Problem Statement

Given an array nums containing n integers and integer k, return the total number of subarrays having sum equal to k.

A subarray is defined as a contiguous non-empty sequence of the array elements.

Examples

Example 1:

• Input: nums = [1, 2, 3], k = 3
• Expected Output: 2
• Justification: There are two subarrays that sum to 3: [1, 2] and [3].

Example 2:

• Input: nums = [10, 2, -2, -20, 10], k = -10
• Expected Output: 3
• Justification: Three subarrays sum up to -10: [10, 2, -2, -20], [2, -2, -20, 10], and [-20, 10].

Example 3:

• Input: nums = [5, 1, 2, -3, 4, -2], k = 3
• Expected Output: 2
• Justification: There are two subarrays that sum to 3: [2, -3, 4], and [1, 2].

Solution

To solve this problem, we'll employ a technique that involves using a hashmap to efficiently track the cumulative sum of elements as we iterate through the array. The core idea is that if the cumulative sum up to two indices, say i and j, differs by the target value k, then the sum of the elements lying between i and j is k.

The algorithm will iterate through the array, calculating the cumulative sum at each step. We then check if (cumulative sum - k) is present in the hashmap. If it is, it means there exists a previous cumulative sum such that the difference between the current sum and that sum equals k, indicating a valid subarray. We add the count of these occurrences to our total. Additionally, we keep updating the hashmap with the count of each cumulative sum encountered. This approach is effective as it allows us to find the required subarrays in a single pass through the array, making it efficient in terms of time complexity.

Step-by-step Algorithm

• Initialize a variable count to 0 and a hashmap cumulativeSumFrequency with a key-value pair (0:1) to handle edge cases.
• Iterate through the array nums while keeping track of the cumulativeSum.
• For each element num in nums:
  • Add num to cumulativeSum.
  • Check if (cumulativeSum - k) is in cumulativeSumFrequency. If it is, add its frequency to count.
  • Update cumulativeSumFrequency by incrementing the count of cumulativeSum.
• Return the value of count.

Algorithm Walkthrough

Let's walk through the algorithm with Example 2: nums = [10, 2, -2, -20, 10], k = -10.

• Start with cumulativeSum = 0, count = 0, cumulativeSumFrequency = {0: 1}.
• For num = 10: cumulativeSum = 10. (cumulativeSum - k) = 20 is not in cumulativeSumFrequency. Update cumulativeSumFrequency to {0: 1, 10: 1}.
• For num = 2: cumulativeSum = 12. (cumulativeSum - k) = 22 is not in cumulativeSumFrequency. Update cumulativeSumFrequency to {0: 1, 10: 1, 12: 1}.
• For num = -2: cumulativeSum = 10. (cumulativeSum - k) = 20 is not in cumulativeSumFrequency. Update cumulativeSumFrequency to {0: 1, 10: 2, 12: 1}.
• For num = -20: cumulativeSum = -10. (cumulativeSum - k) = 0 is in cumulativeSumFrequency. Add 1 to count. Update cumulativeSumFrequency to {0: 1, 10: 2, 12: 1, -10: 1}.
• For num = 10: cumulativeSum = 0. (cumulativeSum - k) = 10 is in cumulativeSumFrequency with frequency 2.

Add 2 to count. Update cumulativeSumFrequency to {0: 2, 10: 2, 12: 1, -10: 1}.

• Final count is 3.

Code

Complexity Analysis

• Time Complexity: O(n), where n is the length of the input array. This is due to a single loop through the array and constant-time hashmap operations.
• Space Complexity: O(n), where n is the length of the input array. The primary space usage is the hashmap storing the cumulative sum frequencies, which in the worst case can grow to the size of the array.