Cycle Detection in Directed Graph in DSA Typescript - Time & Space Complexity

Time Complexity: Cycle Detection in Directed Graph

O(n + e)

Understanding Time Complexity

We want to understand how long it takes to check if a directed graph has a cycle.

This helps us know how the work grows as the graph gets bigger.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function hasCycle(graph: Map<number, number[]>): boolean {
  const visited = new Set<number>();
  const recStack = new Set<number>();

  function dfs(node: number): boolean {
    if (recStack.has(node)) return true;
    if (visited.has(node)) return false;

    visited.add(node);
    recStack.add(node);

    for (const neighbor of graph.get(node) || []) {
      if (dfs(neighbor)) return true;
    }

    recStack.delete(node);
    return false;
  }

  for (const node of graph.keys()) {
    if (dfs(node)) return true;
  }
  return false;
}
    

This code checks if a directed graph has a cycle using depth-first search and tracking recursion stack.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Depth-first search (DFS) visits each node and its neighbors.
• How many times: Each node is visited once, and each edge is checked once during DFS.

How Execution Grows With Input

As the number of nodes and edges grow, the work grows roughly in proportion to their total.

Input Size (n nodes, e edges)	Approx. Operations
10 nodes, 15 edges	About 25 operations (visiting nodes + edges)
100 nodes, 200 edges	About 300 operations
1000 nodes, 5000 edges	About 6000 operations

Pattern observation: The operations grow roughly linearly with the sum of nodes and edges.

Final Time Complexity

Time Complexity: O(n + e)

This means the time to detect a cycle grows linearly with the number of nodes plus edges in the graph.

Common Mistake

[X] Wrong: "The DFS visits nodes multiple times, so time is O(n * e)."

[OK] Correct: Each node is marked visited and processed once, so edges are checked only once, keeping time linear.

Interview Connect

Understanding this time complexity helps you explain how your cycle detection scales with graph size, a key skill in graph problems.

Self-Check

"What if we used an adjacency matrix instead of adjacency lists? How would the time complexity change?"

Practice

1. What does cycle detection in a directed graph help us find?

easy

A. Whether there is a path that starts and ends at the same node following edge directions

B. The shortest path between two nodes

C. The total number of nodes in the graph

D. The maximum degree of any node

Solution

1. Step 1: Understand cycle in directed graph

   A cycle means a path that starts and ends at the same node following the direction of edges.

2. Step 2: Identify what cycle detection finds

   Cycle detection checks if such a path exists in the graph.

3. Final Answer:

   Whether there is a path that starts and ends at the same node following edge directions -> Option A

4. Quick Check:

   Cycle detection = finding directed loops [OK]

Hint: Cycle means path loops back following edge directions [OK]

Common Mistakes:

• Confusing cycle with shortest path
• Ignoring edge directions
• Thinking cycle means any repeated node without direction

2. Which TypeScript data structure is commonly used to track nodes currently in recursion stack during cycle detection?

easy

A. Queue

B. Array

C. Map

D. Set

Solution

1. Step 1: Identify recursion stack tracking

   We need to quickly check if a node is in the current recursion path.

2. Step 2: Choose data structure for fast membership check

   A Set allows O(1) membership check, ideal for tracking nodes in recursion stack.

3. Final Answer:

   Set -> Option D

4. Quick Check:

   Recursion stack tracking uses Set [OK]

Hint: Use Set for fast membership check in recursion stack [OK]

Common Mistakes:

• Using Array and checking membership with loops (slow)
• Using Queue which is FIFO, not suitable here
• Using Map without key-value need

3. What will be the output of the following TypeScript code snippet detecting cycle in a directed graph?

const graph = new Map([
  [1, [2]],
  [2, [3]],
  [3, [1]]
]);

function hasCycle(node: number, visited: Set<number>, recStack: Set<number>): boolean {
  if (!visited.has(node)) {
    visited.add(node);
    recStack.add(node);
    for (const neighbor of graph.get(node) || []) {
      if (!visited.has(neighbor) && hasCycle(neighbor, visited, recStack)) {
        return true;
      } else if (recStack.has(neighbor)) {
        return true;
      }
    }
  }
  recStack.delete(node);
  return false;
}

console.log(hasCycle(1, new Set(), new Set()));

medium

A. false

B. undefined

C. true

D. Error

Solution

1. Step 1: Analyze graph edges

   Graph edges: 1->2, 2->3, 3->1 form a cycle.

2. Step 2: Trace recursion and cycle detection

   Starting at 1, recursion visits 2, then 3, then back to 1 which is in recursion stack, so returns true.

3. Final Answer:

   true -> Option C

4. Quick Check:

   Cycle exists in 1->2->3->1 path [OK]

Hint: Cycle if recursion stack revisits a node [OK]

Common Mistakes:

• Ignoring recursion stack check
• Assuming no cycle if visited is true
• Confusing visited and recursion stack sets

4. Identify the bug in this TypeScript cycle detection function for a directed graph:

function detectCycle(node: number, visited: Set<number>, recStack: Set<number>): boolean {
  if (visited.has(node)) {
    return false;
  }
  visited.add(node);
  recStack.add(node);
  for (const neighbor of graph.get(node) || []) {
    if (detectCycle(neighbor, visited, recStack)) {
      return true;
    }
  }
  recStack.delete(node);
  return false;
}

medium

A. Missing check if neighbor is in recursion stack to detect cycle

B. Visited set is not updated correctly

C. recStack is never cleared

D. Function does not return boolean

Solution

1. Step 1: Review cycle detection logic

   Cycle detection requires checking if neighbor is already in recursion stack to detect back edge.

2. Step 2: Identify missing condition

   The code calls recursion but does not check if neighbor is in recStack, so misses cycles.

3. Final Answer:

   Missing check if neighbor is in recursion stack to detect cycle -> Option A

4. Quick Check:

   Check recStack for back edges to detect cycles [OK]

Hint: Always check recursion stack for back edges [OK]

Common Mistakes:

• Skipping recursion stack membership check
• Confusing visited and recursion stack roles
• Not removing node from recursion stack after recursion

5. Given a directed graph represented as adjacency list:

const graph = new Map([
  [1, [2, 3]],
  [2, [4]],
  [3, [4, 5]],
  [4, [6]],
  [5, [3]],
  [6, []]
]);

Which nodes are part of at least one cycle?

hard

A. [1, 2, 4, 6]

B. [3, 5]

C. [2, 4, 6]

D. [1, 3, 5]

Solution

1. Step 1: Analyze edges for cycles

   Edges: 3->5 and 5->3 form a cycle between nodes 3 and 5.

2. Step 2: Check other nodes

   Nodes 1, 2, 4, 6 do not form cycles as no back edges lead to them.

3. Final Answer:

   [3, 5] -> Option B

4. Quick Check:

   Cycle nodes = 3 and 5 only [OK]

Hint: Look for back edges forming loops between nodes [OK]

Common Mistakes:

• Including nodes not in any cycle
• Missing nodes that form cycles
• Confusing path with cycle