#410

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Split Array Largest Sum

Solve the 'Split Array Largest Sum' problem by minimizing the largest sum across k subarrays using dynamic programming and binary search.

array binary search dynamic programming greedy prefix sum

Problem Statement

Given an array of integers nums and an integer k, you are tasked with splitting the array into k non-empty contiguous subarrays. The objective is to minimize the largest sum among these subarrays. The array must be split in such a way that the sum of the subarrays is as balanced as possible, with the largest sum minimized.

The problem requires careful partitioning of nums and is typically approached using dynamic programming or binary search. The solution should ensure that no subarray sum exceeds a certain value while keeping the partitioning efficient. The challenge lies in finding the optimal partition strategy that minimizes the largest sum of the subarrays.

Examples

Example 1

Input: nums = [7,2,5,10,8], k = 2

Output: 18

There are four ways to split nums into two subarrays. The best way is to split it into [7,2,5] and [10,8], where the largest sum among the two subarrays is only 18.

Example 2

Input: nums = [1,2,3,4,5], k = 2

Output: 9

There are four ways to split nums into two subarrays. The best way is to split it into [1,2,3] and [4,5], where the largest sum among the two subarrays is only 9.

Constraints

1 <= nums.length <= 1000
0 <= nums[i] <= 106
1 <= k <= min(50, nums.length)

Solution Approach

Binary Search on the Maximum Subarray Sum

A binary search is used to find the minimum possible largest sum by iterating over the possible sums. The search space is between the maximum element of nums and the total sum of nums. For each mid-value, check if it's possible to partition the array into k subarrays with that sum using a greedy approach.

Greedy Partitioning Approach

For a given sum from the binary search, the array is greedily partitioned. Starting from the first element, keep adding elements to the current subarray until adding the next element would exceed the target sum. Then, start a new subarray. If more than k subarrays are required, the current sum is too small.

State Transition Dynamic Programming

In dynamic programming, we maintain an array where each element represents the minimum largest sum for a given number of partitions. The state transitions are based on selecting partitions at different points in the array, ensuring that the largest sum for the subarrays is minimized.

Complexity Analysis

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

The time complexity of this solution is O(n * log(sum(nums))), where n is the length of nums and sum(nums) is the total sum of the array. This comes from performing binary search on the sum and iterating through the array for each search step. The space complexity is O(n), which is needed for dynamic programming table storage and partitioning checks.

What Interviewers Usually Probe

Understanding how binary search can be applied in optimization problems is crucial.
Recognizing when to switch between greedy algorithms and dynamic programming is key for this problem.
The candidate should be able to efficiently identify the correct partitioning strategy based on dynamic constraints.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the binary search boundaries. Ensure that the lower boundary starts from the maximum element in the array.
Incorrect greedy approach leading to too many subarrays. It's important to balance the subarray sums correctly.
Failure to optimize dynamic programming transitions efficiently, leading to unnecessary computations and increased complexity.

Follow-up variants

Allowing k to be a dynamic value, where it changes based on additional constraints or conditions.
Adapting the problem to handle negative numbers in nums, which may change how partitions are handled.
Optimizing the solution further using a more efficient greedy strategy or advanced dynamic programming techniques.

FAQ

What is the main algorithmic approach for Split Array Largest Sum?

The problem is best tackled with a combination of binary search and dynamic programming to minimize the largest sum across k subarrays.

Can the problem be solved using only greedy algorithms?

While a greedy approach helps, it must be combined with binary search to efficiently find the minimum largest sum for k subarrays.

How do I optimize the binary search in this problem?

The binary search should focus on finding the smallest possible largest sum by adjusting the boundaries between the maximum element and the total sum of nums.

What are the space and time complexities of the solution?

The time complexity is O(n * log(sum(nums))) and the space complexity is O(n) due to the dynamic programming table used for state transitions.

How can GhostInterview help with this problem?

GhostInterview provides a clear breakdown of how to approach the problem step-by-step, including insights into binary search, dynamic programming, and efficient partitioning.

terminal

Solution

Solution 1: Binary Search

We notice that the larger the maximum sum of the subarrays, the fewer the number of subarrays. When there is a maximum sum of the subarrays that meets the condition, then a larger maximum sum of the subarrays will definitely meet the condition. This means that we can perform a binary search for the maximum sum of the subarrays to find the smallest value that meets the condition.

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class Solution:
    def splitArray(self, nums: List[int], k: int) -> int:
        def check(mx):
            s, cnt = inf, 0
            for x in nums:
                s += x
                if s > mx:
                    s = x
                    cnt += 1
            return cnt <= k

        left, right = max(nums), sum(nums)
        return left + bisect_left(range(left, right + 1), True, key=check)

Continue Practicing

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