841. Keys and Rooms

Description of Problem

There are n rooms labeled from 0 to n - 1 and all the rooms are locked except for room 0. Your goal is to visit all the rooms. However, you cannot enter a locked room without having its key.

When you visit a room, you may find a set of distinct keys in it. Each key has a number on it, denoting which room it unlocks, and you can take all of them with you to unlock the other rooms.

Given an array rooms where rooms[i] is the set of keys that you can obtain if you visited room i, return true if you can visit all the rooms, or false otherwise.

Example 1:

Input: rooms = [[1],[2],[3],[]]
Output: true
Explanation: 
We visit room 0 and pick up key 1.
We then visit room 1 and pick up key 2.
We then visit room 2 and pick up key 3.
We then visit room 3.
Since we were able to visit every room, we return true.

Example 2:

Input: rooms = [[1,3],[3,0,1],[2],[0]]
Output: false
Explanation: We can not enter room number 2 since the only key that unlocks it is in that room.

Constraints:

  • n == rooms.length
  • 2 <= n <= 1000
  • 0 <= rooms[i].length <= 1000
  • 1 <= sum(rooms[i].length) <= 3000
  • 0 <= rooms[i][j] < n
  • All the values of rooms[i] are unique.

Solution

Tags: Depth First Search Graph Theory

Explanation

We use the same manner in 547. Number of Provinces. Perform depth-first-serach on node 0 and see whether all nodes are black.

Code (Rust)

#[derive(Clone, Copy, Debug, PartialEq)]
enum Colour{
    White,
    Gray,
    Black,
}
use Colour::{White, Gray, Black};

impl Solution {
    pub fn can_visit_all_rooms(rooms: Vec<Vec<i32>>) -> bool {
        let n = rooms.len();
        let mut colour = vec![White ; n];
        Self::dfs(0, &rooms, &mut colour);
        return colour.into_iter().all(|c| c == Black);
    }

    fn dfs(n : usize, rooms : &Vec<Vec<i32>>, colour : &mut Vec<Colour>) {
        colour[n] = Gray;

        for &i in rooms[n].iter() {
            let i = i as usize;
            if colour[i] == White {
                Self::dfs(i, rooms, colour);
            }
        }

        colour[n] = Black;

    }

}

Complexity

  • \(|V|\) is the number of nodes
  • \(|E|\) is the number of edges

Time Complexity

  • \( T(|V|,|E|) = \Theta(|V|) \)

Auxiliary Space

  • \(S(|V|,|E|) = \Theta(|V|)\)
    • Keep track of node colours