Lecture Notes on Dijkstra's Shortest Path Algorithm

Overview of Dijkstra's Algorithm

• Purpose: Finds the shortest path between one node and all other nodes in a weighted graph.
• Background:
  • Developed by Edsger Dijkstra.
  • It is a special case of a more general algorithm but developed first.
  • It operates similarly to a breadth-first search.
• Limitation: Does not work with negative weight edges.
• Alternative: Bellman-Ford algorithm addresses this limitation but is not covered here.

Applications

• GPS navigation.
• IP routing.
• Telephone networking.

Implementation Steps

1. Initialization:
   • Use a table or array to record distances.
   • Set initial distances from start node to all nodes as infinity, except the start node itself, which is zero.
2. Algorithm Execution:
   • Find the node with the shortest distance from the start that hasn't been visited.
   • Update distances for each connected node that hasn't been visited:
     • New distance = Current node's distance + Edge weight.
     • Update if this new distance is smaller.
   • Mark the node as visited.
   • Repeat until all nodes are visited.

Example Walkthrough

• Nodes: A through G.
• Steps:
  • Start with node A, calculate and update distances for nodes B, C, and D.
  • Continue with the nearest unvisited node, update connections and mark visited.
  • Repeat the process until all nodes are visited.

Finding the Shortest Path

• Follow the previous nodes back from the goal node to the start node to get the shortest path.
• Example path from A to G: A → D → F → G.

Practical Application Example

• Map Abstraction:
  • Nodes represent towns.
  • Edges represent routes.
  • Edge weightings represent distances.
• Algorithm Usage:
  • Identify shortest paths without considering actual geography.

Pseudo Code and Coding Aspects

• Pseudo Code:
  • Similar to Structured English.
  • Can be implemented in various programming languages.
• Infinity in Programming:
  • Use a very large number to represent infinity.
  • In Python, use float('inf').
  • Other languages might use the maximum possible integer value.

Conclusion

• Understand and trace Dijkstra's algorithm.
• Recognize that multiple shortest paths can exist.
• Importance of choosing appropriate data structures for implementation.

Additional Resources

• A dedicated book covering data structures and algorithms is available on Amazon.
• The book provides detailed explanations, applications, and code examples in Python, C, and Visual Basic.

Note: These notes are intended to provide a high-level understanding and tracing capability for Dijkstra's algorithm, useful for exams and practical application in software development.