Preparing NOJ

Consider a sequence of digits of length $$$2^k$$$ $$$[a_1, a_2, \ldots, a_{2^k}]$$$. We perform the following operation with it: replace pairs $$$(a_{2i+1}, a_{2i+2})$$$ with $$$(a_{2i+1} + a_{2i+2})\bmod 10$$$ for $$$0\le i<2^{k-1}$$$. For every $$$i$$$ where $$$a_{2i+1} + a_{2i+2}\ge 10$$$ we get a candy! As a result, we will get a sequence of length $$$2^{k-1}$$$.

Less formally, we partition sequence of length $$$2^k$$$ into $$$2^{k-1}$$$ pairs, each consisting of 2 numbers: the first pair consists of the first and second numbers, the second of the third and fourth $$$\ldots$$$, the last pair consists of the ($$$2^k-1$$$)-th and ($$$2^k$$$)-th numbers. For every pair such that sum of numbers in it is at least $$$10$$$, we get a candy. After that, we replace every pair of numbers with a remainder of the division of their sum by $$$10$$$ (and don't change the order of the numbers).

Perform this operation with a resulting array until it becomes of length $$$1$$$. Let $$$f([a_1, a_2, \ldots, a_{2^k}])$$$ denote the number of candies we get in this process.

For example: if the starting sequence is $$$[8, 7, 3, 1, 7, 0, 9, 4]$$$ then:

After the first operation the sequence becomes $$$[(8 + 7)\bmod 10, (3 + 1)\bmod 10, (7 + 0)\bmod 10, (9 + 4)\bmod 10]$$$ $$$=$$$ $$$[5, 4, 7, 3]$$$, and we get $$$2$$$ candies as $$$8 + 7 \ge 10$$$ and $$$9 + 4 \ge 10$$$.

After the second operation the sequence becomes $$$[(5 + 4)\bmod 10, (7 + 3)\bmod 10]$$$ $$$=$$$ $$$[9, 0]$$$, and we get one more candy as $$$7 + 3 \ge 10$$$.

After the final operation sequence becomes $$$[(9 + 0) \bmod 10]$$$ $$$=$$$ $$$[9]$$$.

Therefore, $$$f([8, 7, 3, 1, 7, 0, 9, 4]) = 3$$$ as we got $$$3$$$ candies in total.

You are given a sequence of digits of length $$$n$$$ $$$s_1, s_2, \ldots s_n$$$. You have to answer $$$q$$$ queries of the form $$$(l_i, r_i)$$$, where for $$$i$$$-th query you have to output $$$f([s_{l_i}, s_{l_i+1}, \ldots, s_{r_i}])$$$. It is guaranteed that $$$r_i-l_i+1$$$ is of form $$$2^k$$$ for some nonnegative integer $$$k$$$.

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the length of the sequence.

The second line contains $$$n$$$ digits $$$s_1, s_2, \ldots, s_n$$$ ($$$0 \le s_i \le 9$$$).

The third line contains a single integer $$$q$$$ ($$$1 \le q \le 10^5$$$) — the number of queries.

Each of the next $$$q$$$ lines contains two integers $$$l_i$$$, $$$r_i$$$ ($$$1 \le l_i \le r_i \le n$$$) — $$$i$$$-th query. It is guaranteed that $$$r_i-l_i+1$$$ is a nonnegative integer power of $$$2$$$.

Output $$$q$$$ lines, in $$$i$$$-th line output single integer — $$$f([s_{l_i}, s_{l_i + 1}, \ldots, s_{r_i}])$$$, answer to the $$$i$$$-th query.

8 8 7 3 1 7 0 9 4 3 1 8 2 5 7 7

3 1 0

6 0 1 2 3 3 5 3 1 2 1 4 3 6

0 0 1

The first example illustrates an example from the statement.

$$$f([7, 3, 1, 7]) = 1$$$: sequence of operations is $$$[7, 3, 1, 7] \to [(7 + 3)\bmod 10, (1 + 7)\bmod 10]$$$ $$$=$$$ $$$[0, 8]$$$ and one candy as $$$7 + 3 \ge 10$$$ $$$\to$$$ $$$[(0 + 8) \bmod 10]$$$ $$$=$$$ $$$[8]$$$, so we get only $$$1$$$ candy.

$$$f([9]) = 0$$$ as we don't perform operations with it.

Info

Provider CodeForces

Origin Codeforces Round #572 (Div. 2)

Code CF1189C

Tags

data structuresdpimplementationmath

Submitted 204

Passed 85

AC Rate 41.67%

Date 07/21/2019 13:45:09

Related