A15538. BinaryStringForces

You are given a binary string $s$ of length $n$ . We define a maximal substring as a substring that cannot be extended while keeping all elements equal. For example, in the string $11000111$ there are three maximal substrings: $11$ , $000$ and $111$ .

In one operation, you can select two maximal adjacent substrings. Since they are maximal and adjacent, it's easy to see their elements must have different values. Let $a$ be the length of the sequence of ones and $b$ be the length of the sequence of zeros. Then do the following:

- If $a \ge b$ , then replace $b$ selected zeros with $b$ ones.
- If $a < b$ , then replace $a$ selected ones with $a$ zeros.

As an example, for $1110000$ we make it $0000000$ , for $0011$ we make it $1111$ . We call a string being good if it can be turned into $1111...1111$ using the aforementioned operation any number of times (possibly, zero). Find the number of good substrings among all $\frac{n(n+1)}{2}$ non-empty substrings of $s$ .

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 $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — the length of the string $s$ .

The second line of each test case contains the binary string $s$ of length $n$ .

It is guaranteed that sum of $n$ across all test cases doesn't exceed $2 \cdot 10^5$ .

For each test case, print a single integer — the number of good substrings.

Let's define a substring from index $l$ to index $r$ as $[l, r]$ .

For the first test case, the good substrings are:

- $[1,1]$ ,
- $[1,2]$ ,
- $[3,6]$ ,
- $[4,5]$ ,
- $[4,6]$ ,
- $[5,5]$ ,
- $[5,6]$ ,
- $[6,6]$ .

In the second test case, all substrings are good except $[2,2]$ .

In the third test case, all substrings are good.