data structures using example consider the graph given in fig taking as the initial nodeexecute the dijkstra' algorithm on it step set the label of and { step label of and thereforen {dfa step label of has been re-labelled because minimum ( has been re-labelled ( thereforen {dc fc step label of thereforen {dfcbf step label of and ( thereforen {dfcbgfigure graph step label of and thereforen {dfcbganote that we have no labels for node ethis means that is not reachable from only the nodes that are in are reachable from the running time of dijkstra' algorithm can be given as (| | +| |)= (| | where is the set of vertices and in the graph difference between dijkstra' algorithm and minimum spanning tree minimum spanning tree algorithm is used to traverse graph in the most efficient mannerbut dijkstra' algorithm calculates the distance from given vertex to every other vertex in the graph dijkstra' algorithm is very similar to prim' algorithm both the algorithms begin at specific node and extend outward within the graphuntil all other nodes in the graph have been reached the point where these algorithms differ is that while prim' algorithm stores minimum cost edgedijkstra' algorithm stores the total cost from source node to the current node moreoverdijkstra' algorithm is used to store the summation of minimum cost edgeswhile prim' algorithm stores at most one minimum cost edge warshall' algorithm if graph is given as =(ve)where is the set of vertices and is the set of edgesthe path matrix of can be found asp an this is lengthy processso warshall has given very efficient algorithm to calculate the path matrix warshall' algorithm defines matrices [if there is path from vi to vj opn as given in fig the path should not use any pk[ ][jthis means that if [ ][ then there other nodes except vkexists an edge from node vi to vj [otherwiseif [ ][ then there exists an edge from vi to vj that does not use any other vertex figure path matrix entry except if [ ][ then there exists an edge from vi to vj that does not use any other vertex except and note that is equal to the adjacency matrix of if contains nodesthen pn which is the path matrix of the graph from the above discussionwe can conclude that pk[ ][jis equal to only when either of the two following cases occurthere is path from vi to vj that does not use any other node except vk- thereforepk- [ ][