Partition Equal Subset Sum

LeetCode 416 | Difficulty: Medium

Medium

Problem Description

Given an integer array nums, return true if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or false otherwise.

Example 1:

Input: nums = [1,5,11,5]
Output: true
Explanation: The array can be partitioned as [1, 5, 5] and [11].

Example 2:

Input: nums = [1,2,3,5]
Output: false
Explanation: The array cannot be partitioned into equal sum subsets.

Constraints:

- `1 <= nums.length <= 200`

- `1 <= nums[i] <= 100`

Topics: Array, Dynamic Programming

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

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Solutions

Solution 1: C# (Best: 96 ms)

Metric	Value
Runtime	96 ms
Memory	42.1 MB
Date	2021-12-13

Solution

public class Solution {
    public bool CanPartition(int[] nums) {
    int sum = nums.Sum();
    int n = nums.Length;
    if((sum&1) != 0) return false;
    sum /= 2;
    
    bool?[,] memo = new bool?[n+1, sum+1];
    return subsetSum(nums, 0, sum, memo);   
}

bool subsetSum(int[] nums, int pos, int sum, bool?[,] memo)
{
    if(sum == 0) return true;
    if(pos>= nums.Length || sum < 0) return false;
    if(memo[pos,sum] != null) return memo[pos,sum].Value;
    memo[pos,sum] = subsetSum(nums, pos+1, sum-nums[pos], memo) || subsetSum(nums, pos+1, sum, memo);
    return memo[pos, sum].Value;
}
}

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Complexity Analysis

Approach	Time	Space
DP (2D)	$O(n × m)$	$O(n × m)$

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Interview Tips

Key Points

• Discuss the brute force approach first, then optimize. Explain your thought process.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.