Math Problem Statement

In Exercises 8-9, use Dijkstra’s algorithm to find the shortest path from vertex a to vertex z in the given weighted graph. Make a table to summarize each step, as we did in class. 8. Vertex set {a, b, c, d, e, z}, Edge set {{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

Solution

Let's walk through the process of applying Dijkstra's algorithm to find the shortest path from vertex aa to vertex zz 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 aa to zz, updating the distances to each vertex iteratively.

Step-by-Step Process:

  1. Initialize the table:

    • Set the distance from the source aa to aa as 0, and the distance to all other vertices as infinity.
    • Start from vertex aa, 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:

StepCurrent NodeNeighbors to CheckUpdated DistancesNotes
1ab, da = 0, b = 2, d = 1, c = ∞, e = ∞, z = ∞Start from node a. Distances to b and d updated.
2de, ca = 0, b = 2, d = 1, c = 6, e = 11, z = ∞Node d visited. Distances to e (10) and c (5) updated.
3bca = 0, b = 2, d = 1, c = 5, e = 11, z = ∞Node b visited. Distance to c updated (5).
4ce, za = 0, b = 2, d = 1, c = 5, e = 6, z = 13Node c visited. Distances to e (6) and z (13) updated.
5eza = 0, b = 2, d = 1, c = 5, e = 6, z = 8Node e visited. Distance to z updated (8).
6za = 0, b = 2, d = 1, c = 5, e = 6, z = 8Node z visited. Algorithm ends.

Shortest Path:

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


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.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Dijkstra's Algorithm
Shortest Path Algorithms

Formulas

Dijkstra's algorithm formula (for updating shortest paths)

Theorems

Shortest Path Theorem (used in Dijkstra’s Algorithm)

Suitable Grade Level

Grades 10-12