What is the brute force approach and complexity for Count Subarrays with Zero Sum?

Fix each start, grow the end with a running sum, count every sum of 0 with time complexity O(N²) and space complexity O(1).

What is the optimal approach and complexity for Count Subarrays with Zero Sum?

count += how many earlier prefixes have the same total; seed seen[0] = 1 with optimal time complexity O(N) and space complexity O(N).

How do you identify the pattern for Count Subarrays with Zero Sum?

Look for key signals in the prompt: "contiguous subarrays" indicates Each range sum is a difference of two prefix sums.; "sum exactly 0" indicates Look for earlier prefixes **equal** to the current one.; "count how many" indicates Store how many times each prefix appeared (a map of counts), not just whether it appeared (a set).; "negatives allowed" indicates A sliding window cannot be used; prefix sums do not care about signs..

Count Subarrays with Zero Sum

Medium·GeeksforGeeks Pro OnlyBasic Reinforcement

GeeksforGeeks

Problem Statement

You are given an array of integers arr. The numbers can be positive, negative or zero.

Count how many contiguous, non-empty subarrays add up to exactly 0.

Write long long findSubarray(vector<int>& arr) that returns this count. Two subarrays are different if they start or end at a different index, even if they hold the same values.

Input

An integer array arr of length n. Values may be negative.

Output

The number of non-empty contiguous subarrays whose sum is 0 (as a 64-bit integer).

Example 1

Input: arr = [0, 0, 5, 5, 0, 0]

Output: 6

The zero-sum subarrays are [0] (index 0), [0] (index 1), [0, 0] (0..1), [0] (index 4), [0] (index 5) and [0, 0] (4..5).

Example 2

Input: arr = [6, -1, -3, 4, -2, 2, 4, 6, -12, -7]

Output: 4

They are [-1, -3, 4] (1..3), [-2, 2] (4..5), [-1, -3, 4, -2, 2] (1..5) and [-2, 2, 4, 6, -12] (4..8).

Example 3

Input: arr = [0]

Output: 1

The single element is a subarray with sum 0.

Constraints

• 1 ≤ n ≤ 10^6
• −10^9 ≤ arr[i] ≤ 10^9
• Prefix sums and the count can both exceed the int range — use 64-bit types.

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“contiguous subarrays”

→ Each range sum is a difference of two prefix sums.

“sum exactly 0”

→ Look for earlier prefixes **equal** to the current one.

“count how many”

→ Store how many times each prefix appeared (a map of counts), not just whether it appeared (a set).

“negatives allowed”

→ A sliding window cannot be used; prefix sums do not care about signs.

Approach Complexity Ladder

Brute Force — try every startO(N²)O(1)

Fix each start, grow the end with a running sum, count every sum of 0

Optimal — prefix sum + count mapO(N)O(N)

count += how many earlier prefixes have the same total; seed seen[0] = 1

Brute Force Intuition

For every start, walk the end to the right while keeping a running sum of arr[start..end]. Each time the sum is 0, add one to the count.

Recognition rule for this sub-pattern:

Contiguous + exact sum, negatives allowed. Store how many times each prefix sum appeared; look up prefix − K.

Remember this · Prefix Sum

Trigger

Range sums, "subarray sum equals K" (especially with negatives), equal counts, divisible by K.

Think

If I knew the total up to every index, would the answer become a difference of two totals?

Key question

Which earlier prefix value am I looking for when I stand at index i?

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