200901: [AtCoder]ARC090 D - People on a Line

Memory Limit:256 MB Time Limit:2 S
Judge Style:Text Compare Creator:
Submit:0 Solved:0

Description

Score : $400$ points

Problem Statement

There are $N$ people standing on the $x$-axis. Let the coordinate of Person $i$ be $x_i$. For every $i$, $x_i$ is an integer between $0$ and $10^9$ (inclusive). It is possible that more than one person is standing at the same coordinate.

You will given $M$ pieces of information regarding the positions of these people. The $i$-th piece of information has the form $(L_i, R_i, D_i)$. This means that Person $R_i$ is to the right of Person $L_i$ by $D_i$ units of distance, that is, $x_{R_i} - x_{L_i} = D_i$ holds.

It turns out that some of these $M$ pieces of information may be incorrect. Determine if there exists a set of values $(x_1, x_2, ..., x_N)$ that is consistent with the given pieces of information.

Constraints

  • $1 \leq N \leq 100$ $000$
  • $0 \leq M \leq 200$ $000$
  • $1 \leq L_i, R_i \leq N$ ($1 \leq i \leq M$)
  • $0 \leq D_i \leq 10$ $000$ ($1 \leq i \leq M$)
  • $L_i \neq R_i$ ($1 \leq i \leq M$)
  • If $i \neq j$, then $(L_i, R_i) \neq (L_j, R_j)$ and $(L_i, R_i) \neq (R_j, L_j)$.
  • $D_i$ are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$L_1$ $R_1$ $D_1$
$L_2$ $R_2$ $D_2$
$:$
$L_M$ $R_M$ $D_M$

Output

If there exists a set of values $(x_1, x_2, ..., x_N)$ that is consistent with all given pieces of information, print Yes; if it does not exist, print No.


Sample Input 1

3 3
1 2 1
2 3 1
1 3 2

Sample Output 1

Yes

Some possible sets of values $(x_1, x_2, x_3)$ are $(0, 1, 2)$ and $(101, 102, 103)$.


Sample Input 2

3 3
1 2 1
2 3 1
1 3 5

Sample Output 2

No

If the first two pieces of information are correct, $x_3 - x_1 = 2$ holds, which is contradictory to the last piece of information.


Sample Input 3

4 3
2 1 1
2 3 5
3 4 2

Sample Output 3

Yes

Sample Input 4

10 3
8 7 100
7 9 100
9 8 100

Sample Output 4

No

Sample Input 5

100 0

Sample Output 5

Yes

Input

题意翻译

## 题目描述 在一根数轴上站有 $n$ 个人,我们称第 $i$ 个人的坐标为 $x_i(x_i\in [0,10^{9}],x_i\in Z)$,同一个坐标点上可能有多个人。 你现在手上有 $m$ 条信息,第 $i$ 条信息形如 $(li,ri,di)$,含义是第 $r_i$ 个人在第 $l_i$ 个人右数第 $d_i$ 个坐标点上,换言之,$x_{r_i} - x_{l_i} = d_i$。 不幸的是,这 $m$ 条信息中的一些可能有误,请你求出是否存在一组 $x (x_1,x_2,x_3,\dots ,x_n)$ 满足所有信息。 ## 输入格式 输入格式如下: 输入数据的第一行包含两个以空格分开的整数 $n$ 和 $m$,分别表示总人数和信息条数; 接下来的 $m$ 行中第 $i(1\le i\le m)$ 行包含三个以空格分开的整数 $l_i,r_i,d_i$,表示第 $i$ 条信息是 $r_i$ 号人在 $l_i$ 号人右边 $d_i$ 个位置上。 ## 输出格式 若存在一组合法的 $x$,输出一行 ``Yes``;否则输出一行 ``No``。 ## 说明 全部的输入数据满足以下条件: - $1 \le n \le 100000$; - $0 \le m \le 200000$; - $1\le l_i,r_i\le n (1\le i\le m)$; - $0\le d_i\le 10000 (1\le i\le m)$; - $l_i \ne r_i (1 \le i \le m)$; - 如果 $i\le j$,则有 $(l_i,r_i)\ne (l_j,r_j),(l_i,r_i)\ne (r_j,l_j)$; - $d_i$ 为整数。 #### 样例说明 1 $(0,1,2)$ 与 $(101,102,103)$ 都是合法的解。 #### 样例说明 2 若前两条信息是正确的,则有 $x_3 - x_1 = 2$,那么第三条信息就是错误的。 感谢@fbhou 提供的翻译

加入题单

上一题 下一题 算法标签: