IEOR 6614, Spring 2007 : Homework 4

General Instructions

1. Please review the course information.
2. You must write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.
3. Numbered problems are all from the textbook Network Flows .

Problems

1. Problem 4.12. Telephone operator scheduling.
2. Suppose that you have a graph that is not strongly connected. Show that Johnson's algorithm for all-pairs shortest (Theorem 5.4 in AMO) might no longer be correct. Then give a modification that leads to a correct algorithm.
3. Problem 5.51 and 5.52. Scaling for shortest paths. It would be helpful to read Section 3.3.
4. Problem 5.18. Reoptimizing shortest paths.
5. We define a shortest path algorithm to be oblivious if given a graph G=(V,E), with vertices numbered 1..n, it decides to a execute a series of relax statements based only the values n = |V| and m = |E|, and not based on the particular structure of the graph. For example, Bellman-Ford algorithm is oblivious, but Dijkstra's algorithm is not.

   Prove the following statement: Any oblivious algorithm that correctly computes single source shortest paths for all graphs G, must have a worst-case running time of Ω(nm).

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• Homework 3
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