646539

#	Submission time	Handle	Problem	Language	Result	Execution time	Memory
646539	2022-09-30T02:37:03 Z	alvinpiter	Knapsack (NOI18_knapsack)	C++17	100 / 100	64 ms	19524 KB

knapsack

/*
Observations:
* Since the weight of each item is >= 1, we can take at most 2000 items.
* The weight of each item is <= 2000. This allow us to group the items and only take the top 2000 most expensive ones.
  We can have tighter bound tho, for example, when considering items with weight x, we can take at most 2000/x items.

Dynamic programming solution:
dp[w][c] -> maximum price achievable if we only consider items whose weight is between 1..w and our
bag capacity is c.

The complexity of the solution is around W^2 log W.
The log factor comes from the second observation.
*/

#include<bits/stdc++.h>
using namespace std;
#define MAXW 2000

int S, N, dp[MAXW + 3][MAXW + 3];
vector<pair<int, int> > itemsWithWeight[MAXW + 3];
vector<int> itemPricesWithWeight[MAXW + 3];

int main() {
	ios_base::sync_with_stdio(false);
	cin.tie(nullptr);
	cout.tie(nullptr);

  cin >> S >> N;
  for (int i = 1; i <= N; i++) {
    int v, w, k;
    cin >> v >> w >> k;

    itemsWithWeight[w].push_back({v, k});
  }

  for (int w = 1; w <= MAXW; w++) {
    sort(itemsWithWeight[w].rbegin(), itemsWithWeight[w].rend());
    for (auto [price, count]: itemsWithWeight[w]) {
      for (int i = 0; i < count && itemPricesWithWeight[w].size() * w <= MAXW; i++) {
        itemPricesWithWeight[w].push_back(price);
      }
    }
  }

  memset(dp, 0, sizeof(dp));
  for (int w = 1; w <= MAXW; w++) {
    for (int cap = 1; cap <= S; cap++) {
      // Skip all items with weight w.
      dp[w][cap] = dp[w - 1][cap];

      int sumPrice = 0;
      int sumWeight = 0;
      for (auto price: itemPricesWithWeight[w]) {
        sumPrice += price;
        sumWeight += w;

        if (sumWeight <= cap) {
          dp[w][cap] = max(dp[w][cap], sumPrice + dp[w - 1][cap - sumWeight]);
        } else {
          break;
        }
      }
    }
  }

  cout << dp[MAXW][S] << endl;
}

Subtask #1
12.0 / 12.0

#	Verdict	Execution time	Memory	Grader output
1	Correct	12 ms	16084 KB	Output is correct
2	Correct	9 ms	16016 KB	Output is correct
3	Correct	12 ms	16116 KB	Output is correct
4	Correct	13 ms	16084 KB	Output is correct

Subtask #2
17.0 / 17.0

#	Verdict	Execution time	Memory	Grader output
1	Correct	7 ms	16036 KB	Output is correct
2	Correct	12 ms	16084 KB	Output is correct
3	Correct	11 ms	16084 KB	Output is correct
4	Correct	11 ms	16084 KB	Output is correct
5	Correct	12 ms	16084 KB	Output is correct
6	Correct	11 ms	16124 KB	Output is correct
7	Correct	14 ms	16128 KB	Output is correct
8	Correct	14 ms	16084 KB	Output is correct
9	Correct	13 ms	16044 KB	Output is correct
10	Correct	11 ms	16040 KB	Output is correct

Subtask #3
20.0 / 20.0

#	Verdict	Execution time	Memory	Grader output
1	Correct	7 ms	16036 KB	Output is correct
2	Correct	12 ms	16084 KB	Output is correct
3	Correct	11 ms	16084 KB	Output is correct
4	Correct	11 ms	16084 KB	Output is correct
5	Correct	12 ms	16084 KB	Output is correct
6	Correct	11 ms	16124 KB	Output is correct
7	Correct	14 ms	16128 KB	Output is correct
8	Correct	14 ms	16084 KB	Output is correct
9	Correct	13 ms	16044 KB	Output is correct
10	Correct	11 ms	16040 KB	Output is correct
11	Correct	7 ms	16084 KB	Output is correct
12	Correct	13 ms	16028 KB	Output is correct
13	Correct	11 ms	16016 KB	Output is correct
14	Correct	12 ms	16040 KB	Output is correct
15	Correct	12 ms	16084 KB	Output is correct
16	Correct	14 ms	16128 KB	Output is correct
17	Correct	12 ms	16112 KB	Output is correct
18	Correct	11 ms	16128 KB	Output is correct
19	Correct	11 ms	16084 KB	Output is correct
20	Correct	14 ms	16128 KB	Output is correct

