Dijkstra’s, Kruskals and Floyd-Warshall Algorithms

8/12/2019 Dijkstras, Kruskals and Floyd-Warshall Algorithms

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Dijkstras, Kruskals and Floyd-

Warshall Algorithms


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Single-Source Shortest Path Problem

Single-Source Shortest Path Problem - Theproblem of finding shortest paths from a sourcevertex vto all other vertices in the graph.


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Dijkstra's algorithm

Dijkstra's algorithm -is a solution to the single-source

shortest path problem in graph theory.

Works on both directed and undirected graphs.However, all edges must have nonnegative weights.

Approach:Greedy

Input:Weighted graph G={E,V} and source vertex vV,such that all edge weights are nonnegative

Output:Lengths of shortest paths (or the shortestpaths themselves) from a given source vertexvV toall other vertices


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Dijkstra's algorithm - Pseudocode

dist[s]0 (distance to source vertex is zero)for allv V{s}

do dist[v] (set all other distances to infinity)S (S, the set of visited vertices is initially empty)QV (Q, the queue initially contains all

vertices)while Q (while the queue is not empty)do umindistance(Q,dist) (select the element of Q with the min. distance)

SS{u} (add u to list of visited vertices)for allv neighbors[u]

do if dist[v] > dist[u] + w(u, v) (if new shortest path found)

then d[v]

d[u] + w(u, v) (set new value of shortest path)(if desired, add traceback code)return dist


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Dijkstra Animated Example


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Kruskal Algorithm

Minimum Spanning Tree

Disjoint Sets


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Sort Edges

(in reality they are placed in a priority

queue - not sorted - but sorting them

makes the algorithm easier to visualize)


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Minimum Spanning Tree Complete Graph


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FloydWarshall algorithm


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Thank You

Dijkstra’s, Kruskals and Floyd-Warshall Algorithms

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