Shortest Path Algorithms. Definitions Variants  Single-source shortest-paths problem: Given a graph, finding a shortest path from a given source.

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Presentation transcript:

Shortest Path Algorithms

Definitions

Variants  Single-source shortest-paths problem: Given a graph, finding a shortest path from a given source vertex to all the vertices in the graph (Our problem)  Single-destination shortest-paths problem: If you reverse the directions of all edges, you can solve it as a single-source problem.

Variants

 All-pairs shortest-path problems: Find a shortest part from each vertex to each vertex. Although this problem can be solved by running a single-source algorithm once from each vertex, it can usually be solved faster.

Structure of a Shortest Path  Subpaths of shortest paths are shortest paths. (WHY?)  Negative-weight edges?  Cycles?  Negative-weight cycles?

Representing Shortest Paths

Initialization

Relaxation

The Bellman-Ford Algorithm  The Bellman-Ford Algorithm solves the single- source shortest-path problems in which the edge weights may be negative.  It also alerts us if there is a negative-weight cycle that is reachable from the source.  If there is no such cycle, the algorithm produces the shortest paths and their weights.

Example and Implementation Issues

Correctness

Computational Complexity

Dijkstra’s Algorithm

Example

Implementation Issues

Approach

Correctness

Proof:

Computational Complexity