Score : $500$ points

问题描述

Takahashi 拥有一个秘密整数序列 $a$。已知该序列的长度为 $N$。

你想要猜出 $a$ 的具体内容。他承诺会给你以下 $Q$ 条额外信息。

• 第 $i$ 条信息：$a_{l_i}+a_{l_i+1}+\cdots+a_{r_i}$ 的值。

如果给出这 $Q$ 条承诺的信息，是否可能确定序列 $a$ 中所有元素之和，即 $a_1+a_2+\cdots+a_N$ 呢？

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Takahashi has a secret integer sequence $a$. You know that the length of $a$ is $N$.

You want to guess the contents of $a$. He has promised to give you the following $Q$ additional pieces of information.

• The $i$-th information: the value $a_{l_i}+a_{l_i+1}+\cdots+a_{r_i}$.

Is it possible to determine the sum of all elements in $a$, $a_1+a_2+\cdots+a_N$, if the $Q$ pieces of promised information are given?

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq Q \leq \min(2 \times 10^5,\frac{N(N+1)}{2})$
• $1 \leq l_i \leq r_i \leq N$
• $(l_i,r_i) \neq (l_j,r_j)\ (i \neq j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $Q$

$l_1$ $r_1$

$l_2$ $r_2$

$\hspace{0.4cm}\vdots$

$l_Q$ $r_Q$

Output

If it is possible to determine the sum of all elements in $a$, print Yes; otherwise, print No.

Sample Input 1

3 3
1 2
2 3
2 2

Sample Output 1

Yes

From the first and second information, we can find the value $a_1+a_2+a_2+a_3$. By subtracting the value of $a_2$ from it, we can determine the value $a_1+a_2+a_3$.

Sample Input 2

4 3
1 3
1 2
2 3

Sample Output 2

No

We can determine the sum of the first $3$ elements of $a$, but not the sum of all elements.

Sample Input 3

4 4
1 1
2 2
3 3
1 4

Sample Output 3

Yes

The fourth information directly gives us the sum of all elements.

update @ 2024/3/10 10:17:33