Maximum Subarray Sum With Length Divisible by K

Problem Description

Given an array of integers nums and an integer k, return the maximum sum of a non-empty subarray of nums such that the size of the subarray is divisible by k.

Key Insights

• The problem requires finding subarrays whose lengths are multiples of k.
• A prefix sum approach can be utilized to efficiently calculate sums of subarrays.
• Using a hash map can help track the best prefix sums based on their indices modulo k.
• We need to check all possible ending indices of subarrays to find the maximum sum that meets the length condition.

Space and Time Complexity

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

Solution

The solution involves using a prefix sum array to keep track of cumulative sums. We will also use a hash map to store the best prefix sums for indices that share the same remainder when divided by k. This allows us to quickly compute the sums of subarrays whose lengths are divisible by k by leveraging the difference between prefix sums.

1. Initialize a prefix sum variable and a hash map to store the best sum for each remainder modulo k.
2. Iterate through the nums array, updating the prefix sum.
3. For each prefix sum, calculate its remainder when divided by k.
4. If the remainder has been seen before, compute the possible subarray sum using the stored prefix sum and update the maximum if this sum is greater.
5. Store the best prefix sum for each remainder to optimize future calculations.

Code Solutions