Here are some recurrence relations that come up in the analysis of Quicksort and or Select. For each recurrence, write one sentence explaining how it relates to Quicksort or Select and give its asymptotic growth rate in Q notation.

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This problem is about the Floyd-Warshall all pairs shortest paths algorithm

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This is a problem about Network Flows.

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Give an example of a weighted directed graph with at most 5 vertices such that Dijkstra's algorithm will NOT give the correct results for the shortest path lengths from source s to every other vertex. algorithm

Give an example of a weighted directed graph with at most 5 vertices such that Dijkstra's algorithm will NOT give the correct results for the shortest path lengths from source s to every other vertex. Your graph may have negative edge weights but NO negative weight cycles. Indicate what answer Dijkstra's algorithm would give and what the correct answer should be.

Here is a proposed algorithm to solve the single source shortest paths problem in a weighted directed graph G with possibly negative edges weights

Here is a proposed algorithm to solve the single source shortest paths problem in a weighted directed graph G with possibly negative edges weights, but no negative weight cycles: Form the graph G' from G by adding the same positive number, p, to all of the edge weights in the graph. This positive number should be chosen so that all the edge weights in G' are positive. (p = 1 + | min edge weight in G|). Run Dijkstra's algorithm on G'. Is it true that the parent array produced by Dijkstra's algorithm on G' will also give the shortest paths in the original graph, G? Justify your answer with a counterexample or a sketch of a correctness proof.