P13856 [CERC 2023] Labelled Paths

We are given a directed acyclic graph with $n$ vertices and $m$ edges. Each edge has a label (a string of lowercase letters; possibly even an empty string). We can now extend the concept of labels from edges to paths by defining the label of a path as the concatenation of the labels of the edges that constitute this path (in the same order in which they appear in the path). The smallest path from a start vertex $s$ to a destination vertex $t$ is the path (from $s$ to $t$) whose label is lexicographically smallest (i.e. the earliest in lexicographical order) amongst all the paths from $s$ to $t$. Write a program that, for a given $s$, outputs the smallest paths from $s$ to $t$ for all vertices $t$ of the graph.

The first line contains four space-separated integers: $n$ (the number of vertices), $m$ (the number of edges), $d$ (the length of the string $A$, on which see below) and $s$ (the number of the start vertex). The vertices are numbered by integers from $1$ to $n$. The second line contains a string $A$, which is exactly $d$ characters long; all these characters are lowercase letters of the English alphabet. All the edge labels in our graph are substrings of the string $A$. The remaining $m$ lines describe the edges of the graph. The $i$-th of these lines describes the $i$-th edge and contains four space-separated integers: $u_i$ (the start vertex of this edge), $v_i$ (the end vertex of this edge), $p_i$ and $\ell_i$. The last two of these integers indicate that the label of this edge is the substring of $A$ that begins with the $p_i$-th character of $A$ and is $\ell_i$ characters long. For this purpose we consider the characters of $A$ to be indexed by integers from $1$ to $d$.

Output $n$ lines, where the $t$-th line (for $t = 1, \dots, n$) describes the smallest path from $s$ to $t$. If there is no path from $s$ to $t$, the line should contain only the integer $0$ and nothing else. Otherwise the line should start with the number of vertices on the path (including vertices $s$ and $t$), followed by the list of those vertices, separated by spaces. If there are several possible solutions, you may output any of them.

### Comment In this example, the edge $3 \rightarrow 1$ has the label ab; the edge $1 \rightarrow 4$ has the label a; the smallest path from $3$ to $4$ is $3 \rightarrow 1 \rightarrow 4$, whose label is aba. ### Input limits - $1 \leq s \leq n \leq 600$ - $1 \leq m \leq 2000$ - $1 \leq d \leq 10^6$ - $1 \leq u_i \leq n$, $1 \leq v_i \leq n$, $u_i \neq v_i$ (for all $i = 1, \dots, m$) - $1 \leq p_i$, $0 \leq \ell_i$, $p_i + \ell_i - 1 \leq d$ (for all $i = 1, \dots, m$) - The graph is acyclic and has no parallel edges (i.e. from $i \neq j$ it follows that $u_i \neq u_j$ and/or $v_i \neq v_j$).