CS223 Advanced Data Structures and Algorithms 1

Dijkstra’s Shortest Path Algorithm Dijkstra’s Shortest Path Algorithm

Neil TangNeil Tang03/25/200803/25/2008

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Class OverviewClass Overview

The shortest path problem

Applications

Dijkstra’s algorithm

Implementation and time complexities

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Shortest Path ProblemShortest Path Problem

Weighted path length (cost): The sum of the weights of all links on the path.

The single-source shortest path problem: Given a weighted graph G and a source vertex s, find the shortest (minimum cost) path from s to every other vertex in G.

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An ExampleAn Example

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Another ExampleAnother Example

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ApplicationsApplications

Digital map

Computer network

Travel planning

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Dijkstra’s AlgorithmDijkstra’s Algorithm

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Implementation and Time ComplexitiesImplementation and Time Complexities

Trivial: O(|V|2 + |E|) = O(|V|2)

Heap: deleteMin |V| times + decreaseKey |E| times

O(|V|log|V| + |E|log|V|) = O (|E|log|V|)

Fibonacci heap: O(|E| + |V|log|V|)