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

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