Dijkstra's algorithm in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the purpose of Dijkstra's algorithm

What is the main goal of Dijkstra's algorithm in graph theory?

A To find the shortest path from a starting node to all other nodes in a graph with non-negative edge weights.

B To find the longest path between two nodes in any graph.

C To detect cycles in a directed graph.

D To sort the nodes of a graph in topological order.

💡 Hint

Think about what kind of paths Dijkstra's algorithm helps to find and what restrictions it has on edge weights.

Dijkstra's algorithm is designed to find the shortest paths from a single starting node to all other nodes in a graph where all edge weights are zero or positive. It does not work correctly with negative edge weights.

📋 Factual

intermediate

Dijkstra's algorithm and negative edge weights

What happens if Dijkstra's algorithm is applied to a graph that contains negative edge weights?

A It will run infinitely without stopping.

B It will always find the correct shortest paths regardless of edge weights.

C It will detect the negative edges and remove them automatically.

D It may produce incorrect shortest path results because it assumes all edge weights are non-negative.

💡 Hint

Consider the assumptions Dijkstra's algorithm makes about edge weights and how negative weights affect path calculations.

Dijkstra's algorithm assumes all edge weights are non-negative. If negative weights exist, the algorithm can incorrectly calculate shortest paths because it does not revisit nodes once visited, missing shorter paths through negative edges.

🚀 Application

advanced

Output of Dijkstra's algorithm on a weighted graph

Given the following weighted graph edges:

A-B: 4, A-C: 2, B-C: 5, B-D: 10, C-D: 3, D-E: 1, E-F: 2, F-D: 6

What is the shortest distance from node A to node E using Dijkstra's algorithm?

A 6

B 9

C 8

D 7

💡 Hint

Trace the shortest path step-by-step from A to E considering the edge weights.

The shortest path from A to E is A -> C -> D -> E with distances 2 + 3 + 1 = 6. However, the total distance is 6, but option A is 7, so let's re-check carefully.

Paths:

• A-C (2), C-D (3), D-E (1) = 6
• A-B (4), B-D (10), D-E (1) = 15
• A-C (2), C-B (5), B-D (10), D-E (1) = 18

So the shortest distance is 6, which matches option A, but option A is 6, option A is 7. So correct answer is A.

🔍 Analysis

advanced

Time complexity of Dijkstra's algorithm

Which of the following best describes the time complexity of Dijkstra's algorithm when implemented with a binary heap priority queue on a graph with V vertices and E edges?

A O(V + E log V)

B O(V^2)

C O(E + V log V)

D O(V log E)

💡 Hint

Consider how the priority queue operations affect the complexity and how many edges and vertices are processed.

Using a binary heap, Dijkstra's algorithm runs in O((V + E) log V) time because each vertex is inserted into the priority queue once and each edge is relaxed once.

❓ Reasoning

expert

Choosing the correct data structure for Dijkstra's algorithm

Why is a priority queue (min-heap) preferred over a simple queue when implementing Dijkstra's algorithm?

A Because it allows random access to nodes, speeding up the algorithm.

B Because it always selects the next node with the smallest tentative distance, ensuring correct shortest path calculation.

C Because it stores nodes in the order they were discovered, which is necessary for breadth-first search.

D Because it automatically removes cycles from the graph.

💡 Hint

Think about how Dijkstra's algorithm decides which node to explore next.

Dijkstra's algorithm relies on always choosing the node with the smallest current distance to ensure the shortest paths are found. A priority queue efficiently supports this by quickly retrieving the minimum-distance node.

Practice

1. What is the main purpose of Dijkstra's algorithm in graph theory?

easy

A. To detect cycles in a graph

B. To sort nodes in alphabetical order

C. To find the maximum flow in a network

D. To find the shortest path from one node to all other nodes in a weighted graph

Solution

1. Step 1: Understand the goal of Dijkstra's algorithm

   Dijkstra's algorithm is designed to find the shortest distances from a starting node to all other nodes in a graph where edges have weights.

2. Step 2: Compare with other graph algorithms

   Sorting nodes or detecting cycles are different tasks handled by other algorithms, not Dijkstra's.

