2270. Number of Ways to Split Array

2270. Number of Ways to Split Array

Difficulty: Medium

Topics: Array, Prefix Sum

You are given a 0-indexed integer array nums of length n.

nums contains a valid split at index i if the following are true:

• The sum of the first i + 1 elements is greater than or equal to the sum of the last n - i - 1 elements.
• There is at least one element to the right of i. That is, 0 <= i < n - 1.

Return the number of valid splits in nums.

Example 1:

• Input: nums = [10,4,-8,7]
• Output: 2
• Explanation: There are three ways of splitting nums into two non-empty parts:
  • Split nums at index 0. Then, the first part is [10], and its sum is 10. The second part is [4,-8,7], and its sum is 3. Since 10 >= 3, i = 0 is a valid split.
  • Split nums at index 1. Then, the first part is [10,4], and its sum is 14. The second part is [-8,7], and its sum is -1. Since 14 >= -1, i = 1 is a valid split.
  • Split nums at index 2. Then, the first part is [10,4,-8], and its sum is 6. The second part is [7], and its sum is 7. Since 6 < 7, i = 2 is not a valid split.
  • Thus, the number of valid splits in nums is 2.

Example 2:

• Input: nums = [2,3,1,0]
• Output: 2
• Explanation: There are two valid splits in nums:
  • Split nums at index 1. Then, the first part is [2,3], and its sum is 5. The second part is [1,0], and its sum is 1. Since 5 >= 1, i = 1 is a valid split.
  • Split nums at index 2. Then, the first part is [2,3,1], and its sum is 6. The second part is [0], and its sum is 0. Since 6 >= 0, i = 2 is a valid split.

Constraints:

• 2 <= nums.length <= 105
• -105 <= nums[i] <= 105

Hint:

1. For any index i, how can we find the sum of the first (i+1) elements from the sum of the first i elements?
2. If the total sum of the array is known, how can we check if the sum of the first (i+1) elements greater than or equal to the remaining elements?

Solution:

We can approach it using the following steps:

Approach:

1. Prefix Sum: First, we compute the cumulative sum of the array from the left, which helps in checking the sum of the first i+1 elements.
2. Total Sum: Compute the total sum of the array, which is useful in checking if the sum of the remaining elements is less than or equal to the sum of the first i+1 elements.
3. Iterate over the array: For each valid index i (where 0 <= i < n-1), we check if the sum of the first i+1 elements is greater than or equal to the sum of the last n-i-1 elements.
4. Efficiency: Instead of recalculating the sums repeatedly, use the prefix sum and the total sum for efficient comparisons.

Let's implement this solution in PHP: 2270. Number of Ways to Split Array

<?php
/**
 * @param Integer[] $nums
 * @return Integer
 */
function waysToSplitArray($nums) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Example usage:
$nums1 = [10, 4, -8, 7];
echo waysToSplitArray($nums1); // Output: 2

$nums2 = [2, 3, 1, 0];
echo waysToSplitArray($nums2); // Output: 2
?>

Explanation:

1. $totalSum: This variable stores the sum of all elements in the nums array.
2. $prefixSum: This variable keeps track of the cumulative sum of elements from the left (up to index i).
3. $remainingSum: This is the sum of the remaining elements from index i+1 to the end of the array. It is calculated by subtracting $prefixSum from $totalSum.
4. Valid Split Check: For each index i, we check if the prefix sum is greater than or equal to the remaining sum.

Time Complexity:

• O(n): We loop through the array once to compute the sum and once again to check for valid splits. Therefore, the time complexity is linear with respect to the length of the array.

Space Complexity:

• O(1): We are using only a few extra variables ($totalSum, $prefixSum, $remainingSum), so the space complexity is constant.

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