Shortest Subarray With Sum At Least K

Introduction

The shortest subarray with sum at least k is a well‑known algorithmic challenge that asks for the minimal length of a contiguous segment (subarray) whose elements add up to a value of at least k. Now, this problem appears in many domains, from finance—where k might represent a target profit—to computer science contests and real‑world data analysis. In this article we will explore the intuition behind the solution, compare naïve and efficient approaches, and provide a clear, step‑by‑step guide to implement the optimal algorithm. By the end, you will be able to find the shortest subarray with sum at least k in linear time, making your code both fast and reliable Took long enough..

Understanding the Problem

A subarray is a slice of an array that preserves the original order of elements and contains no gaps. The parameter k is a threshold that the sum of the chosen subarray must meet or exceed. Practically speaking, for example, in the array [2, -1, 3, 5], the subarray [ -1, 3, 5] is contiguous and has a sum of 7. e.The goal is to determine the shortest such subarray—i., the one with the smallest number of elements.

Key points to remember:

• The array can contain positive, zero, or negative numbers.
• The length of the subarray is measured by the count of elements, not by the magnitude of the sum.
• If no subarray reaches the sum k, the answer is typically defined as -1 or “not possible”.

Brute Force vs Efficient Methods

The most straightforward way to solve the problem is to examine all possible subarrays. This brute‑force method involves two nested loops:

1. Choose a starting index i.
2. Extend the subarray by increasing the ending index j until the sum meets or exceeds k.

While conceptually simple, this approach has a time complexity of O(n²), which becomes impractical for large arrays (e.g., n > 10⁵).

To improve efficiency, we can exploit the sliding window technique when the array consists only of non‑negative numbers. In that case, once the sum reaches k, moving the left boundary forward can only reduce the window size, guaranteeing that we never miss a shorter valid subarray.

Sliding Window Technique

The sliding window method works in O(n) time and O(1) extra space. It is ideal when all numbers are non‑negative, because the sum is monotonic as the window expands or contracts Most people skip this — try not to..

Steps

1. Initialize two pointers, left = 0 and right = 0, and a variable current_sum = 0.
2. Expand the window by moving right forward and adding arr[right] to current_sum.
3. While current_sum >= k, record the window length (right - left + 1) if it is the smallest found so far, then shrink the window by subtracting arr[left] and incrementing left.
4. Continue until right reaches the end of the array.

Pseudocode

def shortest_subarray_non_negative(arr, k):
    left = 0
    current_sum = 0
    min_length = float('inf')
    for right in range(len(arr)):
        current_sum += arr[right]
        while current_sum >= k:
            min_length = min(min_length, right - left + 1)
            current_sum -= arr[left]
            left += 1
    return min_length if min_length != float('inf') else -1

The bold statement that the algorithm runs in linear time is crucial: each element is added once and removed once, giving a total of 2n operations.

Handling Negative Numbers with Prefix Sum and Deque

When the array may contain negative values, the sliding window no longer guarantees correctness because the sum can decrease when extending the window. Plus, a more general solution uses prefix sums together with a monotonic deque (double‑ended queue). This approach maintains the smallest possible prefix sum for each index, allowing us to find the shortest subarray efficiently Simple, but easy to overlook..

Core Idea

Let prefix[i] be the sum of the first i elements (prefix[0] = 0). Consider this: for any indices i and j (i < j), the sum of subarray arr[i…j‑1] equals prefix[j] - prefix[i]. To satisfy prefix[j] - prefix[i] >= k, we need prefix[i] <= prefix[j] - k.