Dijkstra's algorithm

Dijkstra's algorithm Shortest paths in edge-weighted digraph KrasinGeorgiev Technical University of Sofia g.krasin at gmail com Assistant Professor

Table of Contents • Background • The problem • Properties and assumptions • Applications • Dijkstra'salgorithm and Pseudocode • C# Demo • Related problems and algorithms • Resources

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.

The Problem Single-Source Shortest Path Problem Find the shortest pathsfrom a source vertex to all other vertices in the graph

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

Algorithm Applications

Algorithm Description Dijkstra'salgorithm 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

Algorithm Pseudocode function 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 while return dist;

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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

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.

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 • Finallyand we get time • If unsorted sequence is used instead of priority queue we get O(n2+ e)

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.

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;

Dijkstra's algorithm http://algoacademy.telerik.com

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Dijkstra's algorithm