474. Ones and Zeroes

Problem Description

In this problem from LeetCode, we are tasked with finding the largest subset of binary strings from a given array strs. A binary string consists only of '0's and '1's. We also have two integer constraints, m and n, which are the maximum number of '0's and '1's allowed in our subset, respectively.

The key part to understand here is:

• A subset means every element in our selected group of strings must also exist in the original group (strs), but not necessarily all elements in strs have to be in our subset.
• When creating the largest subset, the total count of '0's in all the strings of our subset cannot exceed m, and the total count of '1's cannot exceed n.

Our goal is to identify the size of this largest subset. Size means the number of strings within it, not the length of each string.

Intuition

To solve this problem, we use a dynamic programming approach.

1. We start by understanding that the choice of including each string can affect our ability to include other strings. If we include a string with many '0's and '1's, it may prevent us from adding additional strings later. Thus, we have to make careful choices.

2. Generally, in dynamic programming, we try to solve smaller subproblems and use their results to construct solutions for larger subproblems. This concept is called optimal substructure.

3. Here, we define a two-dimensional array f where f[i][j] represents the size of the largest subset we can form with at most i '0's and j '1's. The values of i range from 0 to m, and the values of j range from 0 to n, representing all possible constraints we might encounter.

4. We initialize our array with zeros, as the largest subset with zero '0's and '1's is an empty subset, hence zero size.

5. We iterate through each string in strs and count its '0's and '1's. Then, for each string, we update our array f in a decreasing manner, starting from m to the count of '0's in the current string (a) and from n to the count of '1's (b).

6. While updating, we consider two cases for each cell f[i][j]:

   • Not including the current string, which means the f[i][j] would remain unchanged.
   • Including the current string, where we have to look at the value in the cell that represents the leftover capacity (f[i - a][j - b]) after including this string and add 1 to that value to represent the current string being counted.

7. We choose the maximum of these two choices at every step, which ensures that we always have the largest possible subset for a given i and j.

8. After iterating through all strings and updating the array, the value of f[m][n] will give us the size of the largest subset conforming to our constraints.

This dynamic programming solution is efficient as it avoids recalculating the largest subset sizes for every combination of '0's and '1's by building upon previously computed values.

Learn more about Dynamic Programming patterns.

Solution Approach

The implementation of the solution follows the dynamic programming approach to methodically work towards the final answer. Here's an in-depth walk-through of the pattern and the algorithm used:

1. Data Structure: A two-dimensional list f is created with dimensions (m + 1) x (n + 1). In Python, this is realized as a list of lists. Each cell f[i][j] in this array represents the size of the largest subset with i '0's and j '1's. The + 1 in both dimensions is used to include the case where 0 '0's or `1's are used.

2. Initialization: The two-dimensional list f is initialized with zeroes, since the largest subset without considering any strings (and thus having 0 '0's and 1's) has a size of 0`.

3. Counting '0's and '1's: For each string s in strs, s.count("0") and s.count("1") are called to count the number of '0's (a) and '1's (b) respectively.

4. Updating the DP Table: We iterate over the list in reverse for i from m to a - 1 and j from n to b - 1. We do this because we want to make sure that when we account for a new string, we are not overwriting cells that could affect the calculation of cells later in the iteration. This is a common technique in dynamic programming known as avoiding "state contamination."

5. Choice: At each cell f[i][j], we attempt to include the current string. To do this, we compare the existing value f[i][j] (not including the current string) with f[i - a][j - b] + 1 (including the string). f[i - a][j - b] represents the largest subset possible with the remaining capacity after including the current string. We add 1 because we are including the current string in our subset.

6. Taking the Maximum: We use Python's max function to always store the maximum of the two values. Thus, f[i][j] will always hold the size of the largest subset for the specific capacity represented by i and j.

7. Result: After completely filling the two-dimensional list, f[m][n] will give us the maximum size of our desired subset since it represents the size of the largest subset under the full capacity of m '0's and n '1's.

This implementation successfully leverages the central ideas of dynamic programming, namely optimal substructure (solving bigger problems by relying on the solutions to smaller problems) and overlapping subproblems (saving computation by storing intermediate results). The use of a two-dimensional DP table is crucial, as it allows tracking the state of the problem (how many '0's and '1's can still be included) at each step.

Example Walkthrough

Let's take a small example to illustrate the solution approach.

Assume we have the following array of binary strings strs: ["10", "0001", "111001", "1", "0"], and our integer constraints are m = 5 and n = 3. This means we cannot have more than 5 '0's and 3 '1's in our subset.

1. Initialization: We create a two-dimensional list f with dimensions (m + 1) x (n + 1), so f would be a 6x4 matrix, as we include zero counts. It's filled with zeros, like so:

1f = [
2  [0, 0, 0, 0],
3  [0, 0, 0, 0],
4  [0, 0, 0, 0],
5  [0, 0, 0, 0],
6  [0, 0, 0, 0],
7  [0, 0, 0, 0]
8]

2. Counting and updating: We go through each string in strs and update our DP table.

   • For the string "10", we have a=1 (the number of '0's) and b=1 (the number of '1's). We update cells in the range of i = 5 to 1 and j = 3 to 1, comparing the existing value f[i][j] with f[i - 1][j - 1] + 1 (since a=1 and b=1).

   After this string, our DP table update looks like this:

1f = [
2  [0, 0, 0, 0],  // No strings considered yet
3  [0, 1, 1, 1],  // '10' considered for i=1 (up to 1 '0's used)
4  [0, 1, 1, 1],
5  [0, 1, 1, 1],
6  [0, 1, 1, 1],
7  [0, 1, 1, 1]
8]

• Applying a similar process for other strings: "0001", "111001", "1", and "0", updating f for the '0's and '1's in each and choosing the max value for each cell.

3. Final DP Table: Once we process all strings, our DP table will display the maximum number of strings that can be included for a given i number of '0's and j number of '1's. For our case, we getf[m][n] as the result. Let's assume our final table after all updates looks like this:

1f = [
2  [0, 0, 0, 0],
3  [0, 1, 1, 1],
4  [1, 1, 2, 2],
5  [1, 2, 2, 2],
6  [1, 2, 3, 3],
7  [1, 2, 3, 4]
8]

4. Result: From the last entry f[5][3], we see that the maximum size of the subset we can get under the given constraints is 4.

Thus, with m = 5 and n = 3, we are able to include four strings from the array strs in our subset without exceeding the number of '0's and '1's allowed. The subset, in this case, could be ["10", "0001", "1", "0"], which includes 4 strings, adheres to the constraints (`5 '0's and 3 '1's), and is the largest possible subset for these constraints.

Solution Implementation

Time and Space Complexity

Time Complexity:

The time complexity of the given solution is O(k * m * n), where k is the length of the input list strs, m is the maximum number of zeroes, and n is the maximum number of ones that our subsets from strs can contain. This complexity arises because we iterate over all strings in strs, and for each string, we iterate through a 2D array of size m * n in a nested loop fashion.

Space Complexity:

The space complexity of the solution is O(m * n), as we are constructing a 2D array f with m + 1 rows and n + 1 columns to store the intermediate results for dynamic programming. No other data structures are used that grow with the size of the input, so the space complexity is directly proportional to the size of the 2D array.

Learn more about how to find time and space complexity quickly using problem constraints.