Split Array into K Subarrays
Minimize Largest Sum using Binary Search

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Problem Statement

Given an array of non-negative integers and an integer K, split the array into K contiguous subarrays such that the maximum sum among these subarrays is minimized.

You must split the array into exactly K parts.
Each part must be a contiguous segment of the array.
Every element must be used and order must be maintained.

The goal is to make the largest sum among the K subarrays as small as possible.

If the array is empty or K is zero, return 0 as the answer.

Examples❯

Input Array	K	Output	Description
[7, 2, 5, 10, 8]	2	18	Split as [7,2,5] and [10,8]. Max sum = 14 and 18 → answer is 18
[1, 2, 3, 4, 5]	2	9	Best split is [1,2,3] and [4,5] with max sums 6 and 9
[1, 4, 4]	3	4	Split each element as one subarray → max sum is 4
[5]	1	5	Only one element, one group → max sum = 5
[1, 2, 3]	1	6	All elements in one group → total sum = 6
[1, 2, 3]	3	3	Each element in its own group → largest is 3
[]	3	0	Empty array → no subarrays can be formed
[1, 2, 3]	0	0	Zero partitions not allowed → return 0
[10, 20, 30]	5	30	More partitions than elements → some will be empty, max element becomes answer

Solution❯

This problem asks us to split the array into K contiguous subarrays in such a way that the largest sum among these subarrays is as small as possible.

Understanding the Goal

We're not just trying to split the array randomly. We need to find a split where the heaviest load (the subarray with the highest sum) is minimized. Imagine splitting tasks across K workers, and you want to minimize the maximum load any one worker handles.

What Are We Really Searching For?

The answer is somewhere between:

The largest single element in the array (because no subarray can be smaller than this)
And the total sum of the array (if we don’t split it at all)

So, we’re trying to find the minimum possible value of the largest subarray sum — such that we can split the array into exactly K subarrays without any subarray exceeding that sum.

How We Approach It

We use binary search on the answer space. Instead of iterating through all ways to split (which is slow), we guess a value and check if it's possible to split the array using that as the maximum allowed subarray sum. Based on whether it's possible or not, we adjust our guess (just like binary search).

What Happens in Different Cases?

If the array is empty, we return 0 since there’s nothing to split.
If K is 0, we cannot make any valid partitions, so again return 0.
If K ≥ array length, the best case is to assign each number to its own subarray — so the largest element in the array becomes the answer.
If K = 1, we have to take the whole array as one subarray, so the total sum is the answer.
Normal case: We keep checking possible max sums using binary search, and count how many subarrays we need if we restrict the max sum to our current guess. If we need ≤ K subarrays, we try a smaller sum. If we need more than K, we increase the sum.

This method gives us the smallest possible largest subarray sum that still allows splitting the array into exactly K subarrays.

Why Binary Search Works

We’re using binary search over a range of answers — the range between the largest element and the sum of all elements. We're not searching inside the array but rather searching through all possible values the final answer can take. It’s efficient and brings the complexity down to O(n log(sum)).

Visualization

Algorithm Steps

Set low = max(arr) and high = sum(arr).
While low ≤ high:
→ Compute mid = (low + high) / 2.
→ Check if it’s possible to split the array into ≤ K subarrays where no subarray has sum > mid.
→ If possible, update answer = mid and try a smaller value by high = mid - 1.
→ Else, try a larger value by low = mid + 1.
Return the minimum value stored in answer.

Code

Java

Python

JavaScript

C++

Kotlin

Swift

Php

public class SplitArrayLargestSum {
  public static int splitArray(int[] nums, int k) {
    int low = Integer.MIN_VALUE;
    int high = 0;
    for (int num : nums) {
      low = Math.max(low, num); // Minimum possible max sum is the largest element
      high += num;              // Maximum possible max sum is sum of all elements
    }

    int result = high;
    while (low <= high) {
      int mid = low + (high - low) / 2;
      if (canSplit(nums, k, mid)) {
        result = mid;         // Try a smaller maximum sum
        high = mid - 1;
      } else {
        low = mid + 1;        // Increase allowed max sum
      }
    }
    return result;
  }

  private static boolean canSplit(int[] nums, int k, int maxSum) {
    int currentSum = 0;
    int splits = 1;
    for (int num : nums) {
      if (currentSum + num > maxSum) {
        splits++;
        currentSum = num;
        if (splits > k) return false; // Too many splits needed
      } else {
        currentSum += num;
      }
    }
    return true;
  }

  public static void main(String[] args) {
    int[] arr = {7, 2, 5, 10, 8};
    int k = 2;
    System.out.println("Minimum largest sum: " + splitArray(arr, k));
  }
}

Time Complexity

Case	Time Complexity	Explanation
Best Case	O(n log(sum - max))	We perform binary search over the range from max element to sum of all elements, and in each step, we scan the array once.
Average Case	O(n log(sum - max))	For each binary search iteration, we check if the array can be split into K subarrays using a greedy scan.
Worst Case	O(n log(sum - max))	Even in the worst case, binary search narrows down the search space logarithmically, with O(n) work per step.

Space Complexity

O(1)

Explanation: No extra space is used except a few variables for tracking sums and counters.

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Split Array into K Subarrays
Minimize Largest Sum using Binary Search

Problem Statement

Examples❯

Solution❯

Understanding the Goal

What Are We Really Searching For?

How We Approach It

What Happens in Different Cases?

Why Binary Search Works

Visualization

Algorithm Steps

Code

Time Complexity

Space Complexity

Support ProgramGuru.org❯

Split Array into K SubarraysMinimize Largest Sum using Binary Search

Problem Statement

Examples❯

Solution❯

Understanding the Goal

What Are We Really Searching For?

How We Approach It

What Happens in Different Cases?

Why Binary Search Works

Visualization

Algorithm Steps

Code

Time Complexity

Space Complexity

Welcome to ProgramGuru

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Split Array into K Subarrays
Minimize Largest Sum using Binary Search