dp/knapsack-problem/ones-and-zeroes
Ones and Zeroes
描述
You are given an array of binary strings strs
and two integers m
and n
.
Return the size of the largest subset of strs
such that there are at most m
0's and n
1's in the subset.
A set x
is a subset of a set y
if all elements of x
are also elements of y
.
Example 1:
Input: strs = ["10","0001","111001","1","0"], m = 5, n = 3
Output: 4
Explanation: The largest subset with at most 5 0's and 3 1's is {"10", "0001", "1", "0"}, so the answer is 4.
Other valid but smaller subsets include {"0001", "1"} and {"10", "1", "0"}.
{"111001"} is an invalid subset because it contains 4 1's, greater than the maximum of 3.
Example 2:
Input: strs = ["10","0","1"], m = 1, n = 1
Output: 2
Explanation: The largest subset is {"0", "1"}, so the answer is 2.
Constraints:
- 1 <= strs.length <= 600
- 1 <= strs[i].length <= 100
- strs[i] consists only of digits '0' and '1'.
- 1 <= m, n <= 100
分析
0-1 背包问题,每个字符串有两种重量,0 的个数和 1 的个数,每个字符串的价值都是 1,因此这个问题里有两个背包,一个装 0,一个装 1,价值则是背包里字符串的个数。
令f[i][j][k]
表示把前i
个字符串装进容量为j
的背包 0 和容量为k
背包 1,可以获得的最大集合的大小,则状态转移方程是:
$$f[i][j][k]=\max\left{f[i-1][j][k], f[i-1][j-w_0[i][k-w_1[i]]+1\right}$$
代码
# TODO
// Ones and Zeroes
// 0-1 knapsack problem
// Time Complexity: O(l*m*n), Space Complexity: O(m*n)
class Solution {
public int findMaxForm(String[] strs, int m, int n) {
int[][] f = new int[m+1][n+1];
int[] w0 = new int[strs.length];
int[] w1 = new int[strs.length];
for (int i = 0; i < strs.length; ++i) {
w0[i] = numberOfZeroes(strs[i]);
w1[i] = strs[i].length() - w0[i];
}
for (int i = 0; i < strs.length; ++i) {
for(int j = m; j >= w0[i]; --j)
for(int k = n;k >= w1[i]; --k) {
f[j][k] = Math.max(f[j][k], f[j-w0[i]][k-w1[i]]+1);
}
}
return f[m][n];
}
private static int numberOfZeroes(String s) {
int count = 0;
for (char c : s.toCharArray()) {
if (c == '0') count++;
}
return count;
}
}
// TODO