To solve the given problem using Dijkstra's algorithm and find the shortest path between nodes $u$ and $v$:

Steps of Dijkstra's Algorithm:

1. Initialize:

   • Set the distance from $u$ to itself as 0.
   • Set the distance from $u$ to all other vertices as infinity ($\infty$).
   • Use a priority queue (or similar structure) to track the next closest vertex to visit.

2. Relaxation:

   • Pick the vertex with the smallest tentative distance (starting with $u$).
   • Update the distances to its neighbors if a shorter path is found.
   • Repeat for all unvisited vertices.

3. Termination:

   • Stop when the target vertex $v$ has been processed.

Applying the Algorithm:

Step-by-Step Solution:

Graph Representation:

• Vertices: $u, v, w, x, y, z$ (as per the diagram).
• Edge weights as shown.

1. Initialization:

   • Distances: $d[u] = 0$, $d[v] = \infty$, $d[w] = \infty$, $d[x] = \infty$, $d[y] = \infty$, $d[z] = \infty$.
   • Unvisited: $\{ u, v, w, x, y, z \}$.

2. Iteration:

   From $u$:

   • Visit $u$: $d[u] = 0$.
   • Update neighbors:
     • $w: d[w] = \min(\infty, 0 + 6) = 6$,
     • $x: d[x] = \min(\infty, 0 + 5) = 5$.
   • Unvisited: $\{ v, w, x, y, z \}$.

   Next Closest:

   • Visit $x$: $d[x] = 5$.
   • Update neighbors:
     • $w: d[w] = \min(6, 5 + 4) = 6$ (no update),
     • $y: d[y] = \min(\infty, 5 + 1) = 6$.
   • Unvisited: $\{ v, w, y, z \}$.

   Next Closest:

   • Visit $w$: $d[w] = 6$.
   • Update neighbors:
     • $z: d[z] = \min(\infty, 6 + 10) = 16$,
     • $y: d[y] = \min(6, 6 + 3) = 6$ (no update).
   • Unvisited: $\{ v, y, z \}$.

   Next Closest:

   • Visit $y$: $d[y] = 6$.
   • Update neighbors:
     • $v: d[v] = \min(\infty, 6 + 7) = 13$,
     • $z: d[z] = \min(16, 6 + 5) = 11$.
   • Unvisited: $\{ v, z \}$.

   Next Closest:

   • Visit $z$: $d[z] = 11$.
   • Update neighbors:
     • $v: d[v] = \min(13, 11 + 2) = 13$.
   • Unvisited: $\{ v \}$.

   Final Visit:

   • Visit $v$: $d[v] = 13$.

3. Shortest Path: The shortest path distance from $u$ to $v$ is 13.

Backtracking the Path:

1. Start at $v$ (distance = 13).
2. From $z$ to $v$: cost $2$ (path includes $z$).
3. From $y$ to $z$: cost $5$ (path includes $y$).
4. From $x$ to $y$: cost $1$ (path includes $x$).
5. From $u$ to $x$: cost $5$.

Path: $u \to x \to y \to z \to v$.

Would you like a detailed explanation of any step, or do you need a graphical illustration of this process?

Related Questions:

1. How is Dijkstra's algorithm different from Bellman-Ford?
2. Can this graph be solved with a different algorithm like A*?
3. How would you modify this process if negative weights existed?
4. How does the priority queue improve the efficiency of Dijkstra's algorithm?
5. How would the algorithm change if the goal was to find all-pairs shortest paths?

Tip:

When solving with Dijkstra's algorithm, always double-check updates to ensure you account for the lowest-cost path for each step!