Dijkstra's algorithm in Data Structures Theory - Mini Project: Build & Apply

Dijkstra's Algorithm Step-by-Step

📖 Scenario: You are helping a delivery company find the shortest path between cities on a map. Each city is connected by roads with different distances. Your task is to use Dijkstra's algorithm to find the shortest distance from the starting city to all other cities.

🎯 Goal: Build a simple implementation of Dijkstra's algorithm that calculates the shortest distances from a starting city to all other cities in a given graph.

📋 What You'll Learn

Create a graph as a dictionary with cities as keys and their neighbors with distances as values

Set up a dictionary to hold the shortest distances initialized to infinity except the start city

Implement the core logic of Dijkstra's algorithm using a loop and updating distances

Complete the algorithm by returning the dictionary of shortest distances

💡 Why This Matters

🌍 Real World

Dijkstra's algorithm helps find the shortest routes in maps, GPS navigation, and network routing.

💼 Career

Understanding this algorithm is important for roles in software development, data science, and network engineering.

1

Create the graph data structure

Create a dictionary called graph representing the map with these exact entries: 'A': {'B': 5, 'C': 1}, 'B': {'A': 5, 'C': 2, 'D': 1}, 'C': {'A': 1, 'B': 2, 'D': 4, 'E': 8}, 'D': {'B': 1, 'C': 4, 'E': 3, 'F': 6}, 'E': {'C': 8, 'D': 3}, 'F': {'D': 6}

Data Structures Theory

# Create the graph dictionary with the exact city connections and distances
# Your code here

Hint

Use a dictionary where each city points to another dictionary of neighbors and distances.

2

Initialize distances dictionary

Create a dictionary called distances with all cities from graph as keys and set their values to float('inf'). Then set the distance for the start city 'A' to 0.

Data Structures Theory

graph = {
    'A': {'B': 5, 'C': 1},
    'B': {'A': 5, 'C': 2, 'D': 1},
    'C': {'A': 1, 'B': 2, 'D': 4, 'E': 8},
    'D': {'B': 1, 'C': 4, 'E': 3, 'F': 6},
    'E': {'C': 8, 'D': 3},
    'F': {'D': 6}
}
# Create distances dictionary with all cities set to float('inf')
# Set distances['A'] to 0
# Your code here

Hint

Use a dictionary comprehension to set all distances to infinity, then update the start city distance.

3

Implement the core Dijkstra's algorithm logic

Create a set called unvisited containing all cities from graph. Use a while loop that runs while unvisited is not empty. Inside the loop, find the city current_city in unvisited with the smallest distance in distances. Then, for each neighbor and distance in graph[current_city].items(), calculate a new distance and update distances[neighbor] if the new distance is smaller. Finally, remove current_city from unvisited.

Data Structures Theory

graph = {
    'A': {'B': 5, 'C': 1},
    'B': {'A': 5, 'C': 2, 'D': 1},
    'C': {'A': 1, 'B': 2, 'D': 4, 'E': 8},
    'D': {'B': 1, 'C': 4, 'E': 3, 'F': 6},
    'E': {'C': 8, 'D': 3},
    'F': {'D': 6}
}
distances = {city: float('inf') for city in graph}
distances['A'] = 0
# Create unvisited set with all cities
# Use while loop to process unvisited cities
# Find current_city with smallest distance
# Update distances for neighbors
# Remove current_city from unvisited
# Your code here

Hint

Use min() with a key function to find the city with the smallest distance. Update neighbors only if the new distance is smaller.

4

Return the shortest distances

Add a final line to return the distances dictionary after the while loop ends.

Data Structures Theory

graph = {
    'A': {'B': 5, 'C': 1},
    'B': {'A': 5, 'C': 2, 'D': 1},
    'C': {'A': 1, 'B': 2, 'D': 4, 'E': 8},
    'D': {'B': 1, 'C': 4, 'E': 3, 'F': 6},
    'E': {'C': 8, 'D': 3},
    'F': {'D': 6}
}
distances = {city: float('inf') for city in graph}
distances['A'] = 0
unvisited = set(graph.keys())
while unvisited:
    current_city = min(unvisited, key=lambda city: distances[city])
    for neighbor, distance in graph[current_city].items():
        new_distance = distances[current_city] + distance
        if new_distance < distances[neighbor]:
            distances[neighbor] = new_distance
    unvisited.remove(current_city)
# Return the distances dictionary
# Your code here

Hint

Simply add return distances after the loop to finish the algorithm.

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