Subtask #4
24.0 / 24.0

#	Verdict	Execution time	Memory	Grader output
1	Correct	12 ms	16084 KB	Output is correct
2	Correct	9 ms	16016 KB	Output is correct
3	Correct	12 ms	16116 KB	Output is correct
4	Correct	13 ms	16084 KB	Output is correct
5	Correct	7 ms	16036 KB	Output is correct
6	Correct	12 ms	16084 KB	Output is correct
7	Correct	11 ms	16084 KB	Output is correct
8	Correct	11 ms	16084 KB	Output is correct
9	Correct	12 ms	16084 KB	Output is correct
10	Correct	11 ms	16124 KB	Output is correct
11	Correct	14 ms	16128 KB	Output is correct
12	Correct	14 ms	16084 KB	Output is correct
13	Correct	13 ms	16044 KB	Output is correct
14	Correct	11 ms	16040 KB	Output is correct
15	Correct	7 ms	16084 KB	Output is correct
16	Correct	13 ms	16028 KB	Output is correct
17	Correct	11 ms	16016 KB	Output is correct
18	Correct	12 ms	16040 KB	Output is correct
19	Correct	12 ms	16084 KB	Output is correct
20	Correct	14 ms	16128 KB	Output is correct
21	Correct	12 ms	16112 KB	Output is correct
22	Correct	11 ms	16128 KB	Output is correct
23	Correct	11 ms	16084 KB	Output is correct
24	Correct	14 ms	16128 KB	Output is correct
25	Correct	6 ms	16084 KB	Output is correct
26	Correct	16 ms	16132 KB	Output is correct
27	Correct	14 ms	16040 KB	Output is correct
28	Correct	12 ms	16104 KB	Output is correct
29	Correct	15 ms	16028 KB	Output is correct
30	Correct	17 ms	16036 KB	Output is correct
31	Correct	12 ms	16028 KB	Output is correct
32	Correct	13 ms	16124 KB	Output is correct
33	Correct	14 ms	16120 KB	Output is correct
34	Correct	12 ms	16132 KB	Output is correct

Subtask #5
27.0 / 27.0

#	Verdict	Execution time	Memory	Grader output
1	Correct	12 ms	16084 KB	Output is correct
2	Correct	9 ms	16016 KB	Output is correct
3	Correct	12 ms	16116 KB	Output is correct
4	Correct	13 ms	16084 KB	Output is correct
5	Correct	7 ms	16036 KB	Output is correct
6	Correct	12 ms	16084 KB	Output is correct
7	Correct	11 ms	16084 KB	Output is correct
8	Correct	11 ms	16084 KB	Output is correct
9	Correct	12 ms	16084 KB	Output is correct
10	Correct	11 ms	16124 KB	Output is correct
11	Correct	14 ms	16128 KB	Output is correct
12	Correct	14 ms	16084 KB	Output is correct
13	Correct	13 ms	16044 KB	Output is correct
14	Correct	11 ms	16040 KB	Output is correct
15	Correct	7 ms	16084 KB	Output is correct
16	Correct	13 ms	16028 KB	Output is correct
17	Correct	11 ms	16016 KB	Output is correct
18	Correct	12 ms	16040 KB	Output is correct
19	Correct	12 ms	16084 KB	Output is correct
20	Correct	14 ms	16128 KB	Output is correct
21	Correct	12 ms	16112 KB	Output is correct
22	Correct	11 ms	16128 KB	Output is correct
23	Correct	11 ms	16084 KB	Output is correct
24	Correct	14 ms	16128 KB	Output is correct
25	Correct	6 ms	16084 KB	Output is correct
26	Correct	16 ms	16132 KB	Output is correct
27	Correct	14 ms	16040 KB	Output is correct
28	Correct	12 ms	16104 KB	Output is correct
29	Correct	15 ms	16028 KB	Output is correct
30	Correct	17 ms	16036 KB	Output is correct
31	Correct	12 ms	16028 KB	Output is correct
32	Correct	13 ms	16124 KB	Output is correct
33	Correct	14 ms	16120 KB	Output is correct
34	Correct	12 ms	16132 KB	Output is correct
35	Correct	34 ms	18076 KB	Output is correct
36	Correct	40 ms	18376 KB	Output is correct
37	Correct	49 ms	18012 KB	Output is correct
38	Correct	38 ms	18380 KB	Output is correct
39	Correct	45 ms	19140 KB	Output is correct
40	Correct	64 ms	19524 KB	Output is correct
41	Correct	48 ms	18600 KB	Output is correct
42	Correct	51 ms	18516 KB	Output is correct
43	Correct	51 ms	18640 KB	Output is correct
44	Correct	53 ms	18636 KB	Output is correct
45	Correct	45 ms	19448 KB	Output is correct
46	Correct	47 ms	18856 KB	Output is correct
47	Correct	49 ms	19388 KB	Output is correct
48	Correct	50 ms	19224 KB	Output is correct
49	Correct	39 ms	19112 KB	Output is correct
50	Correct	49 ms	19024 KB	Output is correct

Submission #646539

Compilation message

Subtask #1 12.0 / 12.0

Subtask #2 17.0 / 17.0

Subtask #3 20.0 / 20.0

Subtask #4 24.0 / 24.0

Subtask #5 27.0 / 27.0

Subtask #1
12.0 / 12.0

Subtask #2
17.0 / 17.0

Subtask #3
20.0 / 20.0

Subtask #4
24.0 / 24.0

Subtask #5
27.0 / 27.0