Can you suggest a LeetCode Medium problem similar to permutations using dynamic programming?

Question

Could you please provide a detailed explanation or solution approach for a LeetCode Medium problem that is similar to finding permutations using dynamic programming?

Accepted Answer

A LeetCode Medium problem that is conceptually similar to finding permutations using dynamic programming is the "Partition Equal Subset Sum" problem (LeetCode Problem 416). This problem requires using a subset of numbers to achieve a certain target sum, which shares similarities with dynamic programming strategies needed for permutations where you build up solutions incrementally.

### Problem Statement:
You are given a list of positive integers `nums`. Determine if you can partition the list into two subsets such that the sum of the elements in both subsets is equal.

### Solution Approach:
To decide if the list can be partitioned into two subsets with equal sum, the total sum of the elements in `nums` needs to be even (since only then can it be divided into two equal parts). Let's denote this target sum as `S = sum(nums) / 2`. The task is then to find if there exists a subset of `nums` that sums to `S`.

Here’s a detailed explanation of the dynamic programming approach:

1. **State Definition**:
   Define a boolean DP array `dp` where `dp[i]` will indicate if a subset with sum `i` can be formed using elements from `nums`.

2. **Initial State**:
   `dp[0] = True` because a subset sum of 0 can always be achieved with an empty subset.

3. **State Transition**:
   Iterate through each number in `nums` and update the DP array backward (from `S` down to the number). This is done to prevent overwriting results of the previous iteration, thereby maintaining valid answers.
   
   For each number `num` in the list, update the DP table as follows:
   ```python
   for num in nums:
       for i in range(S, num - 1, -1):
           dp[i] = dp[i] or dp[i - num]
   ```

This process uses the property that if a subset sum `i - num` can be achieved (`dp[i - num]` is True), then a subset sum `i` can also be achieved by including `num`.

4. **Final Check**:
   After processing all elements, check `dp[S]`. If it is `True`, it indicates a subset with the sum `S` can be formed, thus implying the original set can be partitioned into two subsets of equal sum.

### Example:

Suppose `nums = [1, 5, 11, 5]`.

1. Total sum = 22, which is even, so the target `S = 11`.
2. Initialize `dp = [True, False, ..., False]` of length 12 (0 to 11).
3. For `num = 1`, update: `dp[1]` becomes True.
4. For `num = 5`, update: `dp[6]` and `dp[5]` become True.
5. For `num = 11`, update: `dp[11]` becomes True, which means a subset sum of 11 is possible.

Finally, because `dp[11]` is True, the input list can indeed be partitioned into subsets of equal sum.

This dynamic programming approach efficiently determines the possibility of partitioning within `O(n*S)` time complexity where `n` is the number of elements in `nums` and `S` is half the total sum, making it a typical dynamic programming problem akin to permutation considerations.

Can you suggest a LeetCode Medium problem similar to permutations using dynamic programming?

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