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- 1. DIJKSTRAS ALGORITHM,BYDR HJH RAHMAH BT MURSHIDI
- 2. INTRODUCTION Dijkstras algorithm, named after itsdiscoverer, Dutch computer scientist EdsgerDijkstra A greedy algorithm that solves the single-sourceshortest path problem for a directed graph withnon negative edge weights.
- 3. INTRODUCTION For example, if the vertices of the graphrepresent cities and edge weights representdriving distances between pairs of citiesconnected by a direct road, Dijkstras algorithmcan be used to find the shortest route betweentwo cities.
- 4. The input of the algorithm consists of a weighteddirected graph G and a source vertex s in G Denote V as the set of all vertices in the graph G. Each edge of the graph is an ordered pair ofvertices (u,v) This representings a connection from vertex u tovertex v
- 5. The set of all edges is denoted E Weights of edges are given by a weight functionw: E → [0, ∞) Therefore w(u,v) is the cost of moving directlyfrom vertex u to vertex v The cost of an edge can be thought of as (ageneralization of) the distance between thosetwo vertices
- 6. The cost of a path between two vertices is thesum of costs of the edges in that path For a given pair of vertices s and t in V, thealgorithm finds the path from s to t with lowestcost (i.e. the shortest path) It can also be used for finding costs of shortestpaths from a single vertex s to all other verticesin the graph.
- 7. BOXES AT EACH NODEOrder oflabellingLabel (i.ePermanentlabel)Workingvalues
- 8. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM FROM S TO TSABTDC3115864124Step 1Label start node Swith permanent label(P-label) of 0.1 04
- 9. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM431 0SABTDC3115864124Step 2For all nodes thatcan be reacheddirectly from S,assign temporarylabels (T-labels)equal to their directdistance from S64
- 10. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM42 31 0SABTDC3115864124Step 3Select the node withsmallest T label andmakes its labelpermanent. In thiscase the node is A.The P-labelrepresents theshortest distancefrom S to that node.Put the order oflabeling as 2.6
- 11. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM42 31 0SABTDC3115864124Step 4Consider all nodesthat can be reachedfrom A, that are Band T. Shortestroute from S to B viaA is 3+4 =7, but B isalready labelled as 6and it’s the best sofar. The shortestroute from S to T viaA is 3 + 11 = 14. PutT-label as 14 in T6414
- 12. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM3 42 31 0SABTDC3115864124Step 5Compare node T, Band C. The smallestT label is now 4 at C.Since this valuecannot be improved,it becomes P-label of4. Put the order oflabeling at C as 36414
- 13. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM3 42 31 0SABTDC3115864124Step 6Consider all nodes thatcan be reached from C,that are B and D.Shortest route from S to Bvia C is 4+1 =5 which isshorter than 6. Change Tlabel 6 to P label 5. Theshortest route from S to Dvia C is 4 + 4 = 8. Put T-label as 8 in D. CompareB and D. B is less than D,so the next node is B. Putthe order of labeling at Bas 44 54148
- 14. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM3 442 331 0SABTDC3115864124Step 7Consider all nodes thatcan be reached from B;that are D and T.Shortest route from S toD via B is 5+2 =7 whichis shorter than 8. ChangeT label 8 to P label 7. Theshortest route from S to Tvia B is 5 + 8 = 13. This issmaller than 14 sochange to 13 in T as Tlabel. Compare T and D.D is less than T so choseD as the next node. Putthe order of labeling as 5in D4 56,546 1314,135 78,7
- 15. TO FIND THE SHORTEST ROUTE BYDIJKSTRA’S ALGORITHM3 442 331 0SABTDC3115864124Step 8The last node is T. Put theorder of labeling as 6 in T.Compare the routes fromS to T via A (3 + 11 =14),via B ( 5 + 8 =13) and viaD (7 + 5 =12). It seemsthat the shortest routefrom S to T is via D.Change the T label in T(13) to P label with thevalue 12. Therefore theshortest way from S to Tis SCBDT which is 124 56,546 1214,13,125 78,7

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