Banh-mi
Contest: Codeforces Round 520 (Div. 2)
Difficulty rating: 1600
Tags: greedy, implementation, math

JATC loves Banh-mi (a Vietnamese food). His affection for Banh-mi is so much that he always has it for breakfast. This morning, as usual, he buys a Banh-mi and decides to enjoy it in a special way.

First, he splits the Banh-mi into $$$n$$$ parts, places them on a row and numbers them from $$$1$$$ through $$$n$$$. For each part $$$i$$$, he defines the deliciousness of the part as $$$x_i \in \{0, 1\}$$$. JATC's going to eat those parts one by one. At each step, he chooses arbitrary remaining part and eats it. Suppose that part is the $$$i$$$-th part then his enjoyment of the Banh-mi will increase by $$$x_i$$$ and the deliciousness of all the remaining parts will also increase by $$$x_i$$$. The initial enjoyment of JATC is equal to $$$0$$$.

For example, suppose the deliciousness of $$$3$$$ parts are $$$[0, 1, 0]$$$. If JATC eats the second part then his enjoyment will become $$$1$$$ and the deliciousness of remaining parts will become $$$[1, \_, 1]$$$. Next, if he eats the first part then his enjoyment will become $$$2$$$ and the remaining parts will become $$$[\_, \_, 2]$$$. After eating the last part, JATC's enjoyment will become $$$4$$$.

However, JATC doesn't want to eat all the parts but to save some for later. He gives you $$$q$$$ queries, each of them consisting of two integers $$$l_i$$$ and $$$r_i$$$. For each query, you have to let him know what is the maximum enjoyment he can get if he eats all the parts with indices in the range $$$[l_i, r_i]$$$ in some order.

All the queries are independent of each other. Since the answer to the query could be very large, print it modulo $$$10^9+7$$$.

Input format:
The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 100\,000$$$).

The second line contains a string of $$$n$$$ characters, each character is either '0' or '1'. The $$$i$$$-th character defines the deliciousness of the $$$i$$$-th part.

Each of the following $$$q$$$ lines contains two integers $$$l_i$$$ and $$$r_i$$$ ($$$1 \le l_i \le r_i \le n$$$) — the segment of the corresponding query.

Output format:
Print $$$q$$$ lines, where $$$i$$$-th of them contains a single integer — the answer to the $$$i$$$-th query modulo $$$10^9 + 7$$$.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first example:

- For query $$$1$$$: One of the best ways for JATC to eats those parts is in this order: $$$1$$$, $$$4$$$, $$$3$$$, $$$2$$$.
- For query $$$2$$$: Both $$$3$$$, $$$4$$$ and $$$4$$$, $$$3$$$ ordering give the same answer.

In the second example, any order of eating parts leads to the same answer.

Example input:
4 2
1011
1 4
3 4

Example output:
14
3

Example input:
3 2
111
1 2
3 3

Example output:
3
1

Editorial:
For each part that we choose, we need to calculate how many times that element is added to our score. You can see that, the first element that we choose is added $$$2^{k-1}$$$ times in our score ($$$k=r-l+1$$$), the second element is added $$$2^{k-2}$$$ times and so on. Therefore, we just need to choose all the $$$1s$$$ first and then all the remaining parts. The final score is $$$(2^x-1) \cdot 2^y$$$, where $$$x$$$ is the number of $$$1s$$$ and $$$y$$$ is the number of $$$0s$$$. Complexity: $$$O(n + Q)$$$.