Prime Routing

Time limit2sMemory limit512 MB

Summary

Find the minimum prime length of a walk from S to T in an undirected graph, where cables may be reused; print -1 if impossible.

Level: Medium7 of 10
Topics: Graph, BFS, Number theory, Shortest path
Solved: No attempts yet

Problem

Fox Jiro is one of the staff members of the ACM-ICPC 2018 Asia Yokohama Regional Contest and is responsible for designing the network for the venue of the contest. His network consists of $N$ computers, which are connected by $M$ cables. The $i$ -th cable connects the $a_i$ -th computer and the $b_i$ -th computer, and it carries data in both directions. Your team will use the $S$ -th computer in the contest, and a judge server is the $T$ -th computer.

He decided to adjust the routing algorithm of the network to maximize the performance of the contestants through the magical power of prime numbers. In this algorithm, a packet (a unit of data carried by the network) is sent from your computer to the judge server passing through the cables a prime number of times if possible. If it is impossible, the contestants cannot benefit by the magical power of prime numbers. To accomplish this target, a packet is allowed to pass through the same cable multiple times.

You decided to write a program to calculate the minimum number of times a packet from $S$ to $T$ needed to pass through the cables. If the number of times a packet passes through the cables cannot be a prime number, print $-1$ .

Input

The input consists of a single test case, formatted as follows.

$N$ $M$ $S$ $T$ $a_1$ $b_1$ $\vdots$ $a_M$ $b_M$

The first line consists of four integers $N$ , $M$ , $S$ , and $T$ ( $2 \leq N \leq 10^5$ , $1 \leq M \leq 10^5$ , $1 \leq S, T \leq N$ , $S \neq T$ ). The $i$ -th line of the following $M$ lines consists of two integers $a_i$ and $b_i$ ( $1 \leq a_i < b_i \leq N$ ), which means that the $i$ -th cable connects the $a_i$ -th computer and the $b_i$ -th computer in the network. You can assume that the network satisfies the following conditions.

The network has no multi-edge, i.e., $(a_i, b_i) \neq (a_j, b_j)$ for all $i$ , $j$ ( $1 \leq i < j \leq M$ ).
The packets from $N$ computers are reachable to $T$ by passing through some number of cables. The number is not necessarily a prime.

Output

If there are ways such that the number of times a packet sent from $S$ to $T$ passes through the cables is a prime number, print the minimum prime number of times in one line. Otherwise, print $-1$ .

Examples4

Example 1
Input
```
5 5 1 5
1 2
1 4
2 3
3 4
3 5
```
Expected output
```
3
```
Example 2
Input
```
5 4 1 5
1 2
2 3
3 4
4 5
```
Expected output
```
-1
```
Example 3
Input
```
2 1 1 2
1 2
```
Expected output
```
3
```
Example 4
Input
```
3 3 1 2
1 2
1 3
2 3
```
Expected output
```
2
```