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3432. Count Partitions with Even Sum Difference

Description

You are given an integer array nums of length n.

A partition is defined as an index i where 0 <= i < n - 1, splitting the array into two non-empty subarrays such that:

  • Left subarray contains indices [0, i].
  • Right subarray contains indices [i + 1, n - 1].

Return the number of partitions where the difference between the sum of the left and right subarrays is even.

 

Example 1:

Input: nums = [10,10,3,7,6]

Output: 4

Explanation:

The 4 partitions are:

  • [10], [10, 3, 7, 6] with a sum difference of 10 - 26 = -16, which is even.
  • [10, 10], [3, 7, 6] with a sum difference of 20 - 16 = 4, which is even.
  • [10, 10, 3], [7, 6] with a sum difference of 23 - 13 = 10, which is even.
  • [10, 10, 3, 7], [6] with a sum difference of 30 - 6 = 24, which is even.

Example 2:

Input: nums = [1,2,2]

Output: 0

Explanation:

No partition results in an even sum difference.

Example 3:

Input: nums = [2,4,6,8]

Output: 3

Explanation:

All partitions result in an even sum difference.

 

Constraints:

  • 2 <= n == nums.length <= 100
  • 1 <= nums[i] <= 100

Solutions

Solution 1: Prefix Sum

We use two variables \(l\) and \(r\) to represent the sum of the left subarray and the right subarray, respectively. Initially, \(l = 0\) and \(r = \sum_{i=0}^{n-1} \textit{nums}[i]\).

Next, we traverse the first \(n - 1\) elements. Each time, we add the current element to the left subarray and subtract it from the right subarray. Then, we check if \(l - r\) is even. If it is, we increment the answer by one.

Finally, we return the answer.

The time complexity is \(O(n)\), where \(n\) is the length of the array \(\textit{nums}\). The space complexity is \(O(1)\).

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class Solution:
    def countPartitions(self, nums: List[int]) -> int:
        l, r = 0, sum(nums)
        ans = 0
        for x in nums[:-1]:
            l += x
            r -= x
            ans += (l - r) % 2 == 0
        return ans

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class Solution {
    public int countPartitions(int[] nums) {
        int l = 0, r = 0;
        for (int x : nums) {
            r += x;
        }
        int ans = 0;
        for (int i = 0; i < nums.length - 1; ++i) {
            l += nums[i];
            r -= nums[i];
            if ((l - r) % 2 == 0) {
                ++ans;
            }
        }
        return ans;
    }
}

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class Solution {
public:
    int countPartitions(vector<int>& nums) {
        int l = 0, r = accumulate(nums.begin(), nums.end(), 0);
        int ans = 0;
        for (int i = 0; i < nums.size() - 1; ++i) {
            l += nums[i];
            r -= nums[i];
            if ((l - r) % 2 == 0) {
                ++ans;
            }
        }
        return ans;
    }
};

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func countPartitions(nums []int) (ans int) {
    l, r := 0, 0
    for _, x := range nums {
        r += x
    }
    for _, x := range nums[:len(nums)-1] {
        l += x
        r -= x
        if (l-r)%2 == 0 {
            ans++
        }
    }
    return
}

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function countPartitions(nums: number[]): number {
    let l = 0;
    let r = nums.reduce((a, b) => a + b, 0);
    let ans = 0;
    for (const x of nums.slice(0, -1)) {
        l += x;
        r -= x;
        ans += (l - r) % 2 === 0 ? 1 : 0;
    }
    return ans;
}

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