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[ARC144E]GCD of Path Weights

时间:2023-01-23 11:11:06浏览次数:61  
标签:10 GCD leq beautifulness Vertex maximum Sample ARC144E Weights

Problem Statement

You are given a directed graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1, 2, \ldots, N$. The $i$-th edge is directed from Vertex $a_i$ to Vertex $b_i$, where $a_i < b_i$.

The beautifulness of a sequence of positive integers $W = (W_1, W_2, \ldots, W_N)$ is defined as the maximum positive integer $x$ that satisfies the following:

  • For every path $(v_1, \ldots, v_k)$ ($v_1 = 1, v_k = N$) from Vertex $1$ to Vertex $N$ in $G$, $\sum_{i=1}^k W_{v_i}$ is a multiple of $x$.

You are given an integer sequence $A = (A_1, A_2, \ldots, A_N)$. Find the maximum beautifulness of a sequence of positive integers $W = (W_1, \ldots, W_N)$ such that $A_i \neq -1 \implies W_i = A_i$. If the maximum beautifulness does not exist, print -1.

Constraints

  • $2\leq N\leq 3\times 10^5$
  • $1\leq M\leq 3\times 10^5$
  • $1\leq a_i < b_i \leq N$
  • $(a_i,b_i)\neq (a_j,b_j)$ if $i\neq j$
  • In the given graph $G$, there is a path from Vertex $1$ to Vertex $N$.
  • $A_i = -1$ or $1\leq A_i\leq 10^{12}$

Input

Input is given from Standard Input in the following format:

$N$ $M$
$a_1$ $b_1$
$\vdots$
$a_M$ $b_M$
$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the maximum beautifulness of a sequence of positive integers $W$. If the maximum beautifulness does not exist, print -1.


Sample Input 1

4 4
1 2
1 3
2 4
3 4
-1 3 7 -1

Sample Output 1

4

There are two paths from Vertex $1$ to Vertex $N$: $(1,2,4)$ and $(1,3,4)$. For instance, $W = (5, 3, 7, 8)$ has a beautifulness of $4$. Indeed, both $W_1 + W_2 + W_4 = 16$ and $W_1 + W_3 + W_4 = 20$ are multiples of $4$.


Sample Input 2

4 5
1 2
1 3
2 4
3 4
1 4
-1 3 7 -1

Sample Output 2

1

There are three paths from Vertex $1$ to Vertex $N$: $(1,2,4)$, $(1,3,4)$, and $(1,4)$. For instance, $W = (5, 3, 7, 8)$ has a beautifulness of $1$.


Sample Input 3

4 4
1 2
1 3
2 4
3 4
3 -1 -1 7

Sample Output 3

-1

For instance, $W = (3, 10^{100}, 10^{100}, 7)$ has a beautifulness of $10^{100}+10$. Since you can increase the beautifulness of $W$ as much as you want, there is no maximum beautifulness.


Sample Input 4

5 5
1 3
3 5
2 3
3 4
1 4
2 -1 3 -1 4

任意一条从 1 到 \(n\) 的路径都是 \(x\) 的倍数,这是一个很固定的要求。比如现在有这样的两条路径:
image
那么 \(w_2+w_3+w_n=w_2+w_4+w_n(\bmod x)\)
\(w_2+w_3+w_n-w_2-w_4-w_n=0(\bmod x)\)

这启发我们建一条反边,权值取反。

但权值在点上啊

拆点,把一个点拆成两个,中间的边的正的方向边权是点的权值,反的方向是点的权值取负。原来的单向边改为双向边,权值为0。此时我们想要让图只有 0 环。

只有0环的图有什么特点?从任意一个点出发,到某一个点的任意路径长度相等(又绕回了开头)。

那么如果此时从 \(1\) 开始搜索,到达某一个点有一条路径距离 \(d\),另一条是 \(w\),那么 \(x\) 整除 \(|d-w|\)

这样子不断求gcd,可以得到 \(x\) 的限制。

但是如果点权为 \(-1\)?拆出来的两个点不连边。因为中间这条边可以随意决定。

注意要特判如果 \(1\) 到 \(n\) 是一个连通块,那么 \(x\) 要和 \(d_n\) 求一个 gcd.

小细节:所有 1 到不了的点和到不了 \(n\) 的点都是不影响 \(x\) 的,要去掉。

真的算一个神题了,知识点最难的也就dfs,但 rated 直飙 3280

标签:10,GCD,leq,beautifulness,Vertex,maximum,Sample,ARC144E,Weights
From: https://www.cnblogs.com/mekoszc/p/17065049.html

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