Anna, Svyatoslav and Maps
Contest: Codeforces Round 581 (Div. 2)
Difficulty rating: 1700
Tags: dp, graphs, greedy, shortest paths

The main characters have been omitted to be short.

You are given a directed unweighted graph without loops with $$$n$$$ vertexes and a path in it (that path is not necessary simple) given by a sequence $$$p_1, p_2, \ldots, p_m$$$ of $$$m$$$ vertexes; for each $$$1 \leq i < m$$$ there is an arc from $$$p_i$$$ to $$$p_{i+1}$$$.

Define the sequence $$$v_1, v_2, \ldots, v_k$$$ of $$$k$$$ vertexes as good, if $$$v$$$ is a subsequence of $$$p$$$, $$$v_1 = p_1$$$, $$$v_k = p_m$$$, and $$$p$$$ is one of the shortest paths passing through the vertexes $$$v_1$$$, $$$\ldots$$$, $$$v_k$$$ in that order.

A sequence $$$a$$$ is a subsequence of a sequence $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) elements. It is obvious that the sequence $$$p$$$ is good but your task is to find the shortest good subsequence.

If there are multiple shortest good subsequences, output any of them.

Input format:
The first line contains a single integer $$$n$$$ ($$$2 \le n \le 100$$$) — the number of vertexes in a graph.

The next $$$n$$$ lines define the graph by an adjacency matrix: the $$$j$$$-th character in the $$$i$$$-st line is equal to $$$1$$$ if there is an arc from vertex $$$i$$$ to the vertex $$$j$$$ else it is equal to $$$0$$$. It is guaranteed that the graph doesn't contain loops.

The next line contains a single integer $$$m$$$ ($$$2 \le m \le 10^6$$$) — the number of vertexes in the path.

The next line contains $$$m$$$ integers $$$p_1, p_2, \ldots, p_m$$$ ($$$1 \le p_i \le n$$$) — the sequence of vertexes in the path. It is guaranteed that for any $$$1 \leq i < m$$$ there is an arc from $$$p_i$$$ to $$$p_{i+1}$$$.

Output format:
In the first line output a single integer $$$k$$$ ($$$2 \leq k \leq m$$$) — the length of the shortest good subsequence. In the second line output $$$k$$$ integers $$$v_1$$$, $$$\ldots$$$, $$$v_k$$$ ($$$1 \leq v_i \leq n$$$) — the vertexes in the subsequence. If there are multiple shortest subsequences, print any. Any two consecutive numbers should be distinct.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
Below you can see the graph from the first example:

The given path is passing through vertexes $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$. The sequence $$$1-2-4$$$ is good because it is the subsequence of the given path, its first and the last elements are equal to the first and the last elements of the given path respectively, and the shortest path passing through vertexes $$$1$$$, $$$2$$$ and $$$4$$$ in that order is $$$1-2-3-4$$$. Note that subsequences $$$1-4$$$ and $$$1-3-4$$$ aren't good because in both cases the shortest path passing through the vertexes of these sequences is $$$1-3-4$$$.

In the third example, the graph is full so any sequence of vertexes in which any two consecutive elements are distinct defines a path consisting of the same number of vertexes.

In the fourth example, the paths $$$1-2-4$$$ and $$$1-3-4$$$ are the shortest paths passing through the vertexes $$$1$$$ and $$$4$$$.

Example input:
4
0110
0010
0001
1000
4
1 2 3 4

Example output:
3
1 2 4

Example input:
4
0110
0010
1001
1000
20
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Example output:
11
1 2 4 2 4 2 4 2 4 2 4

Editorial:
Firstly, find the matrix of the shortest paths using Floyd-Warshall algortihm or running dfs from all vertexes; then a greedy approach works here: add $$$p_1$$$ to the answer and then go along the path; if the distance from the last vertex in the answer to the current vertex $$$p_i$$$ is shorter than in given path, add the vertex $$$p_{i-1}$$$ to the answer and continue traversing the path. Don't forget to add $$$p_m$$$ in the end!