The statement "One can compute shortest paths from a single source s to all vertices v faster than Bellman-Ford by re-weighting the edges to be non-negative and then running Dijkstra's algorithm" is False.
The statement "The path between any two vertices s and t in the minimum spanning tree of a graph G must be a shortest path from s to t in G" is False.
The statement "If the weight of each edge in the graph is increased by one, the shortest path from s to t will still be a shortest path" is True.
The statement is False. Although re-weighting the edges to be non-negative and running Dijkstra's algorithm is faster than the Bellman-Ford algorithm for finding shortest paths in graphs with non-negative edge weights, it does not hold for graphs with negative-weight edges.
The reason is that Dijkstra's algorithm relies on the property of selecting the smallest edge weight at each step, which may not work correctly in the presence of negative-weight edges.
The statement is False. While the minimum spanning tree of a graph connects all vertices with the minimum total edge weight, it does not guarantee that the path between any two vertices in the minimum spanning tree is the shortest path in the original graph.
The minimum spanning tree focuses on minimizing the total weight of the tree, not necessarily considering individual shortest paths between pairs of vertices.
The statement is True. If the weight of each edge in a graph is increased by one, the relative order of the edge weights remains the same. Therefore, the shortest path from a vertex s to another vertex t will still be the shortest path even after increasing the edge weights.
The increased weights simply shift the absolute values of the weights, but the relative differences between the weights remain unchanged, ensuring that the shortest path remains the same.
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