Challenge: Subarray Sum Zero

Find the number of subarrays summing to $0$. This is the $k=0$ case of Subarray Sum Equals K. observation: subarray $arr[i..j]$ sums to $0$ iff $prefix[j+1] = prefix[i]$. Count pairs of equal prefix sums.

If a prefix sum $s$ appears $c$ times, it contributes $\binom{c}{2} = c(c-1)/2$ zero-sum subarrays. Try it on $[-1, 1, -1, 1]$. Prefix sums: $0, -1, 0, -1, 0$. Sum $0$ appears $3$ times: $3 \cdot 2 / 2 = 3$ subarrays. Sum $-1$ appears $2$ times: $1$ subarray. Total: $4$.