[LeetCode] 207. Course Schedule 課程清單

 

There are a total of n courses you have to take, labeled from 0 to n-1.

Some courses may have prerequisites, for example to take course 0 you have to first take course 1, which is expressed as a pair: [0,1]

Given the total number of courses and a list of prerequisite pairs, is it possible for you to finish all courses?

Example 1:

Input: 2, [[1,0]] 
Output: true
Explanation: There are a total of 2 courses to take. 
             To take course 1 you should have finished course 0. So it is possible.

Example 2:

Input: 2, [[1,0],[0,1]]
Output: false
Explanation: There are a total of 2 courses to take. 
             To take course 1 you should have finished course 0, and to take course 0 you should
             also have finished course 1. So it is impossible.

Note:

1. The input prerequisites is a graph represented by a list of edges, not adjacency matrices. Read more about how a graph is represented.
2. You may assume that there are no duplicate edges in the input prerequisites.

Hints:

1. This problem is equivalent to finding if a cycle exists in a directed graph. If a cycle exists, no topological ordering exists and therefore it will be impossible to take all courses.
2. There are several ways to represent a graph. For example, the input prerequisites is a graph represented by a list of edges. Is this graph representation appropriate?
3. Topological Sort via DFS - A great video tutorial (21 minutes) on Coursera explaining the basic concepts of Topological Sort.
4. Topological sort could also be done via BFS.

 

這道課程清單的問題對於咱們學生來講應該不陌生，由於咱們在選課的時候常常會遇到想選某一門課程，發現選它以前必須先上了哪些課程，這道題給了不少提示，第一條就告訴咱們了這道題的本質就是在有向圖中檢測環。 LeetCode 中關於圖的題不多，有向圖的僅此一道，還有一道關於無向圖的題是 Clone Graph。我的認爲圖這種數據結構相比於樹啊，鏈表啊什麼的要更爲複雜一些，尤爲是有向圖，很麻煩。第二條提示是在講如何來表示一個有向圖，能夠用邊來表示，邊是由兩個端點組成的，用兩個點來表示邊。第三第四條提示揭示了此題有兩種解法，DFS 和 BFS 均可以解此題。咱們先來看 BFS 的解法，咱們定義二維數組 graph 來表示這個有向圖，一維數組 in 來表示每一個頂點的入度。咱們開始先根據輸入來創建這個有向圖，並將入度數組也初始化好。而後咱們定義一個 queue 變量，將全部入度爲0的點放入隊列中，而後開始遍歷隊列，從 graph 裏遍歷其鏈接的點，每到達一個新節點，將其入度減一，若是此時該點入度爲0，則放入隊列末尾。直到遍歷完隊列中全部的值，若此時還有節點的入度不爲0，則說明環存在，返回 false，反之則返回 true。代碼以下：

 

解法一：

class Solution {
public:
    bool canFinish(int numCourses, vector<vector<int>>& prerequisites) {
        vector<vector<int>> graph(numCourses, vector<int>());
        vector<int> in(numCourses);
        for (auto a : prerequisites) {
            graph[a[1]].push_back(a[0]);
            ++in[a[0]];
        }
        queue<int> q;
        for (int i = 0; i < numCourses; ++i) {
            if (in[i] == 0) q.push(i);
        }
        while (!q.empty()) {
            int t = q.front(); q.pop();
            for (auto a : graph[t]) {
                --in[a];
                if (in[a] == 0) q.push(a);
            }
        }
        for (int i = 0; i < numCourses; ++i) {
            if (in[i] != 0) return false;
        }
        return true;
    }
};

 

下面咱們來看 DFS 的解法，也須要創建有向圖，仍是用二維數組來創建，和 BFS 不一樣的是，咱們像如今須要一個一維數組 visit 來記錄訪問狀態，這裏有三種狀態，0表示還未訪問過，1表示已經訪問了，-1 表示有衝突。大致思路是，先創建好有向圖，而後從第一個門課開始，找其可構成哪門課，暫時將當前課程標記爲已訪問，而後對新獲得的課程調用 DFS 遞歸，直到出現新的課程已經訪問過了，則返回 false，沒有衝突的話返回 true，而後把標記爲已訪問的課程改成未訪問。代碼以下：

 

解法二：

class Solution {
public:
    bool canFinish(int numCourses, vector<vector<int>>& prerequisites) {
        vector<vector<int>> graph(numCourses, vector<int>());
        vector<int> visit(numCourses);
        for (auto a : prerequisites) {
            graph[a[1]].push_back(a[0]);
        }
        for (int i = 0; i < numCourses; ++i) {
            if (!canFinishDFS(graph, visit, i)) return false;
        }
        return true;
    }
    bool canFinishDFS(vector<vector<int>>& graph, vector<int>& visit, int i) {
        if (visit[i] == -1) return false;
        if (visit[i] == 1) return true;
        visit[i] = -1;
        for (auto a : graph[i]) {
            if (!canFinishDFS(graph, visit, a)) return false;
        }
        visit[i] = 1;
        return true;
    }
};

 

Github 同步地址：

https://github.com/grandyang/leetcode/issues/207

 

相似題目：

Minimum Height Trees

Course Schedule II

Course Schedule III

Graph Valid Tree

 

參考資料：

https://leetcode.com/problems/course-schedule/

https://leetcode.com/problems/course-schedule/discuss/58524/Java-DFS-and-BFS-solution

https://leetcode.com/problems/course-schedule/discuss/58516/Easy-BFS-Topological-sort-Java

https://leetcode.com/problems/course-schedule/discuss/162743/JavaC%2B%2BPython-BFS-Topological-Sorting-O(N-%2B-E)

 

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