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Is the following statement valid about shortest paths? Given a graph, suppose we have calculated shortest path from a source to all other vertices. If we modify the graph such that weights of all edges is becomes double of the original weight, then the shortest path remains same only the total weight of path changes.
- a)True
- b)False
Correct answer is option 'A'. Can you explain this answer?
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Is the following statement valid about shortest paths?Given a graph, s...
The shortest path remains same. It is like if we change unit of distance from meter to kilo meter, the shortest paths don't change.
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Is the following statement valid about shortest paths?Given a graph, s...
Shortest Paths in a Graph
- In graph theory, a shortest path is defined as the path between two vertices in a graph that has the minimum total weight or cost.
- The weight of a path is the sum of the weights of its constituent edges.
- There can be multiple shortest paths between two vertices in a graph, but all of them will have the same total weight.
Effects of Doubling Edge Weights
- Suppose we have calculated the shortest path from a source vertex to all other vertices in a given graph.
- If we modify the graph such that the weights of all edges become double of their original weights, the shortest path will remain the same, but the total weight of the path will change.
Explanation
- When we calculate the shortest path using algorithms like Dijkstra's algorithm or Bellman-Ford algorithm, the path is determined based on the relative weights of the edges.
- Doubling the weights of all edges does not change their relative weights, so the shortest path remains the same.
- However, since the weights are now doubled, the total weight of the path will also be doubled compared to the original graph.
Example
- Let's consider an example to illustrate this concept.
- Suppose we have a graph with three vertices A, B, and C, and the following edge weights:
- A to B: 3
- B to C: 4
- If we calculate the shortest path from A to C using Dijkstra's algorithm, the path will be A -> B -> C with a total weight of 7.
- Now, if we double the edge weights, the new weights will be:
- A to B: 6
- B to C: 8
- The shortest path from A to C will still be A -> B -> C, but the total weight of the path will now be 14.
Conclusion
- In summary, when we calculate the shortest path in a graph and then modify the graph by doubling the weights of all edges, the shortest path remains the same, but the total weight of the path changes.
- Therefore, the given statement is valid: the shortest path remains the same, only the total weight of the path changes.
- In graph theory, a shortest path is defined as the path between two vertices in a graph that has the minimum total weight or cost.
- The weight of a path is the sum of the weights of its constituent edges.
- There can be multiple shortest paths between two vertices in a graph, but all of them will have the same total weight.
Effects of Doubling Edge Weights
- Suppose we have calculated the shortest path from a source vertex to all other vertices in a given graph.
- If we modify the graph such that the weights of all edges become double of their original weights, the shortest path will remain the same, but the total weight of the path will change.
Explanation
- When we calculate the shortest path using algorithms like Dijkstra's algorithm or Bellman-Ford algorithm, the path is determined based on the relative weights of the edges.
- Doubling the weights of all edges does not change their relative weights, so the shortest path remains the same.
- However, since the weights are now doubled, the total weight of the path will also be doubled compared to the original graph.
Example
- Let's consider an example to illustrate this concept.
- Suppose we have a graph with three vertices A, B, and C, and the following edge weights:
- A to B: 3
- B to C: 4
- If we calculate the shortest path from A to C using Dijkstra's algorithm, the path will be A -> B -> C with a total weight of 7.
- Now, if we double the edge weights, the new weights will be:
- A to B: 6
- B to C: 8
- The shortest path from A to C will still be A -> B -> C, but the total weight of the path will now be 14.
Conclusion
- In summary, when we calculate the shortest path in a graph and then modify the graph by doubling the weights of all edges, the shortest path remains the same, but the total weight of the path changes.
- Therefore, the given statement is valid: the shortest path remains the same, only the total weight of the path changes.
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