Knapsack 1

Points: 7 (partial)

Time limit: 0.25s

Python 1.0s

Memory limit: 64M

Problem type

These problems are from the AtCoder DP contest, and were transferred onto DMOJ. All problem statements were made by several AtCoder users. As there is no access to the test data, all data is randomly generated. If there are issues with the statement or data, please open a ticket by clicking the "Report an issue" button at the bottom of the page.

There are $N$ ~N~ items, numbered $1, 2, \dots, N$ ~1, 2, \dots, N~. For each $i$ ~i~ $(1 \le i \le N)$ ~(1 \le i \le N)~, item $i$ ~i~ has a weight of $w_i$ ~w_i~ and a value of $v_i$ ~v_i~.

Taro has decided to choose some of the $N$ ~N~ items and carry them home in a knapsack. The capacity of the knapsack is $W$ ~W~, which means that the sum of the weights of items taken must be at most $W$ ~W~.

Find the maximum possible sum of the values of items that Taro takes home.

Constraints

All values in input are integers.
$1 \le N \le 100$ ~1 \le N \le 100~
$1 \le W \le 10^5$ ~1 \le W \le 10^5~
$1 \le w_i \le W$ ~1 \le w_i \le W~
$1 \le v_i \le 10^9$ ~1 \le v_i \le 10^9~

Input Specification

The first line of input will contain 2 space separated integers, $N$ ~N~ and $W$ ~W~.

The next $N$ ~N~ lines will contain 2 space separated integers, $w_i$ ~w_i~ and $v_i$ ~v_i~, the weight and value of item $i$ ~i~.

Output Specification

You are to output a single integer, the maximum possible sum of the values of items that Taro takes home.

Sample Input 1

Sample Output 1

Sample Input 2

5 5
1 1000000000
1 1000000000
1 1000000000
1 1000000000
1 1000000000

Sample Output 2

5000000000

Sample Input 3

Sample Output 3

Sample Explanations

For the first sample, items $1$ ~1~ and $3$ ~3~ should be taken. Then, the sum of the weights is $3+5 = 8$ ~3+5 = 8~, and the sum of the values is $30+60 = 90$ ~30+60 = 90~.

For the second sample, it is important to note that the answer may not fit in a 32-bit integer type.

For the third sample, items $2$ ~2~, $4$ ~4~, and $5$ ~5~ should be taken. Then, the sum of the weights is $5+6+3 = 14$ ~5+6+3 = 14~, and the sum of the values is $6+6+5 = 17$ ~6+6+5 = 17~.

Comments

4

OneYearOld commented on July 9, 2020, 10:34 a.m.

Does anyone know why I am getting std:bad_alloc on a few cases? My code works in my IDE without any errors. Any help would be very appreciated.

-6

maxcruickshanks commented on July 9, 2020, 3:11 p.m.

Instead of pushing back 0s for every value of W (when initializing your DP vector), try using a large array and declare it outside of your main function (example). Also, don't hard-code cases when your code is WAing; it doesn't help you debug your code.

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8

wleung_bvg commented on July 9, 2020, 3:37 p.m.

The cause of std::bad_alloc is unrelated to where the vector is located (in general, only fixed sized arrays have issues when allocated locally). The issue is that your code is exceeding the memory limit for this problem. A two dimensional vector of size $N \times W$ ~N \times W~ $64$ ~64~-bit integers can take up to $80$ ~80~ MB of storage. In addition, vector::push_back can result in the actual capacity of the vector being up to twice as large as the actual size (assuming vector::reserve is not called), thus resulting in $160$ ~160~ MB of storage in the worst case.

You should aim for a solution that uses $\mathcal{O}(N + W)$ ~\mathcal{O}(N + W)~ memory.