Topological sorting in Data Structures Theory - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to identify the type of graph suitable for topological sorting.

Data Structures Theory

A topological sort can only be performed on a [1] graph.

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Adirected acyclic

Bcyclic

Cundirected

Dweighted

Attempts:

3 left

💡 Hint

Common Mistakes

Choosing cyclic graphs because they have directions.

Thinking undirected graphs can be topologically sorted.

Topological sorting is only possible on directed acyclic graphs (DAGs) because cycles prevent a linear ordering.

2fill in blank

medium

Complete the code to describe the output of a topological sort.

Data Structures Theory

The output of a topological sort is a [1] of the vertices such that for every directed edge u -> v, u comes before v.

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Atree structure

Brandom order

Creverse order

Dlinear ordering

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing topological sort with sorting by value.

Thinking the output is a tree or graph structure.

Topological sorting produces a linear ordering of vertices respecting the direction of edges.

3fill in blank

hard

Fix the error in the statement about cycle detection in topological sorting.

Data Structures Theory

If a graph contains a cycle, topological sorting [1] be performed.

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Acan always

Bmight

Ccannot

Dshould

Attempts:

3 left

💡 Hint

Common Mistakes

Assuming cycles can be ignored.

Thinking topological sort can handle any graph.

Topological sorting cannot be performed on graphs with cycles because no linear ordering exists that respects all edges.

4fill in blank

hard

Fill both blanks to complete the description of a common algorithm for topological sorting.

Data Structures Theory

One common method uses Depth-First Search (DFS) and adds each vertex to a stack after visiting all its [1]. Then, the stack is [2] to get the topological order.

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Aneighbors

Bchildren

Creversed

Dsorted

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'neighbors' instead of 'children' which is less precise here.

Sorting the stack instead of reversing it.

The DFS method visits all children (outgoing edges) before adding a vertex to the stack. Reversing the stack gives the topological order.

5fill in blank

hard

Fill all three blanks to complete the pseudocode for Kahn's algorithm for topological sorting.

Data Structures Theory

Initialize a queue with all vertices having [1] zero. While the queue is not empty, remove a vertex [2] and add it to the result. Then, decrease the [3] of its neighbors by one.

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Ain-degree

Bfront

Dout-degree

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing in-degree with out-degree.

Removing vertices from the back instead of the front.

Kahn's algorithm starts with vertices with in-degree zero, processes them from the front of the queue, and decreases the in-degree of their neighbors.

Practice

1. What is the main requirement for a graph to have a valid topological sorting?

easy

A. The graph must be a Directed Acyclic Graph (DAG)

B. The graph must be undirected

C. The graph must contain cycles

D. The graph must be complete

Solution

1. Step 1: Understand the definition of topological sorting

   Topological sorting orders nodes so that every directed edge from node A to node B means A comes before B in the order.

2. Step 2: Identify the graph type needed

   This ordering is only possible if the graph has no cycles, meaning it must be a Directed Acyclic Graph (DAG).

3. Final Answer:

   The graph must be a Directed Acyclic Graph (DAG) -> Option A

4. Quick Check:

   DAG = Required for topological sorting [OK]

Hint: Topological sort needs no cycles, so graph must be DAG [OK]

Common Mistakes:

• Thinking topological sort works on undirected graphs
• Assuming cycles are allowed
• Confusing DAG with any directed graph

2. Which of the following is the correct way to represent the order of nodes after a topological sort?

easy

A. A list where each node appears before all nodes it points to

B. A list where nodes appear in random order

C. A list where each node appears after all nodes it points to

D. A list sorted by node values numerically

Solution

1. Step 1: Recall the property of topological order

   In topological sorting, every node appears before all nodes it has edges to (its dependencies come first).

2. Step 2: Match this property to the options

   A list where each node appears before all nodes it points to correctly states that each node appears before all nodes it points to, which is the definition of topological order.

3. Final Answer:

   A list where each node appears before all nodes it points to -> Option A

4. Quick Check:

   Node order respects edges direction = A list where each node appears before all nodes it points to [OK]

Hint: Topological order means dependencies come first [OK]

Common Mistakes:

• Confusing order with numerical sorting
• Thinking nodes appear after their dependencies
• Assuming random order is valid

3. Given the directed edges: A -> B, B -> C, and A -> C, which of the following is a valid topological order?

medium

A. [C, A, B]

B. [B, A, C]

C. [C, B, A]

D. [A, B, C]

Solution

1. Step 1: Analyze the edges and dependencies

   Node A points to B and C, so A must come before B and C. Node B points to C, so B must come before C.

2. Step 2: Check each option against these rules

   [A, B, C] respects A before B and C, and B before C. The other options violate these dependencies.

3. Final Answer:

   [A, B, C] -> Option D

4. Quick Check:

   Order respects edges = [A, B, C] [OK]

Hint: Place nodes before their dependents in order [OK]

Common Mistakes:

• Ignoring edge directions
• Placing dependent nodes before their prerequisites
• Choosing reverse order

4. Consider the following graph edges: 1 -> 2, 2 -> 3, 3 -> 1. What is the problem when trying to perform topological sorting on this graph?

medium

A. The graph has too many nodes

B. The graph is disconnected

C. The graph contains a cycle, so topological sorting is impossible

D. The graph is undirected

Solution

1. Step 1: Identify the cycle in the graph

   The edges form a cycle: 1 -> 2 -> 3 -> 1, which means the graph is not acyclic.

2. Step 2: Understand topological sorting requirements

   Topological sorting requires the graph to be acyclic. A cycle makes it impossible to order nodes without violating dependencies.

3. Final Answer:

   The graph contains a cycle, so topological sorting is impossible -> Option C

4. Quick Check:

   Cycle present = no topological sort [OK]

Hint: Cycles block topological sorting; check for cycles first [OK]

Common Mistakes:

• Ignoring cycles and trying to sort anyway
• Confusing disconnected graphs with cycles
• Assuming undirected graphs can be topologically sorted

5. You have tasks with dependencies: Task 1 before Task 3, Task 2 before Task 3, and Task 3 before Task 4. Which of the following is a valid topological order for these tasks?

hard

A. [3, 1, 2, 4]

B. [1, 2, 3, 4]

C. [4, 3, 2, 1]

D. [2, 1, 4, 3]

Solution

1. Step 1: List the dependencies clearly

   Task 1 and Task 2 must come before Task 3, and Task 3 must come before Task 4.

2. Step 2: Validate each option against dependencies

   [1, 2, 3, 4] respects all dependencies. The other options violate at least one dependency.

3. Final Answer:

   [1, 2, 3, 4] -> Option B

4. Quick Check:

   Order respects all dependencies = [1, 2, 3, 4] [OK]

Hint: Place all prerequisites before dependent tasks [OK]

Common Mistakes:

• Placing dependent tasks before prerequisites
• Ignoring order of multiple dependencies
• Assuming any order is valid