Dijkstra's algorithmShortest paths in edge-weighted digraph

Krasin Georgiev

Technical University of Sofiag.krasin at gmail com

Assistant Professor

Table of Contents1. Background2. The problem3. Properties and assumptions4. Applications5. Dijkstra's algorithm and

Pseudocode6. C# Demo7. Related problems and algorithms8. Resources

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Background Let G=(V,E) be a (di)graph. In directed graphs, edges areone-way

An edge-weighted graph is a graphwhere we associate weights or costs with each edge (or other attributes).

A shortest path from vertex s to vertex t is a directed path from s to t with the property that no other such path has a lower weight. 3

The ProblemSingle-Source Shortest Path

Problem

Find the shortest paths from a source vertex

to all other vertices in the graph

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Properties and Assumptions

Paths are directed. The weights are not necessarily distances (could be time or cost).

Edge weights are positive (or zero)

Shortest paths are not necessarily unique

Not all vertices need be reachable

Parallel edges may be present

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Algorithm ApplicationsApplication

Vertex Edge Solutions

Map Intersection

Road Find the shortest routeFind the fastest routeFind the best route

Network Router Connection Internet routingFlight Agenda

Airports Flights Find earliest time to reach destination

Epidemiology

Individuals Possible contacts

Model the spread of infectious diseases

Arbitrage Currency Exchange rate

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Algorithm Description

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Dijkstra's algorithm first initiates all vertex distances with preliminary values and puts all vertexes in a priority queue. Then picks the unvisited vertex with the lowest-distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited when done with neighbors.

Algorithm Trace Trough

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Algorithm Trace Trough

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Algorithm Pseudocodefunction Dijkstra(Graph, source): for each vertex v in Graph: // Initializations dist[v] := infinity ; // Unknown distance function from // source to v previous[v] := undefined ; // Previous node in optimal path end for // from source dist[source] := 0 ; // Distance from source to source Q := the set of all nodes in Graph ; // All nodes in the graph are put in a // Priority Queue Q while Q is not empty: // The main loop u := vertex in Q with smallest distance in dist[] ; // Start node in first case remove u from Q ; if dist[u] = infinity: break ; // all remaining vertices are end if // inaccessible from source for each neighbor v of u in Q: // where v has not yet been // removed from Q. alt := dist[u] + dist_between(u, v) ; if alt < dist[v]: // Relax (u,v,a) dist[v] := alt ; previous[v] := u ; decrease-key v in Q; // Reorder v in the Queue end if end for end whilereturn dist;

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C# Demo0

1

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3

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Modifications We can solve different problems using modified Dijkstra algorithm elements: Graph –vertices, edges and weights

meanings Distance – definition Priority Queue Relaxation and Distance

Initialization

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Some Theory Dijkstra algorithm is based on the following Lemmas: Shortest paths are composed of

shortest paths. It is based on the fact that if there was a shorter path than any sub-path, then the shorter path should replace that sub-path to make the whole path shorter.

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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Some Theory Analysis of Dijkstra’s Algorithm:

The initialization uses only O(n) time. Each vertex is processed exactly once. The inner loop is

called once for each edge in the graph. Each call of the inner loop does O(1) work plus, possibly, one Decrease-Key operation.

All of the priority queue operations require time

Finally

and we get time If unsorted sequence is used instead of priority queue

we get O(n2 + e) 14

Related algorithms Breadth-first search - special-case of

Dijkstra's algorithm on unweighted graphs when all edge costs are positive and identical. The priority queue degenerates into a FIFO queue.

Uniform-cost search - the shortest path to a particular node

The A* algorithm - generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target.

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Resources

http://algs4.cs.princeton.edu/44sp/ http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm

http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-046JFall-2005/CourseHome/

https://www.udacity.com/course/cs271

Nakov’s book: Programming = ++Algorithms;

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