PROOFS Obtain full marks by completing two of the following three tasks: (a) Formulate the Floyd-Warshall...

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PROOFS Obtain full marks by completing two of the following three tasks: (a) Formulate the Floyd-Warshall algorithm and prove its correctness. The proof should include the definition of the height k of a path from a node u, to node u,, the lemma about Dk [i, j] and its proof and how the correctness of the Floyd-Warshall algorithm follows from the lemma. (b) Vertices of a directed acyclic graph (DAG) can be sorted such that all directed edges point to the right. Describe the algorithm Topological Sorting which achieves that and prove that this algorithm works. (c) Formulate and prove the Max-Flow-Min-Cut-Theorem. PROOFS Obtain full marks by completing two of the following three tasks: (a) Formulate the Floyd-Warshall algorithm and prove its correctness. The proof should include the definition of the height k of a path from a node u, to node u,, the lemma about Dk [i, j] and its proof and how the correctness of the Floyd-Warshall algorithm follows from the lemma. (b) Vertices of a directed acyclic graph (DAG) can be sorted such that all directed edges point to the right. Describe the algorithm Topological Sorting which achieves that and prove that this algorithm works. (c) Formulate and prove the Max-Flow-Min-Cut-Theorem.