Solution Approach

Let's walk through the implementation step by step, using the binary search template.

Step 1: Build the Prefix Sum Array

We create a prefix sum array s where s[i] represents the sum of the first i elements:

s = list(accumulate(nums, initial=0))

The initial=0 ensures s[0] = 0, making our array have length n+1. For example, if nums = [2, 3, 1, 2], then s = [0, 2, 5, 6, 8].

Step 2: Initialize the Answer

We set ans = n + 1 as an initial value larger than any possible valid subarray length.

Step 3: Binary Search Template

For each starting position i, we use the binary search template to find the first j where s[j] >= s[i] + target:

left, right = i, n
first_true_index = -1

while left <= right:
    mid = (left + right) // 2
    if prefix_sums[mid] >= prefix_sums[i] + target:  # feasible condition
        first_true_index = mid
        right = mid - 1
    else:
        left = mid + 1

Step 4: Update Minimum Length

If first_true_index != -1 (meaning we found a valid ending position), we update our answer:

if first_true_index != -1:
    ans = min(ans, first_true_index - i)

The length of the subarray is first_true_index - i.

Step 5: Return the Result

If ans is still greater than n, no valid subarray exists, so we return 0.

Time Complexity: O(n log n) - We iterate through n+1 positions and perform O(log n) binary search for each. Space Complexity: O(n) - For storing the prefix sum array.

Example Walkthrough

Let's walk through a concrete example with nums = [2, 3, 1, 2, 4, 3] and target = 7.

Step 1: Build the Prefix Sum Array

Starting with nums = [2, 3, 1, 2, 4, 3], we create the prefix sum array: s = [0, 2, 5, 6, 8, 12, 15]

Step 2: Initialize Answer

Set ans = 7 (which is n + 1, since n = 6)

Step 3-5: Iterate and Search with Template

Now we iterate through each position and use the binary search template:

• i = 4, s[4] = 8:

  • Need to find first j where s[j] >= 8 + 7 = 15
  • Binary search with template:
    • left = 4, right = 6, first_true_index = -1
    • mid = 5: s[5] = 12 < 15 → left = 6
    • mid = 6: s[6] = 15 >= 15 → first_true_index = 6, right = 5
    • left > right, exit loop
  • first_true_index = 6, subarray length = 6 - 4 = 2
  • Update: ans = min(3, 2) = 2

• i = 5, s[5] = 12:

  • Need to find first j where s[j] >= 12 + 7 = 19
  • Binary search with template:
    • left = 5, right = 6, first_true_index = -1
    • mid = 5: s[5] = 12 < 19 → left = 6
    • mid = 6: s[6] = 15 < 19 → left = 7
    • left > right, exit loop
  • first_true_index = -1, no valid subarray starting at position 5

Step 6: Return Result

The minimum length found is 2, which corresponds to the subarray from index 4 to 5 in the original array: [4, 3] with sum = 7.

This matches our expected answer! The subarray [4, 3] has exactly the target sum of 7 and is the shortest possible.

Time and Space Complexity

Time Complexity: O(n × log n)

The algorithm consists of two main operations:

• Creating the prefix sum array using accumulate(): O(n) time
• Iterating through each element in the prefix sum array: O(n) iterations
  • For each iteration, performing a binary search using bisect_left() on the prefix sum array of length n+1: O(log n) time per search
• Total time: O(n) + O(n × log n) = O(n × log n)

Space Complexity: O(n)

The algorithm uses additional space for:

• The prefix sum array s which stores n+1 elements (including the initial 0): O(n) space
• A few scalar variables (ans, i, j, x): O(1) space
• Total space: O(n) + O(1) = O(n)

Common Pitfalls

Pitfall 1: Using the Wrong Binary Search Template Variant

The Issue: Using while left < right with right = mid instead of the correct template with first_true_index.

Incorrect Implementation:

# WRONG: Using incorrect template
while left < right:
    mid = (left + right) // 2
    if prefix_sums[mid] >= target_sum:
        right = mid
    else:
        left = mid + 1
return left  # May return out-of-bounds index

Correct Implementation:

first_true_index = -1
while left <= right:
    mid = (left + right) // 2
    if prefix_sums[mid] >= target_sum:
        first_true_index = mid
        right = mid - 1
    else:
        left = mid + 1
# Check first_true_index != -1 before using

Pitfall 2: Misunderstanding the Binary Search Target

The Issue: Searching for target directly instead of prefix_sums[i] + target.

Incorrect Implementation:

# WRONG: This searches for target, not the required sum
if prefix_sums[mid] >= target:  # Should be prefix_sums[i] + target

Correct Implementation:

# CORRECT: Search for prefix_sums[i] + target
if prefix_sums[mid] >= prefix_sums[i] + target:

Pitfall 3: Off-by-One Errors with Array Indexing

The Issue: The prefix sum array has length n+1 while the original array has length n. This can cause confusion when checking bounds.

Common Mistakes:

# WRONG: Using wrong initial value
min_length = n  # Should be n + 1 to distinguish "not found" case

# WRONG: Checking first_true_index incorrectly
if first_true_index <= n:  # Should check first_true_index != -1

Correct Implementation:

min_length = n + 1
if first_true_index != -1:
    min_length = min(min_length, first_true_index - i)

Pitfall 4: Not Handling Edge Cases Properly

The Issue: Forgetting to handle cases where no valid subarray exists.

Incomplete Solution:

# WRONG: Returns n + 1 instead of 0 when no valid subarray exists
return min_length

Complete Solution:

# CORRECT: Properly handle the "no solution" case
return min_length if min_length <= n else 0