# Python solution (linear time, and linear space) with dummies explanation

• ``````# linear time, and linear space complexity
def maxSubArrayLen(self, nums, k):
if not nums:
return 0
prefix_sums = [0]
number_sum = 0
# generate a list of the prefix sums
for i in range(len(nums)):
number_sum += nums[i]
prefix_sums.append(number_sum)
ht = {}
max_length = 0
for i in range(len(prefix_sums)):
# We want the first (left-most) instance of a sum because we are looking
# for the longest length subarray
if prefix_sums[i] not in ht:
ht[prefix_sums[i]] = i
# If a previously seen sum that equates to a difference of k, then determine
# if it is the longest subarray length
if (prefix_sums[i] - k) in ht:
max_length = max(max_length, i - ht[prefix_sums[i]-k])
return max_length
``````

Explanation:
I'll explain the algorithm using the following example:
input nums = [-2, 7, 1,-1, 5,-2, 3]
prefix_sums = [0,-2, 5, 6, 5,10, 8,11]
At curr_index 7 in the prefix_sums list, the sum from the beginning to curr_index is 8, and if you subtract 8 with the target of 3 then you get 5. From that we are looking for an index to the left of curr_index that equates to that 5. To the left of curr_index because whatever index we find represents at that specific point in time, that was the sum. If you do find that index to the left of curr_index that is equal to 5, then you have found two indexes that have changed a difference of 3 (target) and therefore have sumed to 3. then the difference between the indexes represents the length of that subarray.

This is using the prefix sum algorithm to achieve the linear time complexity.

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