This approach involves topologically sorting the courses based on their prerequisites and then calculating the minimal time to complete each course using dynamic programming.
Create a directed graph where each course is a node, and a prerequisite relationship (a,b) is a directed edge from a to b.
Perform a topological sort on the courses. This step ensures that all prerequisites of a course are considered before the course itself.
Use dynamic programming to calculate the minimum time required to complete each course by iteratively adding its prerequisite's completion time.
Time Complexity: O(n + E), where n is the number of courses and E is the number of prerequisite relationships, due to traversing all nodes and edges once. Space Complexity: O(n + E), needed to store the graph and additional arrays.
1```javascript
2function minTime(n, relations, time) {
3 const indegree = new Array(n + 1).fill(0);
4 const adj = Array.from({ length: n + 1 }, () => []);
5 const dp = new Array(n + 1).fill(0);
67 for (const [u, v] of relations) {
8 adj[u].push(v);
9 indegree[v]++;
Explanation
This JavaScript implementation constructs an adjacency list and degrees array to use in topological sorting. By processing with a queue, it determines course completion order, updating a `dp` to store the minimum computation times and ultimately derives the result as the maximum time needed.
DFS with Memorization
This approach uses Depth-First Search (DFS) with memorization to efficiently compute the completion time for courses.
Model the courses and prerequisites as a directed graph.
Utilize DFS to explore courses, where each exploration path calculates the time required as we resolve prerequisites recursively.
Utilize memorization to store already computed completion times, avoiding redundant calculations and ensuring optimal performance.
Time Complexity: O(n + E), visiting each node and edge at least once. Space Complexity: O(n + E), due to storage for graph data and function call stack.
The JavaScript solution advances DFS through recursive capability, retaining computed course times and dependencies in a combination of arrays. Memorization through dp helps the solution operate more effectively by caching path findings.
