Balanced Sub arrays

Do not assume what balanced means. Ask whether it refers to equal zeros and ones, equal counts of two values, a sum target, parentheses balance, or a modulo condition. Many balanced-subarray tasks reduce to equal prefix states: for equal zeros and ones, map 0 to -1 and 1 to +1; two equal prefix sums delimit a balanced subarray. Keep a frequency map of prefix sums, seed sum zero with count one, and for each new prefix add its prior frequency to the answer before incrementing it. The same pattern works for other equality constraints once the state is defined correctly. If balance means a nonnegative property instead, that recurrence may be wrong, so clarification is the central answer. Test the empty prefix, all-one-type input, negative or large values if allowed, and an answer that exceeds 32-bit range. The prefix-map form is O(n) expected time and O(n) space.