Shortest path algorithm
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Dijkstra's algorithm is used to find the shortest paths from a source node to all other nodes in a network. It works by marking all nodes as tentative with initial distances from the source set to 0 and others to infinity. It then extracts the closest node, adds it to the shortest path tree, and relaxes distances of its neighbors. This process repeats until all nodes are processed. When applied to the example network, Dijkstra's algorithm finds the shortest path from node A to all others to be A-B=4, A-C=6, A-D=8, A-E=7, A-F=7, A-G=7, and A-H=9.
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Shortest path algorithm
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- 1. Shortest Path Problem • Given: network topology with link costs – c(x , y): link cost from node x to node y – Infinity if x and y are not direct neighbors • Compute: least-cost paths to all nodes – From a given source u to all other nodes. 3 2 1 1 4 1 4 5 3 2 A B C D E F G H
- 2. Subrata paul Roll No :- 13015411. Department Of Computer Science & Engineering , University Of Rajshahi.
- 3. What Is Routing ?? •A famous quotation from RFC 791 “A name indicates what we seek. An address indicates where it is. A route indicates how we get there.” -- Jon Postel • Routing is the process of selecting best paths.
- 4. Goals Of Routing Algorithm • We want any several properties of any routing schema. Property Meaning Correctness Finds paths that work Efficient Paths Use network bandwidth well Fair Paths Doesn’t starve any nodes Fast Convergence Recovers quickly after changes Salability Works well as network grows large
- 5. What Are The Best Paths ?? • Many Possibilities :- - Latency, avoid circuitous paths. - Bandwidth, avoid slow links . - Money, avoid expensive links . - Hops, reduce switching . A B C D E F G
- 6. Shortest Paths (1) • Find the shortest path from A E. • All links are bidirectional , with equal cost in both direction.
- 7. Shortest Path(2) • ABCE is a shortest path • Dist(ABCE) = 4 + 2 + 1 = 7 • This is less than : - Dist(ABE) = 8 - Dist(ABFE) = 9 - Dist(AE) = 10 - Dist(ABCDE) = 10
- 8. How To Compute ?? • Using Dijkstra’s Algorithm :- - Given a network topology . We are about to compute shortest path from a source to all other nodes of the network.
- 9. Dijkstra’s Algorithm • Algorithm :- - Mark all nodes tentative , set distances from source to 0(Zero) for source and ∞(infinity) for all other nodes - While tentative nodes remain : a ) Extract N , a node with lowest distance b) Add link to N to the shortest path tree C) Relax the distances of neighbors of N by lowering any better distance estimates
- 10. Dijkstra’s Algorithm A B C D E F G H 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞
- 11. Dijkstra’s Algorithm A B C D E F G H 0 4 ∞ ∞ 10 ∞ ∞ ∞
- 12. Dijkstra’s Algorithm A B C D E F G H 0 4 6 ∞ 8 7 7 ∞
- 13. Dijkstra’s Algorithm A B C D E F G H 0 4 6 8 7 7 7 9
- 14. Dijkstra’s Algorithm A B C D E F G H 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ A B C D E F G H 0 4 6 8 7 7 7 9
- 15. Dijkstra’s Algorithm A B C D E F G H 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ A B C D E F G H 0 4 6 8 7 7 7 9
- 16. Dijkstra’s Algorithm A B C D E F G H 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ A B C D E F G H 0 4 6 8 7 7 7 9
- 17. Dijkstra’s Algorithm A B C D E F G H 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ A B C D E F G H 0 4 6 8 7 7 7 9
- 18. Dijkstra’s Algorithm A B C D E F G H 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ A B C D E F G H 0 4 6 8 7 7 7 9 All nodes aredonenow.
- 19. Dijkstra’s Algorithm • So , the shortest path from source node A to all other nodes are – A – B = 4 A – C = 6 A – D = 8 A – E = 7 A – F = 7 A – G = 7 A – H = 9
- 20. Conclusion • Routing is a distributed algorithm – React to changes in the topology – Compute the shortest paths • Two main shortest-path algorithms – Dijkstra – Bellman-Ford • Convergence process – Changing from one topology to another – Transient periods of inconsistency across routers