Understanding the Problem Step-by-Step
We are given an array that can contain both positive and negative numbers, and a target sum k
. Our goal is to find the length of the longest subarray whose elements sum up exactly to k
.
Let’s say the input is:
arr = [1, -1, 5, -2, 3], k = 3
We are to find the longest continuous section (subarray) in arr
where the numbers add up to 3. The expected output is 4
because the subarray [1, -1, 5, -2]
adds up to 3 and has length 4.
Step-by-Step Strategy
We will use the concepts of prefix sum and a hash map to solve this efficiently.
The idea is to keep a running total (prefix sum) while iterating through the array. For each index i
, we calculate the sum from the beginning up to that index, called curr_sum
. If we have seen a previous prefix sum value such that:
curr_sum - previous_sum = k
then the subarray between those two indices has a total sum of k
.
We store the first occurrence of each prefix sum in a hash map, so we can quickly check if a subarray with sum k
ends at the current index.
Working Through the Example
arr = [1, -1, 5, -2, 3], k = 3
- At index 0: curr_sum = 1 → not equal to k, store (1, 0)
- At index 1: curr_sum = 0 → not equal to k, store (0, 1)
- At index 2: curr_sum = 5 → 5 - 3 = 2, not in map, store (5, 2)
- At index 3: curr_sum = 3 → 3 - 3 = 0, 0 is in map at index 1 → length = 3 - 1 = 2
- At index 4: curr_sum = 6 → 6 - 3 = 3, 3 is in map at index 3 → length = 4 - 3 = 1
- Also, at index 4, curr_sum = 6 is a new sum → store (6, 4)
- Final answer is max length = 4 (from index 0 to 3)
Edge Case Considerations
Case 1: Prefix sum equals k
If curr_sum
itself is equal to k
at any index, it means the subarray from index 0 to current index is valid.
Case 2: Target sum is 0
Even if the target is 0, the logic works. We just look for previously seen prefix sums that equal the current prefix sum.
Case 3: All elements are zero and target is zero
The entire array is valid. For example, [0, 0, 0]
with k = 0
should return 3.
Case 4: No valid subarray found
If we go through the array and never find a suitable subarray, the answer is 0
.
Case 5: Empty array
No elements to process means no valid subarray. So, return 0
.
Instead of checking every possible subarray, which takes O(n^2)
time, we process the array once and use a hash map to find matching prefix sums. This brings down the time complexity to O(n)
, making it highly efficient even for large arrays.
This approach works well even when the array has negative numbers, which is a limitation for techniques like the sliding window method.
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