Topological Sort Using DFS in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main idea behind Topological Sort using DFS?

We visit each node and explore all its neighbors first. After visiting all neighbors, we add the node to the result. This way, nodes appear after all nodes they depend on.

beginner

What kind of graph can we apply Topological Sort to?

Topological Sort works only on Directed Acyclic Graphs (DAGs), which means the graph has directed edges and no cycles.

beginner

In Topological Sort using DFS, when do we add a node to the sorted list?

We add a node to the sorted list only after all its neighbors (nodes it points to) have been visited and added.

intermediate

Why do we use a stack or reverse the order of nodes after DFS in Topological Sort?

Because nodes are added after visiting neighbors, the order is reversed. Using a stack or reversing the list gives the correct topological order.

intermediate

What happens if the graph contains a cycle when performing Topological Sort using DFS?

The algorithm cannot produce a valid topological order because cycles mean some nodes depend on each other in a loop. Detecting cycles is important to avoid incorrect results.

Which graph type is required for Topological Sort using DFS?

A Cyclic Directed Graph

B Undirected Graph

C Directed Acyclic Graph (DAG)

D Any Graph

Topological Sort only works on Directed Acyclic Graphs (DAGs) because cycles prevent a valid ordering.

When do we add a node to the result list in DFS-based Topological Sort?

A After visiting all neighbors

B Before visiting neighbors

C When the node is first discovered

D Randomly

We add the node after visiting all its neighbors to ensure dependencies come first.

What data structure is commonly used to store the order during DFS Topological Sort?

A Queue

B Linked List

C HashSet

D Stack

A stack stores nodes in reverse order of completion, which is reversed later to get the topological order.

What does a cycle in the graph indicate during Topological Sort?

A No valid topological order

B Graph is undirected

C Valid ordering exists

D Graph is disconnected

Cycles mean nodes depend on each other in a loop, so no valid order exists.

What is the time complexity of Topological Sort using DFS for a graph with V vertices and E edges?

A O(V^2)

B O(V + E)

C O(E^2)

D O(V * E)

Each vertex and edge is visited once, so the time complexity is O(V + E).

Explain how Topological Sort using DFS works step-by-step.

Think about visiting neighbors before the node itself.

Describe how cycle detection is important in Topological Sort using DFS and how it can be done.

Consider marking nodes as visiting and visited.

Practice

1. What is the main purpose of using DFS in Topological Sort on a directed graph?

easy

A. To find the shortest path between two nodes

B. To visit nodes in a way that all dependencies come before the node itself

C. To detect cycles in an undirected graph

D. To count the number of connected components

Solution

1. Step 1: Understand DFS role in Topological Sort

   DFS visits nodes deeply, ensuring all nodes reachable from a node are visited before the node itself.

2. Step 2: Relate DFS order to dependencies

   In Topological Sort, nodes must appear after all their dependencies, which DFS achieves by post-order traversal.

3. Final Answer:

   To visit nodes in a way that all dependencies come before the node itself -> Option B

4. Quick Check:

   DFS ensures dependency order [OK]

Hint: DFS post-order gives correct dependency order [OK]

Common Mistakes:

• Confusing DFS with BFS for topological sorting
• Thinking DFS finds shortest paths
• Assuming DFS detects cycles in undirected graphs only

2. Which of the following C# code snippets correctly starts a DFS-based topological sort on all nodes in a graph represented as adjacency list Dictionary?

easy

A. for(int i=0; i

B. while(graph.Count > 0) DFS(graph.First().Key);

C. foreach(var node in graph.Keys) if(!visited[node]) DFS(node);

D. foreach(var node in graph.Values) DFS(node[0]);

Solution

1. Step 1: Identify correct iteration over nodes

   We must iterate over all keys (nodes) and call DFS only if not visited.

2. Step 2: Check each option's correctness

   foreach(var node in graph.Keys) if(!visited[node]) DFS(node); correctly iterates keys and checks visited. for(int i=0; i 0) DFS(graph.First().Key); modifies graph during iteration, unsafe. foreach(var node in graph.Values) DFS(node[0]); iterates values, not nodes.

