(1) Given a weighted directed graph G = (V, E, w), where V = {1, 2, 3, 4}, E = {(1,3), (2,1), (2,4), (3,2), (4,1), (4,3)}, and w(1,3) = −2, w(2,1) = 1, w(2,4) = 2, w(3,2) = 4, w(4,1) = −3, w(4,3) = 3. (a) Represent the graph G graphically; (b) Run SLOW-ALL-PAIRS-SHORTEST-PATHS on the above graph and show the matrices L(k) that result for each iteration of the loop. (2) Discuss how to use the Floyd-Warshall algorithm to detect a negative-weight cycle. (3)  The essence of Johnson’s algorithm is re-weighting so that a transformed graph has no negative weight edges, which enables the use of Dijkstra’s algorithm. Let us modify Johnson’s algorithm such that G' = G and s be any vertex.   (a) Give a counter example (i.e. a simple weighted and directed graph) to show this modification is incorrect assuming ∞ − ∞ is undefined (b) Show (using logical arguments) this modification produces the correct results when a given G is strongly connected.

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