Course Schedule (graph)

本文介绍了一种基于课程及其先修关系的课程调度算法。通过构建有向图并利用拓扑排序来判断是否能完成所有课程,解决课程安排问题。具体方法包括记录每个节点的入度、寻找入度为0的节点并更新相邻节点的入度。

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Course Schedule

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?

For example:

2, [[1,0]]

There are a total of 2 courses to take. To take course 1 you should have finished course 0. So it is possible.

2, [[1,0],[0,1]]

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:
The input prerequisites is a graph represented by a list of edges, not adjacency matrices.
You may assume that there are no duplicate edges in the input prerequisites.

FROM:
LeetCode-Algorithms-graph

Hint:
把每节课作为节点,课程之间的先导关系作为边(基础课程指向后续课程),我们可以画出一个有向图。这显然是一道判断有向图是否为有向无环图的题,只要存在拓扑序列,就是无环图。因此,这道题可以通过求解拓扑序列来获得答案。

显然我们要先找到入度为0的节点,然后把它从图中删除,同时删除相关的边,再在所得新图中继续找寻入度为0的点,重复以上过程,如果最后所有节点都被删除,删除的顺序就是一个拓扑序列,而这个图也是一个有向无环图。反之,就是存在环路,无法找到拓扑序列。

为了在删除节点时方便我们把指向后续课程的边删除,存储时我们将节点指出去的节点放在一个集合之中,每删除一个节点,就把该节点对应集合内所有节点入度-1,达到删除边的效果。

时间复杂度O(V+E)

class Solution {
public:

    bool canFinish(int numCourses, vector<pair<int, int>>& prerequisites) {
        vector<unordered_set<int> > graph(numCourses);
        vector<int> indegree(numCourses);//记录所有节点的入度

        for (auto p : prerequisites) {
            graph[p.second].insert(p.first);
            indegree[p.first]++;
        }

        int finishedCountNum = 0;
        queue<int> canLearnCourses;//存储入度为0的节点
        for (int i = 0; i < numCourses; i++) {
            if (!indegree[i]) {
                canLearnCourses.push(i);
            }
        }
        while(!canLearnCourses.empty()) {
            int p = canLearnCourses.front();
            canLearnCourses.pop();
            finishedCountNum++;
            for (auto j : graph[p]) {
                indegree[j]--;
                if (!indegree[j]) {
                    canLearnCourses.push(j);
                }
            }
        }
        return finishedCountNum >= numCourses;
    }
};
根据引用[1]和引用的描述,设计一个教学计划编制程序的步骤如下: 1. 创建课程列表:将所有的课程作为节点,构建一个课程列表。 2. 添加先修关系:根据每门课程的先修关系,将课程之间的先修关系添加到课程列表中。 3. 构建AOV网:将课程列表中的课程看作AOV网的节点,根据先修关系,构建AOV网。 4. 拓扑排序:对AOV网进行拓扑排序,以确定每门课程的学习顺序。 5. 学期安排:根据每学期的学分上限和学习负担,将课程分配到不同的学期中。 6. 输出教学计划:按照学期顺序,输出每个学期的课程安排。 以下是一个示例的教学计划编制程序的伪代码: ```python # 创建课程列表 courses = ['课程A', '课程B', '课程C', '课程D', '课程E', '课程F'] # 添加先修关系 prerequisites = { '课程A': [], '课程B': ['课程A'], '课程C': ['课程A'], '课程D': ['课程B', '课程C'], '课程E': ['课程D'], '课程F': ['课程E'] } # 构建AOV网 graph = {} for course in courses: graph[course] = [] for course, prereq in prerequisites.items(): for p in prereq: graph[p].append(course) # 拓扑排序 def topological_sort(graph): in_degree = {course: 0 for course in graph} for course in graph: for prereq in graph[course]: in_degree[prereq] += 1 queue = [course for course in graph if in_degree[course] == 0] order = [] while queue: course = queue.pop(0) order.append(course) for prereq in graph[course]: in_degree[prereq] -= 1 if in_degree[prereq] == 0: queue.append(prereq) return order order = topological_sort(graph) # 学期安排 semester_limit = 3 # 每学期的学分上限 workload = 0 # 当前学期的学分总数 semester = 1 # 当前学期 schedule = {semester: []} # 教学计划 for course in order: if workload + course_credits[course] <= semester_limit: schedule[semester].append(course) workload += course_credits[course] else: semester += 1 schedule[semester] = [course] workload = course_credits[course] # 输出教学计划 for semester, courses in schedule.items(): print(f"第{semester}学期:") for course in courses: print(course) ```
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