1. Analysis of Algorithms Uri Zwick March 2014 Dynamic All-Pairs Shortest-Paths 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A

2. Demetrescu-Italiano (2004) Locally Shortest Paths (LSPs) shortest path A LSP is not necessarily a SPs  is a LSP iff l [  ] and r[  ] are SP

3. Shortest Path Extensions shortest path  L[  ] - Left SP extensions R[  ] - Right SP extensions

4. Run Dijkstra “in parallel” from all vertices Consider only LSPs Keep left and right extensions of SPs found Keep candidate SPs in a priority queue (Static version) Demetrescu-Italiano (2004)

5. Combining two paths

6. (Static version) Demetrescu-Italiano (2004) New shortest path 

7. Static APSP algorithm Demetrescu-Italiano (2004) Running time: Uniqueness assumption: All shortest paths are unique

8. SP(u,v) denotes the shortest path from u to v (Not explicitly maintained) p[u,v] – The second vertex on the shortest path from u to v found so far q[u,v] – The penultimate vertex on the shortest path from u to v found so far L[u,v] – A list of vertices w for which w  SP(u,v) is known to be a shortest path R[u,v] – A list of vertices w for which SP(u,v)  w is known to be a shortest path dist[u,v] – The length of the shortest path from u to v found so far

9. Static APSP algorithm

11. Dynamic algorithms An incremental algorithm only allows insertions An decremental algorithm only allows deletions A fully-dynamic algorithm allows both insertion and deletions

12. Example: Many LSPs Every path is an LSP – n 3 paths Two complete layers Can assign weights so that all SPs are unique n

13. Blue edges are significantly lighter than black edges Example: Many LSP changes When red edge inserted, all previous LSPs destroyed When red edge deleted, all previous LSPs recreated 0

14. A path  is historic, if it was a shortest path at some point of time after the last update of one of its vertices Historical Paths (HPs) Let  be a path at time t Let t’ be the time of the last update, before t, on  A path  is historic at time t if and only if there exists t’  t’’  t such that  is a SP at time t’’ A path  stops being historic only as a result of an update on it

15. Demetrescu-Italiano (2004) Locally Historical Paths (LHPs) historical path  l[  ] and r[  ] are not necessarily shortest at the same time  is a LHP iff l[  ] and r[  ] are HP

16. An empty vertex path

17. Converting an edge into a path

18. Combining two paths

19. Dynamic-APSP initialization

20. Insert and delete edges

21. Build-Paths

22. New-Shortest-Path

23. Examine a path

24. Removing a path

25. Update operations

26. A “full update” operation

27. Example: Many HPs and LSPs n Complete n n+1 2n2n 1 2 n

28. Phase 1: Decrease edge weights n Complete n n1n1 1 1 2 n

29. n n n1n1 1 n n1n1 1 Phase 2: Increase edge weights

30. n Complete n n1n1 1 n 2 1 n 3 new LHPs created at each step