1 / 16

Shortest Path Algorithms

Shortest Path Algorithms. Shortest Path Finding. Unweighted Shortest Path All-pairs Shortest-path Single-source Shortest-path Single-source-destination Shortest-path. Single-source Shortest Path. Bellman-Ford algorithm Dijkstra’s algorithm. Dijkstra’s Algorithm. Dijkstra(DG, source)

gavan
Télécharger la présentation

Shortest Path Algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Shortest Path Algorithms //pages.cpsc.ucalgary.ca/~verwaal/335

  2. Shortest Path Finding • Unweighted Shortest Path • All-pairs Shortest-path • Single-source Shortest-path • Single-source-destination Shortest-path //pages.cpsc.ucalgary.ca/~verwaal/335

  3. Single-source Shortest Path • Bellman-Ford algorithm • Dijkstra’s algorithm //pages.cpsc.ucalgary.ca/~verwaal/335

  4. Dijkstra’s Algorithm Dijkstra(DG, source) initialize(DG, source) verticesFound  queue.addAll(DG.vertices) while (queue not empty) v  queue.deleteMin() add v to verticesFound relaxEdges(v) //pages.cpsc.ucalgary.ca/~verwaal/335

  5. Example b c b c  null  null 1 1 10 10 2 a 2 4 4 Initialize 9 9 a 0 null 3 3 6 6 5 5 7 7 d e  null  null 2 2 d e queue: a(0), b(), c(), d(), e() verticesFound: {} //pages.cpsc.ucalgary.ca/~verwaal/335

  6. Example – 1st Iteration 10 a b c b c  null  null  null  null 1 1 10 10 10 a 2 a 2 4 1st Iter 4 9 9 0 null 0 null 3 3 6 6 5 5 5 7 7 5 a  null  null  null  null 2 2 d e d e 10 5 queue: a(0), b(), c(), d(), e() verticesFound: {} queue: b(), c(), d(), e() verticesFound: {a} //pages.cpsc.ucalgary.ca/~verwaal/335

  7. Example – 2nd Iteration 14 d 8 d b c b c 10 a  null 1 10 a  null 1 10 10 10 a 2 4 a 2 2nd Iter 4 9 9 0 null 9 3 3 0 null 3 6 5 6 5 5 7 7 7 d 5 a  null 2 2 5 a  null 2 d e d e 8 14 7 queue: d(5), b(10), c(), e() verticesFound: {a} queue: b(10), c(), e() verticesFound: {a, d} //pages.cpsc.ucalgary.ca/~verwaal/335

  8. Example – 3rd Iteration 13 e b c b c 8 d 14 c 8 d 14 d 1 1 10 10 a 2 a 2 4 3rd Iter 4 9 9 9 9 0 null 0 null 3 3 3 3 6 6 6 5 5 7 7 7 5 a 7 d 5 a 7 d 2 2 2 2 d e d e 13 queue: e(7), b(8), c(14) verticesFound: {a, d} queue: b(8), c(14) verticesFound: {a, d, e} //pages.cpsc.ucalgary.ca/~verwaal/335

  9. Example – 4th Iteration 9 b b c b c 8 d 13 e 8 d 13 e 1 1 1 10 10 a 2 a 2 2 4 4th Iter 4 9 9 9 9 0 null 0 null 3 3 3 3 6 6 6 6 5 5 7 7 7 7 5 a 7 d 5 a 7 d 2 2 2 2 d e d e 9 queue: b(8), c(13) verticesFound: {a, d, e} queue: c(13) verticesFound: {a, d, e, b} //pages.cpsc.ucalgary.ca/~verwaal/335

  10. Example – 5th Iteration b c b c 8 d 9 b 8 d 9 b 1 1 1 1 10 10 a 2 a 2 2 4 4 5th Iter 2 4 9 9 9 9 0 null 0 null 3 3 3 3 6 6 6 6 5 5 7 7 7 7 5 a 7 d 5 a 7 d 2 2 2 2 d e d e queue: c(9) verticesFound: {a, d, e, b} queue: verticesFound: {a, d, e, b, c} //pages.cpsc.ucalgary.ca/~verwaal/335

  11. Analysis of Dijkstra’s Algorithm • Loop executed |V| times • call removeMin in queue |V| times • decreaseKey in queue called |E| times • Priority Queue implementation: • Array – O(|V|2) • Binary Heap – O(|E|log|V|) • Fibonacci Heap – O(|V|log|V| + |E|) //pages.cpsc.ucalgary.ca/~verwaal/335

  12. All-pairs Shortest-path • Run Dijkstra’s algorithm on all vertices • Recursive Matrix Multiplication • Floyd-Warshall Algorithm //pages.cpsc.ucalgary.ca/~verwaal/335

  13. Recursive Solution • Considers length of shortest path • At most |V| - 1 • Path must contain sub path which is also shortest • The shortest path from u to v either • goes directly from u to v (path length 1), or • goes through some other vertex w //pages.cpsc.ucalgary.ca/~verwaal/335

  14. Divide and Conquer Algorithm APSP-Recursive(G, u, v) if (u = v) return 0 minPathLength  for each vertex w in G a  APSP-Recursive(G, u, w) if (a+weight(w,v)< minPathLength) minPath  a + weight(w,v) return minPathLength //pages.cpsc.ucalgary.ca/~verwaal/335

  15. Analysis of Divide and Conquer • Run time O(2n) • Duplicating work • Instead compute from the bottom-up • Dynamic Programming Algorithm • In table store shortest path considering each vertex as intermediate //pages.cpsc.ucalgary.ca/~verwaal/335

  16. Dynamic Programming Algorithm APSP-DP(G ) // G is adjacency matrix D  G for k  1 to |V| do for i  1 to |V| do for j  1 to |V| do if D[i,k] + D(k,j) < D[i,j] D[i,j] = D[i,k] + D(k,j) //pages.cpsc.ucalgary.ca/~verwaal/335

More Related