3. Final Answer:

   foreach(var node in graph.Keys) if(!visited[node]) DFS(node); -> Option C

4. Quick Check:

   Iterate keys and check visited [OK]

Hint: Iterate graph keys, check visited before DFS [OK]

Common Mistakes:

• Iterating over graph.Values instead of Keys
• Assuming node indices are continuous from 0
• Modifying collection during iteration

3. What is the output of the following C# code performing topological sort using DFS on the graph edges: 5->2, 5->0, 4->0, 4->1, 2->3, 3->1?

var graph = new Dictionary {
  {5, new List<int>{2,0}},
  {4, new List<int>{0,1}},
  {2, new List<int>{3}},
  {3, new List<int>{1}},
  {0, new List<int>()},
  {1, new List<int>()}
};
var visited = new HashSet<int>();
var stack = new Stack<int>();
void DFS(int node) {
  visited.Add(node);
  foreach(var n in graph[node]) {
    if(!visited.Contains(n)) DFS(n);
  }
  stack.Push(node);
}
foreach(var node in graph.Keys) {
  if(!visited.Contains(node)) DFS(node);
}
var result = stack.ToArray();
Console.WriteLine(string.Join(", ", result));

medium

A. 4, 5, 0, 2, 3, 1

B. 5, 4, 2, 3, 1, 0

C. 5, 4, 0, 1, 2, 3

D. 4, 5, 0, 1, 2, 3

Solution

1. Step 1: Perform DFS and track stack push order

   Start DFS from 5: visit 2, then 3, then 1; push 1, 3, 2, then 0 from 5's other edge; push 5. Then DFS from 4: visit 0 and 1 already visited, push 4.

2. Step 2: Stack order and output

   Stack push order is 1, 3, 2, 0, 5, 4. Converting stack to array reverses order: 4, 5, 0, 2, 3, 1.

3. Final Answer:

   4, 5, 0, 2, 3, 1 -> Option A

4. Quick Check:

   DFS post-order reversed = topological order [OK]

Hint: Stack reversed after DFS gives topological order [OK]

Common Mistakes:

• Not reversing stack before output
• Visiting nodes in wrong order
• Ignoring nodes with no outgoing edges

4. The following C# DFS-based topological sort code throws a runtime error. What is the likely cause?

void DFS(int node) {
  visited.Add(node);
  foreach(var n in graph[node]) {
    if(!visited.Contains(n)) DFS(n);
  }
  stack.Push(node);
}
foreach(var node in graph.Keys) {
  DFS(node);
}

medium

A. No error; code runs correctly

B. NullReferenceException because graph is empty

C. Stack overflow due to infinite recursion from cycles

D. KeyNotFoundException because some nodes are missing in graph keys

Solution

1. Step 1: Analyze potential runtime errors in DFS

   When recursing to a neighbor n not present as a key in the graph (common for sink nodes), accessing graph[n] throws KeyNotFoundException.

2. Step 2: Outer loop issue

   Missing !visited check before DFS(node) causes duplicate stack pushes (logical error), not runtime error. Cycle detection needs separate recursion stack.

3. Final Answer:

   KeyNotFoundException because some nodes are missing in graph keys -> Option D

4. Quick Check:

   graph[n] for missing sink node key -> KeyNotFoundException [OK]

Hint: Include all nodes as keys with empty lists [OK]

Common Mistakes:

• Omitting sink/leaf nodes from graph keys
• Not initializing empty List<int>() for nodes with no outgoing edges
• Assuming all edge targets are automatically graph keys

5. Given a directed acyclic graph (DAG) with nodes representing tasks and edges representing dependencies, how can you modify the DFS-based topological sort to detect if the graph contains a cycle?

hard

A. Use an additional recursion stack set to track nodes in the current DFS path and detect back edges

B. Count the number of nodes visited and compare with total nodes after DFS

C. Run BFS instead of DFS to detect cycles

D. Sort nodes by their degree before DFS

Solution

1. Step 1: Understand cycle detection in DFS

   Use a recursion stack set to mark nodes currently in the DFS path. If a node is revisited while in this set, a cycle exists.

2. Step 2: Explain why other options fail

   Counting visited nodes doesn't detect cycles. BFS alone doesn't detect cycles in directed graphs. Sorting by degree doesn't help detect cycles.

3. Final Answer:

   Use an additional recursion stack set to track nodes in the current DFS path and detect back edges -> Option A

4. Quick Check:

   Recursion stack detects cycles in DFS [OK]

Hint: Track recursion stack to find back edges indicating cycles [OK]

Common Mistakes:

• Ignoring recursion stack for cycle detection
• Assuming BFS detects cycles in directed graphs
• Using node count to detect cycles