Tokitsukaze and Good 01-String (hard version)

1000ms

262144K

Description:

This is the hard version of the problem. The only difference between the two versions is that the harder version asks additionally for a minimum number of subsegments.

Tokitsukaze has a binary string $$$s$$$ of length $$$n$$$, consisting only of zeros and ones, $$$n$$$ is even.

Now Tokitsukaze divides $$$s$$$ into the minimum number of contiguous subsegments, and for each subsegment, all bits in each subsegment are the same. After that, $$$s$$$ is considered good if the lengths of all subsegments are even.

For example, if $$$s$$$ is "11001111", it will be divided into "11", "00" and "1111". Their lengths are $$$2$$$, $$$2$$$, $$$4$$$ respectively, which are all even numbers, so "11001111" is good. Another example, if $$$s$$$ is "1110011000", it will be divided into "111", "00", "11" and "000", and their lengths are $$$3$$$, $$$2$$$, $$$2$$$, $$$3$$$. Obviously, "1110011000" is not good.

Tokitsukaze wants to make $$$s$$$ good by changing the values of some positions in $$$s$$$. Specifically, she can perform the operation any number of times: change the value of $$$s_i$$$ to '0' or '1' ($$$1 \leq i \leq n$$$). Can you tell her the minimum number of operations to make $$$s$$$ good? Meanwhile, she also wants to know the minimum number of subsegments that $$$s$$$ can be divided into among all solutions with the minimum number of operations.

Input:

The first contains a single positive integer $$$t$$$ ($$$1 \leq t \leq 10\,000$$$) — the number of test cases.

For each test case, the first line contains a single integer $$$n$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$) — the length of $$$s$$$, it is guaranteed that $$$n$$$ is even.

The second line contains a binary string $$$s$$$ of length $$$n$$$, consisting only of zeros and ones.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, print a single line with two integers — the minimum number of operations to make $$$s$$$ good, and the minimum number of subsegments that $$$s$$$ can be divided into among all solutions with the minimum number of operations.

Sample Input:

Sample Output:

Note:

In the first test case, one of the ways to make $$$s$$$ good is the following.

Change $$$s_3$$$, $$$s_6$$$ and $$$s_7$$$ to '0', after that $$$s$$$ becomes "1100000000", it can be divided into "11" and "00000000", which lengths are $$$2$$$ and $$$8$$$ respectively, the number of subsegments of it is $$$2$$$. There are other ways to operate $$$3$$$ times to make $$$s$$$ good, such as "1111110000", "1100001100", "1111001100", the number of subsegments of them are $$$2$$$, $$$4$$$, $$$4$$$ respectively. It's easy to find that the minimum number of subsegments among all solutions with the minimum number of operations is $$$2$$$.

In the second, third and fourth test cases, $$$s$$$ is good initially, so no operation is required.

Informação

Provedor Codeforces

Código CF1678B2