#1712

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Ways to Split Array Into Three Subarrays

The problem requires finding the number of ways to split an array into three subarrays where each split meets a certain sum condition.

array two pointers binary search prefix sum

Problem Statement

Given an integer array nums, you need to find the number of ways to split it into exactly three subarrays such that the sum of each subarray is equal. The split is considered good if the sum of the three resulting subarrays is the same. As the number of ways may be large, return the result modulo 10^9 + 7.

For example, with nums = [1,1,1], the only valid split is [1] [1] [1]. The problem also provides constraints where nums length is between 3 and 10^5, and each element is between 0 and 10^4.

Examples

Example 1

Input: nums = [1,1,1]

Output: 1

The only good way to split nums is [1] [1] [1].

Example 2

Input: nums = [1,2,2,2,5,0]

Output: 3

There are three good ways of splitting nums: [1] [2] [2,2,5,0] [1] [2,2] [2,5,0] [1,2] [2,2] [5,0]

Example 3

Input: nums = [3,2,1]

Output: 0

There is no good way to split nums.

Constraints

3 <= nums.length <= 105
0 <= nums[i] <= 104

Solution Approach

Binary Search on Answer Space

The solution uses binary search over the possible sum values. This allows the efficient determination of possible valid splits by narrowing down the range of sums that might work, optimizing the solution for large inputs.

Prefix Sum for Efficient Calculation

To quickly check if a subarray meets the sum condition, a prefix sum array is created. This allows for constant-time sum calculations over any subarray, making the algorithm efficient when checking each potential split.

Two Pointers for Subarray Validation

Using two pointers, we can efficiently check the potential valid splits by maintaining running sums on both ends of the array. This helps in avoiding redundant calculations and ensures the solution scales well.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the solution is O(n log S), where n is the length of the array and S is the range of possible sums, which is bounded by the sum of all elements in the array. The space complexity is O(n) due to the prefix sum array storing cumulative sums for efficient access.

What Interviewers Usually Probe

The candidate demonstrates the ability to handle large inputs efficiently using binary search and prefix sums.
The candidate identifies and uses a two-pointer technique to validate subarray splits in an optimized way.
The candidate understands the trade-off between different approaches and chooses the most efficient one.

Common Pitfalls or Variants

Common pitfalls

Not utilizing prefix sums, leading to inefficient subarray sum checks.
Failing to optimize the binary search step, resulting in suboptimal performance for larger inputs.
Overlooking edge cases where no valid split exists, returning incorrect results for such inputs.

Follow-up variants

Different variations of the problem could involve finding splits based on conditions other than sum equality, such as minimum or maximum values.
This problem could be extended to more than three subarrays, which would introduce more complexity into the binary search and validation steps.
In some cases, the constraints might change, requiring a solution that works for even larger arrays or different input value ranges.

FAQ

What is the core approach to solve the 'Ways to Split Array Into Three Subarrays' problem?

The core approach is to use binary search over the possible sum values combined with prefix sums to efficiently find valid splits.

How do I optimize the solution for large arrays in the 'Ways to Split Array Into Three Subarrays' problem?

By using binary search over the valid sum space and leveraging prefix sums for constant-time subarray sum calculations, the solution can scale efficiently.

What is the role of the prefix sum in this problem?

The prefix sum allows for quick calculation of subarray sums, which is crucial for efficiently checking if a potential split meets the sum condition.

What are common pitfalls when solving the 'Ways to Split Array Into Three Subarrays' problem?

Common pitfalls include neglecting to use prefix sums, not optimizing the binary search step, and overlooking edge cases where no valid split exists.

Can this problem be generalized to more than three subarrays?

Yes, the problem can be generalized to more subarrays, though this introduces additional complexity in both the binary search and validation steps.

terminal

Solution

Solution 1: Prefix Sum + Binary Search

First, we preprocess the prefix sum array $s$ of the array $nums$, where $s[i]$ represents the sum of the first $i+1$ elements of the array $nums$.

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class Solution:
    def waysToSplit(self, nums: List[int]) -> int:
        mod = 10**9 + 7
        s = list(accumulate(nums))
        ans, n = 0, len(nums)
        for i in range(n - 2):
            j = bisect_left(s, s[i] << 1, i + 1, n - 1)
            k = bisect_right(s, (s[-1] + s[i]) >> 1, j, n - 1)
            ans += k - j
        return ans % mod

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