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Understanding the Shortest Path Problem: Lecture Highlights on Bellman-Ford Algorithm

In this lecture from December 9, 2009, we explore the Shortest Path Problem in directed graphs, focusing on finding the shortest path from a source vertex 's' to a target vertex 't' using the Bellman-Ford algorithm. We discuss how to compute the shortest path costs when edges can have negative weights and no negative cycles are present. The lecture also covers recurrence relations and the consequences of using at most 'i' edges, providing essential insights for understanding how to derive optimal paths in graphs efficiently.

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Understanding the Shortest Path Problem: Lecture Highlights on Bellman-Ford Algorithm

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  1. Lecture 39 CSE 331 Dec 9, 2009

  2. Announcements Please fill in the online feedback form Sample final has been posted Graded HW 9 on Friday

  3. Shortest Path Problem Input: (Directed) Graph G=(V,E) and for every edge e has a cost ce (can be <0) t in V Output: Shortest path from every s to t Assume that G has no negative cycle 1 1 t s Shortest path has cost negative infinity 899 100 -1000

  4. Recurrence Relation OPT(i,v) = cost of shortest path from v to t with at most i edges OPT(i,v) = min {OPT(i-1,v), min(v,w) in E{cv,w + OPT(i-1, w)}} Path uses ≤ i-1 edges Best path through all neighbors

  5. Some consequences OPT(i,v) = shortest path from v to t with at most i edges OPT(i,v) = min {OPT(i-1,v), min(v,w) in E{cv,w + OPT(i-1, w)}} OPT(n-1,v) is shortest path cost between v and t Group talk time: How to compute the shortest path between s and t given all OPT(i,v) values

  6. Today’s agenda Finish Bellman-Ford algorithm Look at a related problem: longest path problem

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