I need help solving the all-pairs shortest-path problem for the digraph with the weight matrix below.
I need help solving the all-pairs shortest-path problem for the digraph with the weight matrix below.
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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I need help solving the all-pairs shortest-path problem for the digraph with the weight matrix below.
![The image depicts a matrix commonly used in graph theory, particularly in representing the adjacency matrix or the weighted adjacency matrix of a graph. The matrix is as follows:
\[
\begin{bmatrix}
0 & 2 & \infty & 1 & 8 \\
6 & 0 & 3 & 2 & \infty \\
\infty & \infty & 0 & 4 & \infty \\
\infty & \infty & 2 & 0 & 3 \\
3 & \infty & \infty & \infty & 0
\end{bmatrix}
\]
### Explanation:
- Each element in the matrix represents the weight or cost of traversing from one node to another in a graph. The rows and columns correspond to nodes.
- A "0" on the diagonal indicates that the cost to reach a node from itself is zero, which is typical since there is no distance to travel.
- The symbol "\(\infty\)" indicates that there is no direct path between the nodes, meaning they might not be directly connected.
- Numbers other than zero on the non-diagonal elements represent the weights or distances between nodes, suggesting the existence of a direct path.
This matrix could be used in algorithms like Dijkstra's or Floyd-Warshall to find shortest paths in weighted graphs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a1cfcd3-5f91-45b5-9e6a-7b9caa54bed1%2Fa5db1af6-9e76-4cc2-82cf-c15433eb654a%2Fku2ie2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image depicts a matrix commonly used in graph theory, particularly in representing the adjacency matrix or the weighted adjacency matrix of a graph. The matrix is as follows:
\[
\begin{bmatrix}
0 & 2 & \infty & 1 & 8 \\
6 & 0 & 3 & 2 & \infty \\
\infty & \infty & 0 & 4 & \infty \\
\infty & \infty & 2 & 0 & 3 \\
3 & \infty & \infty & \infty & 0
\end{bmatrix}
\]
### Explanation:
- Each element in the matrix represents the weight or cost of traversing from one node to another in a graph. The rows and columns correspond to nodes.
- A "0" on the diagonal indicates that the cost to reach a node from itself is zero, which is typical since there is no distance to travel.
- The symbol "\(\infty\)" indicates that there is no direct path between the nodes, meaning they might not be directly connected.
- Numbers other than zero on the non-diagonal elements represent the weights or distances between nodes, suggesting the existence of a direct path.
This matrix could be used in algorithms like Dijkstra's or Floyd-Warshall to find shortest paths in weighted graphs.
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