Maximum Number of Non-Overlapping Subarrays With Sum Equals Target

Maximum Number of Non-Overlapping Subarrays With Sum Equals Target

What is this problem about?

The Maximum Number of Non-Overlapping Subarrays With Sum Equals Target coding problem asks you to find the maximum number of non-overlapping subarrays in a given array nums where each subarray's sum equals a specified target.

Why is this asked in interviews?

Google frequently asks this type of problem to assess a candidate's understanding of prefix sums combined with hash tables for efficient lookups, and greedy approaches for maximizing counts under non-overlapping constraints. It tests the ability to adapt standard sum-equals-target patterns to more complex scenarios.

Algorithmic pattern used

Prefix Sum with Hash Map and Greedy: Maintain a running prefix sum. Store (prefix_sum, index) pairs in a hash map. When iterating through the array, if current_prefix_sum - target is found in the hash map, it means a subarray ending at the current index with sum target exists. To ensure non-overlapping, after finding such a subarray, greedily "reset" your prefix sum and hash map to only consider elements after this found subarray. This greedy choice (taking the earliest possible subarray) maximizes future opportunities.

Example explanation

nums = [1, 1, 1, 1, 1], target = 2.

1. prefix_sum = 0, map = {0: -1} (dummy entry for sum 0)
2. i = 0, num = 1: prefix_sum = 1. 1 - 2 = -1 not in map. map = {0: -1, 1: 0}.
3. i = 1, num = 1: prefix_sum = 2. 2 - 2 = 0 is in map. Found subarray nums[0..1]. Count = 1. Reset: prefix_sum = 0, map = {0: 1} (new dummy starting after current subarray).
4. i = 2, num = 1: prefix_sum = 1. 1 - 2 = -1 not in map. map = {0: 1, 1: 2}.
5. i = 3, num = 1: prefix_sum = 2. 2 - 2 = 0 is in map. Found subarray nums[2..3]. Count = 2. Reset: prefix_sum = 0, map = {0: 3}.
6. i = 4, num = 1: prefix_sum = 1. 1 - 2 = -1 not in map. map = {0: 3, 1: 4}. Result: 2 non-overlapping subarrays.

Common mistakes candidates make

• Not resetting the prefix sum/hash map: Failing to "reset" after finding a valid subarray will lead to overlapping counts. The problem requires non-overlapping.
• Incorrect map usage: Ensure the map stores the prefix_sum to index mapping correctly, especially the initial (0, -1) entry.
• Complexity: Naive nested loops for all subarrays are O(N^2). Prefix sum with a hash map brings it down to O(N).

Interview preparation tip

For the Array Hash Table Greedy Prefix Sum interview pattern, this problem highlights a common strategy: when a greedy choice maximizes future options (like taking the shortest valid subarray to leave maximum remaining elements), it's often optimal. Master prefix sums and hash map applications for range sum queries.