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Maximum Number of Ways to Partition an Array

Difficulty: Hard

Problem Description

You are given a 0-indexed integer array nums of length n. The number of ways to partition nums is the number of pivot indices that satisfy both conditions:

  1. 1 <= pivot < n
  2. nums[0] + nums[1] + ... + nums[pivot - 1] == nums[pivot] + nums[pivot + 1] + ... + nums[n - 1]

You are also given an integer k. You can choose to change the value of one element of nums to k, or to leave the array unchanged. Return the maximum possible number of ways to partition nums to satisfy both conditions after changing at most one element.

Key Insights

  • The problem requires finding pivot indices where the sum of elements before and after the pivot are equal.
  • We need to consider the effect of changing one element to k and how it affects the partitioning conditions.
  • Prefix sums can be utilized to efficiently compute the sums required for checking the partition conditions.
  • We can track the number of valid partitions before and after changing an element to determine the maximum.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(n)

Solution

The solution utilizes a prefix sum approach to efficiently calculate the sums of the two halves of the array around each pivot index. We first compute the total sum of the array and the prefix sums. For each potential pivot, we check if the sums of the two partitions are equal. Additionally, we evaluate the impact of changing each element to k by recalculating the potential partitions. The maximum count of partitions from both scenarios (original and modified) yields the result.

Code Solutions

def maxWaysToPartition(nums, k):
    n = len(nums)
    total_sum = sum(nums)
    prefix_sum = [0] * n
    prefix_sum[0] = nums[0]

    for i in range(1, n):
        prefix_sum[i] = prefix_sum[i - 1] + nums[i]

    count = 0
    valid_pivots = set()

    for pivot in range(1, n):
        left_sum = prefix_sum[pivot - 1]
        right_sum = total_sum - left_sum

        if left_sum == right_sum:
            count += 1
            valid_pivots.add(pivot)

    for i in range(n):
        original_value = nums[i]
        new_value = k
        if original_value != new_value:
            new_total_sum = total_sum - original_value + new_value
            for pivot in range(1, n):
                left_sum = prefix_sum[pivot - 1] - (original_value if i < pivot else 0) + (new_value if i < pivot else 0)
                right_sum = new_total_sum - left_sum

                if left_sum == right_sum and pivot not in valid_pivots:
                    count += 1
                    valid_pivots.add(pivot)

    return count
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