Problem Statement
Ones and Zeroes
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.
Examples
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
Problem Breakdown
To solve this problem, we need to:
- This problem is a variation of the 0/1 Knapsack problem with two constraints: the number of 0's and the number of 1's.
- We can use dynamic programming to solve this problem, where the state is defined by the number of 0's and 1's we have used so far.
- For each string, we have two options: include it in our subset or exclude it.
- We need to keep track of the maximum subset size for each possible combination of 0's and 1's.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Ones and Zeroes
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.
Examples
Example 1:
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:
The largest subset is {"0", "1"}, so the answer is 2.
Problem Breakdown
Constraints:
- 1 <= strs.length <= 600
- 1 <= strs[i].length <= 100
- strs[i] consists only of digits '0' and '1'
- 1 <= m, n <= 100
Key Insights:
This problem is a variation of the 0/1 Knapsack problem with two constraints: the number of 0's and the number of 1's.
We can use dynamic programming to solve this problem, where the state is defined by the number of 0's and 1's we have used so far.
For each string, we have two options: include it in our subset or exclude it.
We need to keep track of the maximum subset size for each possible combination of 0's and 1's.
Real-World Application
This problem has several practical applications:
Resource Allocation
Optimizing the allocation of limited resources (like memory or processing power) to maximize the number of tasks that can be completed.
Data Compression
Selecting a subset of binary data that fits within specific constraints while maximizing the amount of information retained.
Budget Optimization
Maximizing the number of projects that can be undertaken with limited budget constraints for different types of resources.