B. Maximum Substring
brute force; greedy; implementation
A binary string is a string consisting only of the characters 0 and 1. You are given a binary string \(s\).For some non-empty substring\(^\dagger\) \(t\) of string \(s\) containing \(x\) characters 0 and \(y\) characters 1, define its cost as: \(x \cdot y\), if \(x > 0\) and \(y > 0\); \(x^2\), if \(x > 0\) and \(y = 0\); \(y^2\), if \(x = 0\) and \(y > 0\). Given a binary string \(s\) of length \(n\), find the maximum cost across all its non-empty substrings.\(^\dagger\) A string \(a\) is a substring of a string \(b\) if \(a\) can be obtained from \(b\) by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.
Each test consists of multiple test cases. The first line contains a single integer \(t\) (\(1 \leq t \leq 10^5\)) — the number of test cases. The description of test cases follows.The first line of each test case contains a single integer \(n\) (\(1 \le n \le 2 \cdot 10^5\)) — the length of the string \(s\).The second line of each test case contains a binary string \(s\) of length \(n\).It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2 \cdot 10^5\).
For each test case, print a single integer — the maximum cost across all substrings.
In the first test case, we can take a substring \(111\). It contains \(3\) characters 1 and \(0\) characters 0. So \(a = 3\), \(b = 0\) and its cost is \(3^2 = 9\).In the second test case, we can take the whole string. It contains \(4\) characters 1 and \(3\) characters 0. So \(a = 4\), \(b = 3\) and its cost is \(4 \cdot 3 = 12\).In the third test case, we can can take a substring \(1111\) and its cost is \(4^2 = 16\).In the fourth test case, we can take the whole string and cost is \(4 \cdot 3 = 12\).In the fifth test case, we can take a substring \(000\) and its cost is \(3 \cdot 3 = 9\).In the sixth test case, we can only take the substring \(0\) and its cost is \(1 \cdot 1 = 1\).