Let's walk through the process of applying Dijkstra's algorithm to find the shortest path from vertex $a$ to vertex $z$ in the given weighted graph. I'll also summarize the steps in a table.

Graph Details:

• Vertices: {a, b, c, d, e, z}
• Edges:
  • {a, b}, {b, c}, {a, d}, {d, e}, {d, c}, {c, e}, {c, z}, {e, z}
• Weights:
  • w({a, b}) = 2
  • w({b, c}) = 3
  • w({a, d}) = 1
  • w({d, e}) = 10
  • w({d, c}) = 5
  • w({c, e}) = 1
  • w({c, z}) = 8
  • w({e, z}) = 2

Dijkstra’s Algorithm:

The goal is to find the shortest path from $a$ to $z$, updating the distances to each vertex iteratively.

Step-by-Step Process:

1. Initialize the table:

   • Set the distance from the source $a$ to $a$ as 0, and the distance to all other vertices as infinity.
   • Start from vertex $a$, and iteratively explore all connected vertices.

2. Update the distances:

   • For each vertex, check its neighbors and update the distances if a shorter path is found.

3. Track visited nodes:

   • Mark nodes as "visited" once their shortest distance is found.

Table of Steps:

Step	Current Node	Neighbors to Check	Updated Distances	Notes
1	a	b, d	a = 0, b = 2, d = 1, c = ∞, e = ∞, z = ∞	Start from node a. Distances to b and d updated.
2	d	e, c	a = 0, b = 2, d = 1, c = 6, e = 11, z = ∞	Node d visited. Distances to e (10) and c (5) updated.
3	b	c	a = 0, b = 2, d = 1, c = 5, e = 11, z = ∞	Node b visited. Distance to c updated (5).
4	c	e, z	a = 0, b = 2, d = 1, c = 5, e = 6, z = 13	Node c visited. Distances to e (6) and z (13) updated.
5	e	z	a = 0, b = 2, d = 1, c = 5, e = 6, z = 8	Node e visited. Distance to z updated (8).
6	z		a = 0, b = 2, d = 1, c = 5, e = 6, z = 8	Node z visited. Algorithm ends.

Shortest Path:

The shortest path from $a$ to $z$ is: $a \to d \to c \to e \to z$ with a total distance of 8.

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Would you like to go over any of the steps in more detail, or have other questions about Dijkstra's algorithm? Here are some related questions you could explore:

1. How does Dijkstra's algorithm work with negative edge weights?
2. What are the limitations of Dijkstra's algorithm?
3. How does the time complexity of Dijkstra's algorithm change with different data structures?
4. How would you implement Dijkstra’s algorithm in Python?
5. How does the algorithm behave in graphs with cycles?

Tip: Always keep track of the nodes you have visited to avoid revisiting them, as it ensures you find the shortest path efficiently.