210. Course Schedule II

Description of the Problem

There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [a_i, b_i] indicates that you must take course bi first if you want to take course a_i.

• For example, the pair [0, 1], indicates that to take course 0 you have to first take course 1.

Return the ordering of courses you should take to finish all courses. If there are many valid answers, return any of them. If it is impossible to finish all courses, return an empty array.

Example 1:

Input: numCourses = 2, prerequisites = [[1,0]]
Output: [0,1]
Explanation: There are a total of 2 courses to take. To take course 1 you should have finished course 0. So the correct course order is [0,1].

Example 2:

Input: numCourses = 4, prerequisites = [[1,0],[2,0],[3,1],[3,2]]
Output: [0,2,1,3]
Explanation: There are a total of 4 courses to take. To take course 3 you should have finished both courses 1 and 2. Both courses 1 and 2 should be taken after you finished course 0.
So one correct course order is [0,1,2,3]. Another correct ordering is [0,2,1,3].

Example 3:

Input: numCourses = 1, prerequisites = []
Output: [0]

Constraints:

• 1 <= numCourses <= 2000
• 0 <= prerequisites.length <= numCourses * (numCourses - 1)
• prerequisites[i].length == 2
• 0 <= a_i, b_i < numCourses
• a_i != b_i
• All the pairs [a_i, b_i] are distinct.

Solution

Tags: Graph Theory

Explanation

To get the proper order of courses, we use Depth First Search to do Topological Sorting. For the detailed explanation. Please see Introduction to Algorithms (CLRS) 3rd Ed. (Section 22.4)

Code (Rust)

use std::collections::{HashMap, VecDeque};

impl Solution {
    pub fn find_order(num_courses: i32, prerequisites: Vec<Vec<i32>>) -> Vec<i32> {
        let num_courses = num_courses as usize;

        let mut adj_list : Vec<Vec<usize>> = vec![ vec![] ; num_courses ];
        let mut is_gray : Vec<bool> = vec![false; num_courses];
        let mut is_black : Vec<bool> = vec![false; num_courses];
        let mut order : VecDeque<i32> = VecDeque::new();

        for p in prerequisites.into_iter(){
            let (c1, c2) = (p[1] as usize, p[0] as usize);
            adj_list[c1].push(c2);
        }

        let mut has_loop = false;

        for vertex in (0..num_courses) {
            if !is_black[vertex] && !is_gray[vertex] && !has_loop {
                has_loop = Self::visit(vertex, &adj_list, &mut is_gray, &mut is_black, &mut order);
            }
        }

        if !has_loop { order.into_iter().collect::<Vec<i32>>() } else { vec![] }
    }

    fn visit(
        vertex : usize, 
        adj_list : &Vec<Vec<usize>>,
        is_gray : &mut Vec<bool>,
        is_black : &mut Vec<bool>,
        order : &mut VecDeque<i32>
    ) -> bool {
        for &v in adj_list[vertex].iter() {
            match (is_black[v], is_gray[v]){
                (false, false) => {
                    is_gray[v] = true;
                    if Self::visit(v, adj_list, is_gray, is_black, order) == true {
                        return true;
                    }
                    is_gray[v] = false;
                }
                (false, true) => return true,
                _ => {}
            }
        }
        is_black[vertex] = true;
        order.push_front(vertex as i32);
        return false;
    }
}