3. Final Answer:

   To find the shortest path from one node to all other nodes in a weighted graph -> Option D

4. Quick Check:

   Shortest path finding = To find the shortest path from one node to all other nodes in a weighted graph [OK]

Hint: Remember: Dijkstra finds shortest paths from one node [OK]

Common Mistakes:

• Confusing shortest path with sorting or cycle detection
• Thinking it finds maximum flow
• Assuming it works on unweighted graphs only

2. Which of the following is the correct step in Dijkstra's algorithm?

easy

A. Select the unvisited node with the smallest tentative distance

B. Select the visited node with the largest distance

C. Update distances by adding the maximum edge weight

D. Remove nodes with the highest degree first

Solution

1. Step 1: Identify the selection criteria in Dijkstra's algorithm

   The algorithm always picks the unvisited node with the smallest tentative distance to explore next.

2. Step 2: Eliminate incorrect options

   Choosing visited nodes or using maximum edge weights contradicts the algorithm's logic.

3. Final Answer:

   Select the unvisited node with the smallest tentative distance -> Option A

4. Quick Check:

   Pick closest unvisited node = Select the unvisited node with the smallest tentative distance [OK]

Hint: Always pick closest unvisited node next [OK]

Common Mistakes:

• Picking visited nodes again
• Adding maximum instead of minimum edge weights
• Removing nodes based on degree

3. Consider the graph with nodes A, B, C and edges: A-B (4), A-C (2), C-B (1). Starting from A, what is the shortest distance to B using Dijkstra's algorithm?

medium

A. 3

B. 4

C. 2

D. 1

Solution

1. Step 1: Initialize distances from A

   Distance to A is 0, to B is infinity, to C is infinity initially.

2. Step 2: Update distances via neighbors

   From A, distance to C is 2, to B is 4. Next, from C, distance to B is 2 + 1 = 3, which is less than 4, so update B's distance to 3.

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Shortest path A->C->B = 3 [OK]

Hint: Check all paths, pick minimum sum of edges [OK]

Common Mistakes:

• Ignoring the path through C
• Choosing direct edge without comparison
• Mixing up edge weights

4. In the following pseudocode for Dijkstra's algorithm, what is the error?

1. Initialize distances to infinity
2. Set distance to start node as 0
3. While unvisited nodes remain:
4.    Pick any unvisited node
5.    Update distances to neighbors
6.    Mark node as visited

medium

A. Not initializing distances to zero

B. Picking any unvisited node instead of the one with smallest distance

C. Marking nodes as visited before updating distances

D. Setting start node distance to infinity

Solution

1. Step 1: Identify the node selection step

   The algorithm must pick the unvisited node with the smallest tentative distance, not just any node.

2. Step 2: Understand the impact of wrong selection

   Picking any node can lead to incorrect shortest paths because it ignores the priority of closer nodes.

3. Final Answer:

   Picking any unvisited node instead of the one with smallest distance -> Option B

4. Quick Check:

   Must pick closest unvisited node [OK]

Hint: Always pick node with smallest tentative distance [OK]

Common Mistakes:

• Choosing nodes randomly
• Setting wrong initial distances
• Marking nodes visited too early

5. You have a weighted graph with some edges having zero weight. How does Dijkstra's algorithm handle zero-weight edges when finding shortest paths?

hard

A. It treats zero-weight edges as infinite weight edges

B. It ignores zero-weight edges completely

C. It treats zero-weight edges like any other edge and updates distances accordingly

D. It causes the algorithm to fail or loop infinitely

Solution

1. Step 1: Understand edge weights in Dijkstra's algorithm

   Dijkstra's algorithm works with non-negative weights, including zero, treating them as valid edges.

2. Step 2: Effect of zero-weight edges

   Zero-weight edges allow the algorithm to update distances without adding cost, effectively moving to connected nodes at no extra distance.

3. Final Answer:

   It treats zero-weight edges like any other edge and updates distances accordingly -> Option C

4. Quick Check:

   Zero weight edges are valid and processed [OK]

Hint: Zero weight edges count as normal edges with no added cost [OK]

Common Mistakes:

• Ignoring zero-weight edges thinking they cause errors
• Assuming zero weight means infinite cost
• Believing algorithm loops infinitely with zero weights