Dijkstra's Shortest Path Algorithm (Cambridge International A-Level Computer Science)

Dijkstra’s shortest path algorithm

• In Computer Science, an optimisation problem involves finding the most efficient solution to a given problem

• This could mean minimising cost, time, or resource usage, or maximising output, efficiency, or value

• Examples of optimisation problems include:

  • Finding the shortest route between two locations (e.g. Google Maps)
  • Minimising resource usage in manufacturing or video games
  • Creating efficient timetables (e.g. for schools or offices)
  • Scheduling tasks and staff shifts to avoid conflicts or downtime

• One of the most common real-world examples is finding the shortest path from A to B, which is exactly what Dijkstra’s algorithm solves

What is Dijkstra’s shortest path algorithm?

• In A Level Computer Science, Dijkstra’s shortest path algorithm is a classic optimisation algorithm
• It calculates the shortest path from a starting node to all other nodes in a weighted graph
• It is often compared to breadth-first search, but Dijkstra’s includes edge weights and ensures the lowest total cost to reach each destination node

How It works:

• The graph is made up of nodes (vertices) and edges (arcs)

• Each edge has a weight, which can represent:

  • Time
  • Distance
  • Cost

• The algorithm explores all possible routes, keeping track of the shortest known distance to each node

• It guarantees the optimal path from the start node to every other node in the graph

• For revision on graphs, see section 18 - AI & Graphs

• An illustrated example of Dijkstra’s algorithm is shown below:

Performing Dijkstra's shortest path algorithm

• Set A path to 0 and all other path weights to infinity

• Visit A. Update all neighbouring nodes path weights to the distance from A + A's path weight (0)

• Choose the next node to visit that has the lowest path weight (D)
• Visit D
• Update all neighbouring, non-visited nodes with a new path weight if the old path weight is bigger than the new path weight
• C is updated from 5 to 3 as the new path is shorted (A > D > C)

• Choose the next node to visit that has the lowest path weight (E)
• Visit E
• Update all neighbouring, non-visited nodes from E with a new path weight
• F is updated from 7 to 5 as the new path is shorter (A > D > E > F)

• Choose the next node to visit, alphabetically (B)
• Visit B
• Update all neighbouring non-visited nodes from B with a new weight if the path is less than the current path
• C is not updated as A > B > C is 4

• Choose the next lowest node and visit (C)
• Update C's neighbours if the new path is shorter than the old path
• A > D > C > G is 8, so G is not updated

• Choose the next lowest node and visit (F)
• Update F's neighbours if the new path is shorter than the old path
• F has no non-visited neighbours so no updates are made

• Choose the next lowest node and visit (G)
• G has no non-visited neighbours so no updates are made
• All nodes have been visited
• Back tracking from G gives the following order and total distance A-D-E-G (5)

Figure 1: Performing Dijkstra’s Shortest path Algorithm

Programming Dijkstra's shortest path algorithm

How do you write Dijkstra's shortest path algorithm?

Pseudocode

Assumptions:

• graph[node] returns a list of neighbour nodes
• weight(x, y) returns the edge weight between node x and y
• Lists like visited, unvisited, and path support ADD and REMOVE

Python

Example usage:

Output:

• A list of nodes representing the shortest path from 'A' to 'E'