Course Schedule
A medium Graph problem included in Love Babbar 450, Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 2
- Core for
- 11 roles
- Platform
- LeetCode
The problem
There are a total of numCourses courses labeled from 0 to numCourses-1. You are given a list of prerequisite pairs where each pair [a, b] means course b must be taken before course a. Determine if it is possible to finish all courses.
Example 1
- Input
- numCourses = 2, prerequisites = [[1,0]]
- Output
- true
- Why
- Take course 0 first, then course 1. No cycle in prerequisites.
Example 2
- Input
- numCourses = 2, prerequisites = [[1,0],[0,1]]
- Output
- false
- Why
- Course 0 depends on course 1 and vice versa, creating a cycle.
Constraints
- 1 <= numCourses <= 2000
- 0 <= prerequisites.length <= 5000
- prerequisites[i].length == 2
How to think about it
Updated 2026-09-09Finishing all courses is possible if and only if there is no circular dependency anywhere in the prerequisite chain. A cycle in a directed graph traps any progression, so the entire question reduces to cycle detection. If you can peel off nodes with zero prerequisites one by one until none remain, no cycle exists.
Approaches, worst first
DFS three-color cycle detection
time O(V + E) · space O(V + E)
Assign each node one of three states: unvisited, currently visiting in recursion stack, or fully processed. Traversing into an edge whose target is marked as currently visiting flags a back-edge and proves a cycle exists.
Kahn algorithm with indegree arrayWrite this one
time O(V + E) · space O(V + E)
Track in-degrees of all courses and push zero-indegree vertices into a queue. Repeatedly dequeue a course, increment a completed counter, and decrement the in-degrees of dependent courses, enqueueing any that drop to zero. If final count equals numCourses, all courses can be finished.
Where people lose marks · 3
- Reversing the edge direction by mistake: `prerequisites = [a, b]` means `b -> a` (b must precede a), not `a -> b`.
- Disconnected graph components require scanning every course index from 0 to numCourses - 1 as candidate roots, not just launching from course 0.
- Duplicate edges in prerequisites can artificially inflate in-degree counts if not deduplicated or accounted for in adjacency lists.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
What each operation costs
| Operation | Time |
|---|---|
| visit all nodes and edges via search | O(v + e) |
| topological sort using in-degree counts | O(v + e) |
| shortest path using dijkstra with a min-heap | O((v + e) log v) |
What usually goes wrong with Graph
- Adding a node to the visited set when popping from the queue instead of when pushing, which lets neighboring nodes enqueue duplicate entries and wastes memory.
- Failing to check for cycles in directed graphs when finding prerequisite orders, causing topological sort routines to hang or return incomplete lists.
- Assuming an input graph is fully connected and scanning from only a single starting node, missing disconnected islands and isolated components.
Which roles need this problem
Graph is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 6 more roles, including Performance Engineer, Search Engineer, Information Retrieval Engineer.
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