Raging Thunder
Contest: Codeforces Round 654 (Div. 2)
Difficulty rating: 2800
Tags: data structures, divide and conquer, implementation

You are a warrior fighting against the machine god Thor.

Thor challenge you to solve the following problem:

There are $$$n$$$ conveyors arranged in a line numbered with integers from $$$1$$$ to $$$n$$$ from left to right. Each conveyor has a symbol "<" or ">". The initial state of the conveyor $$$i$$$ is equal to the $$$i$$$-th character of the string $$$s$$$. There are $$$n+1$$$ holes numbered with integers from $$$0$$$ to $$$n$$$. The hole $$$0$$$ is on the left side of the conveyor $$$1$$$, and for all $$$i \geq 1$$$ the hole $$$i$$$ is on the right side of the conveyor $$$i$$$.

When a ball is on the conveyor $$$i$$$, the ball moves by the next rules:

If the symbol "<" is on the conveyor $$$i$$$, then:

- If $$$i=1$$$, the ball falls into the hole $$$0$$$.
- If the symbol "<" is on the conveyor $$$i-1$$$, the ball moves to the conveyor $$$i-1$$$.
- If the symbol ">" is on the conveyor $$$i-1$$$, the ball falls into the hole $$$i-1$$$.

If the symbol ">" is on the conveyor $$$i$$$, then:

- If $$$i=n$$$, the ball falls into the hole $$$n$$$.
- If the symbol ">" is on the conveyor $$$i+1$$$, the ball moves to the conveyor $$$i+1$$$.
- If the symbol "<" is on the conveyor $$$i+1$$$, the ball falls into the hole $$$i$$$.

You should answer next $$$q$$$ queries, each query is defined by the pair of integers $$$l, r$$$ ($$$1 \leq l \leq r \leq n$$$):

- First, for all conveyors $$$l,l+1,...,r$$$, the symbol "<" changes to ">" and vice versa. These changes remain for the next queries.
- After that, put one ball on each conveyor $$$l,l+1,...,r$$$. Then, each ball falls into some hole. Find the maximum number of balls in one hole. After the query all balls disappear and don't considered in the next queries.

Input format:
The first line contains two integers $$$n$$$, $$$q$$$ ($$$1 \le n \le 5 \times 10^5 , 1 \le q \le 10^5$$$).

The second line contains a string $$$s$$$ of length $$$n$$$. It consists of characters "<" and ">".

Next $$$q$$$ lines contain the descriptions of the queries, $$$i$$$-th of them contains two integers $$$l$$$, $$$r$$$ $$$(1 \le l \le r \le n)$$$, describing the $$$i$$$-th query.

Output format:
Print $$$q$$$ lines, in the $$$i$$$-th of them print the answer to the $$$i$$$-th query.

Constraints:
Time limit: 3.0 s
Memory limit: 512.0 MB

Note:
- In the first query, the conveyors change to ">><<<". After that, put a ball on each conveyor $$$\{2,3,4\}$$$. All three balls fall into the hole $$$2$$$. So the answer is $$$3$$$.
- In the second query, the conveyors change to ">>>>>". After that, put a ball on each conveyor $$$\{3,4,5\}$$$. All three balls fall into the hole $$$5$$$. So the answer is $$$3$$$.
- In the third query, the conveyors change to "<<<<<". After that, put a ball on each conveyor $$$\{1,2,3,4,5\}$$$. All five balls fall into the hole $$$0$$$. So the answer is $$$5$$$.
- In the fourth query, the conveyors change to ">>><<". After that, put a ball on each conveyor $$$\{1,2,3\}$$$. All three balls fall into the hole $$$3$$$. So the answer is $$$3$$$.
- In the fifth query, the conveyors change to "><<><". After that, put a ball on each conveyor $$$\{2,3,4\}$$$. Two balls fall into the hole $$$1$$$, and one ball falls into the hole $$$4$$$. So, the answer is $$$2$$$.
- In the sixth query, the conveyors change to "<>><>". After that, put a ball on each conveyor $$$\{1,2,3,4,5\}$$$. Three balls fall into the hole $$$3$$$, one ball falls into the hole $$$0$$$ and one ball falls into the hole $$$5$$$. So, the answer is $$$3$$$.

Example input:
5 6
><>><
2 4
3 5
1 5
1 3
2 4
1 5

Example output:
3
3
5
3
2
3

Editorial:
First, observe where the balls fall. When there is a ">>>...>><<...<<<" structure, the balls on there will fall into one hole. So, our goal is handling this structure.
Each query is asking about a segment. Then, to solve this problem, we can use segment tree.
Each node maintains following:
• Prefix structure
• Suffix structure
• The answer for between above structures
For example, a string "><<>><>><<>>>>" will be converted to the following data:
• Prefix structure : "><<"
• Suffix structure : ">>>>"
• The answer for between above structures : take the answer of ">><" or ">><<". the largest answer is $$$4$$$.
And we need to implement that combining two data ( [left data] + [right data] ). Mainly we should merge left suffix and right prefix, and calculate the answer for the segment, but notice that there are some exceptions. The exceptions are in case of there are only one structure in the merged node, like ">>><<" or "<<<". (You may maintain other flags for your implement.)
Then, how to handling each queries? We also keep above data for when '>' are changed into '<' and vice versa on each node of the segment tree. And when a change query actually comes to some nodes, swap these data. Apply lazy propagation for handling this. When calculate the answer for a query, don't forget to consider about the prefix(or the suffix).
Total complexity : $$$O(n + q \log n)$$$