Question

Computer Science

Posted 5 months ago

```
We are given a directed graph G = (V, E) with edge weights w(e) for e E E. The edge weights
are allowed to be negative. Let - C be the minimum value of the edge weights. Create a new set
of edge weights w'(e) = w(e) + C. Thus these new weights are non-negative: w'(e) 0 for all
e E E. Run Dijkstra's algorithm from a specified start vertex S E V on G with these new weights
w'.
For
every graph G, every weights W, and every S E V, the above algorithm is guaranteed to find
the shortest paths in G (not the actual lengths but the paths) from S with respect to the original
weights W:
True
False
If you entered true, provide a short explanation. If you entered false provide a counterexample
(show the graph and what the algorithm produces VS. the correct solution).
```

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Answer from Sia

Posted 5 months ago

False

Key Concept

Edge Weight Modification and Dijkstra's Algorithm

Explanation

Dijkstra's algorithm assumes that all edge weights are non-negative to guarantee the shortest path. If you simply add a constant C to all edge weights to make them non-negative, the shortest paths with respect to the original weights may not be preserved. This is because adding the same value to all edges does not maintain the relative differences in path lengths unless all paths contain the same number of edges. A counterexample can be constructed where a path with more edges becomes shorter than a path with fewer edges after adding C, which would not be the case with the original weights.

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