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7) Use Dijkstra's algorithm to find the length of the shortest path between vertices a and z. Show each step of the calculations. 4 7 4 2 3 4 3. 10

7) Use Dijkstra's algorithm to find the length of the shortest path between vertices a and z. Show each step of the calculations. 4 7 4 2 3 4 3. 10

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Dijkstra's Algorithm to Find the Shortest Path** **Problem Statement:** Use Dijkstra's algorithm to determine the shortest path length between vertices \( a \) and \( z \). Show each step of the calculations. **Diagram Description:** The diagram is a graph of vertices and weighted edges. The vertices, labeled \( a, b, c, d, e, f, g, h, i, \) and \( z \), are connected as follows: - \( a \) to \( b \) with a weight of 4 - \( a \) to \( c \) with a weight of 3 - \( b \) to \( d \) with a weight of 5 - \( b \) to \( e \) with a weight of 2 - \( c \) to \( d \) with a weight of 3 - \( c \) to \( g \) with a weight of 6 - \( d \) to \( e \) with a weight of 1 - \( d \) to \( f \) with a weight of 9 - \( d \) to \( g \) with a weight of 5 - \( e \) to \( f \) with a weight of 3 - \( f \) to \( z \) with a weight of 4 - \( g \) to \( i \) with a weight of 8 - \( h \) to \( i \) with a weight of 3 - \( i \) to \( z \) with a weight of 5 **Note:** Write the vertex set \( S \) and the current shortest path at each step. **Solution Steps:** 1. **Initialization:** - Start at vertex \( a \). - Set \(\text{distance}(a) = 0\) and all other vertices to infinity. - Current vertex set \( S = \{a\} \). 2. **Explore neighbors of \( a \):** - Update distances: \(\text{distance}(b) = 4\), \(\text{distance}(c) = 3\). - Choose the vertex with the smallest distance not in \( S \), which is \( c \). 3. **Add \( c \) to \( S \) and explore neighbors:** - Current
Transcribed Image Text:**Dijkstra's Algorithm to Find the Shortest Path** **Problem Statement:** Use Dijkstra's algorithm to determine the shortest path length between vertices \( a \) and \( z \). Show each step of the calculations. **Diagram Description:** The diagram is a graph of vertices and weighted edges. The vertices, labeled \( a, b, c, d, e, f, g, h, i, \) and \( z \), are connected as follows: - \( a \) to \( b \) with a weight of 4 - \( a \) to \( c \) with a weight of 3 - \( b \) to \( d \) with a weight of 5 - \( b \) to \( e \) with a weight of 2 - \( c \) to \( d \) with a weight of 3 - \( c \) to \( g \) with a weight of 6 - \( d \) to \( e \) with a weight of 1 - \( d \) to \( f \) with a weight of 9 - \( d \) to \( g \) with a weight of 5 - \( e \) to \( f \) with a weight of 3 - \( f \) to \( z \) with a weight of 4 - \( g \) to \( i \) with a weight of 8 - \( h \) to \( i \) with a weight of 3 - \( i \) to \( z \) with a weight of 5 **Note:** Write the vertex set \( S \) and the current shortest path at each step. **Solution Steps:** 1. **Initialization:** - Start at vertex \( a \). - Set \(\text{distance}(a) = 0\) and all other vertices to infinity. - Current vertex set \( S = \{a\} \). 2. **Explore neighbors of \( a \):** - Update distances: \(\text{distance}(b) = 4\), \(\text{distance}(c) = 3\). - Choose the vertex with the smallest distance not in \( S \), which is \( c \). 3. **Add \( c \) to \( S \) and explore neighbors:** - Current
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