Keys

Timothy the architect has designed a new escape game. In this game, there are $n$ rooms numbered from $0$ to $n - 1$ . Initially, each room contains exactly one key. Each key has a type, which is an integer between $0$ and $n - 1$ , inclusive. The type of the key in room $i$ ( $0 \le i \le n - 1$ ) is $r\[i]$ . Note that multiple rooms may contain keys of the same type, i.e., the values $r\[i]$ are not necessarily distinct.

There are also $m$ bidirectional connectors in the game, numbered from $0$ to $m - 1$ . Connector $j$ ( $0 \le j \le m - 1$ ) connects a pair of different rooms $u\[j]$ and $v\[j]$ . A pair of rooms can be connected by multiple connectors.

The game is played by a single player who collects the keys and moves between the rooms by traversing the connectors. We say that the player traverses connector $j$ when they use this connector to move from room $u\[j]$ to room $v\[j]$ , or vice versa. The player can only traverse connector $j$ if they have collected a key of type $c\[j]$ before.

At any point during the game, the player is in some room $x$ and can perform two types of actions:

collect the key in room $x$ , whose type is $r\[x]$ (unless they have collected it already),
traverse a connector $j$ , where either $u\[j] = x$ or $v\[j] = x$ , if the player has collected a key of type $c\[j]$ beforehand. Note that the player never discards a key they have collected.

The player starts the game in some room $s$ not carrying any keys. A room $t$ is reachable from a room $s$ , if the player who starts the game in room $s$ can perform some sequence of actions described above, and reach room $t$ .

For each room $i$ ( $0 \le i \le n - 1$ ), denote the number of rooms reachable from room $i$ as $p\[i]$ . Timothy would like to know the set of indices $i$ that attain the minimum value of $p\[i]$ across $0 \le i \le n - 1$ .

문제

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