Score : $525$ points

问题描述

给定 $Q$ 组整数三元组 $(a_1, b_1, d_1), (a_2, b_2, d_2), \ldots, (a_Q, b_Q, d_Q)$。

集合 $\lbrace 1, 2, \ldots, Q\rbrace$ 的一个子集 $S$ 被定义为 好集合，如果存在一个长度为 $N$ 的整数序列 $(X_1, X_2, \ldots, X_N)$ 满足以下条件：

对于所有 $i \in S$，有 $X_{a_i} - X_{b_i} = d_i$。

从空集开始，按顺序对 $i = 1, 2, \ldots, Q$ 执行以下操作：

如果 $S \cup \lbrace i \rbrace$ 是一个好集合，则用 $S \cup \lbrace i \rbrace$ 替换 $S$。

以 升序 打印最终集合 $S$ 中的所有元素。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given $Q$ triples of integers $(a_1, b_1, d_1), (a_2, b_2, d_2), \ldots, (a_Q, b_Q, d_Q)$.

A subset $S$ of the set $\lbrace 1, 2, \ldots, Q\rbrace$ is defined to be a good set if there exists an integer sequence $(X_1, X_2, \ldots, X_N)$ of length $N$ that satisfies:

$X_{a_i} - X_{b_i} = d_i$ for all $i \in S$.

Starting with $S$ as an empty set, perform the following operation for $i = 1, 2, \ldots, Q$ in this order:

If $S \cup \lbrace i \rbrace$ is a good set, then replace $S$ with $S \cup \lbrace i \rbrace$.

Print all elements of the final set $S$ in ascending order.

Constraints

• All input values are integers.
• $1 \leq N, Q \leq 2 \times 10^5$
• $1 \leq a_i, b_i \leq N$
• $-10^9 \leq d_i \leq 10^9$

Input

The input is given from Standard Input in the following format:

$N$ $Q$

$a_1$ $b_1$ $d_1$

$a_2$ $b_2$ $d_2$

$\vdots$

$a_Q$ $b_Q$ $d_Q$

Output

Print the sequence $(s_1, s_2, \ldots, s_k)$ of all elements of the final set $S$ in ascending order, separated by spaces, in the following format:

$s_1$ $s_2$ $\ldots$ $s_k$

Sample Input 1

3 5
1 2 2
3 2 -3
2 1 -1
3 3 0
1 3 5

Sample Output 1

1 2 4 5

Starting with $S$ as an empty set, perform the operation described in the problem statement for $i = 1, 2, 3, 4, 5$ in this order, as follows.

• For $i = 1$, the set $S \cup \lbrace i \rbrace = \lbrace 1 \rbrace$ is a good set, because $(X_1, X_2, X_3) = (3, 1, 4)$ satisfies the condition in the problem statement, for example, so replace $S$ with $\lbrace 1\rbrace$.
• For $i = 2$, the set $S \cup \lbrace i \rbrace = \lbrace 1, 2 \rbrace$ is a good set, because $(X_1, X_2, X_3) = (3, 1, -2)$ satisfies the condition in the problem statement, for example, so replace $S$ with $\lbrace 1, 2\rbrace$.
• For $i = 3$, the set $S \cup \lbrace i \rbrace = \lbrace 1, 2, 3 \rbrace$ is not a good set.
• For $i = 4$, the set $S \cup \lbrace i \rbrace = \lbrace 1, 2, 4 \rbrace$ is a good set, because $(X_1, X_2, X_3) = (3, 1, -2)$ satisfies the condition in the problem statement, for example, so replace $S$ with $\lbrace 1, 2, 4\rbrace$.
• For $i = 5$, the set $S \cup \lbrace i \rbrace = \lbrace 1, 2, 4, 5 \rbrace$ is a good set, because $(X_1, X_2, X_3) = (3, 1, -2)$ satisfies the condition in the problem statement, for example, so replace $S$ with $\lbrace 1, 2, 4, 5\rbrace$.

Therefore, the final set $S$ is $\lbrace 1, 2, 4, 5\rbrace$.

Sample Input 2

200000 1
1 1 1

Sample Output 2

The final set $S$ is empty.

Sample Input 3

5 20
4 2 125421359
2 5 -191096267
3 4 -42422908
3 5 -180492387
3 3 174861038
2 3 -82998451
3 4 -134761089
3 1 -57159320
5 2 191096267
2 4 -120557647
4 2 125421359
2 3 142216401
4 5 -96172984
3 5 -108097816
1 5 -50938496
1 2 140157771
5 4 65674908
4 3 35196193
4 4 0
3 4 188711840

Sample Output 3

1 2 3 6 8 9 11 14 15 16 17 19

update @ 2024/3/10 01:53:16