1. Yuval Peres Dmitry Sotnikov Benny Sudakov Uri Zwick (武熠)
All-Pairs Shortest Paths in O(n2) time
with high probability
expected
Yuval Peres Dmitry Sotnikov Benny Sudakov Uri Zwick (武熠)
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2. All-Pairs Shortest Paths in weighted directed graphs
Worst case results
Authors
Time bound
n  Dijkstra
mn + n2 log n
Pettie (2004)
mn + n2 log log n
KKP (1993) McGeoch (1995)
m*n + n2 log n
Chan (2007)
 n3 / log2n
m*  no. of edges participating in shortest paths

3. All-Pairs Shortest Paths in randomly weighted complete directed graphs
Expected running times (some with high probability)
Authors
Expected Time bound
Spira (1973)
n2 log2 n
Bloniarz (1983)
n2 log n log* n
Moffat and Takaoka (1987)
Mehlhorn and Priebe (1997)
n2 log n
Hassin and Zemel (1985)
Frieze and Grimmett (1985)
First three results hold in the endpoint independent model
Can we get rid of the log factor ???
Not (only) a data structures problem

4. Distances in randomly weighted graphs [Davis-Prieditis 1993] [Janson 1999]
Uniform distribution U[0,1]
Exp. distribution EXP(1)
Memoryless!

5. Distances

6. All-Pairs Shortest Paths in weighted directed graphs
Dynamic vertex updates
Authors
Update time
Demetrescu-Italiano (2004)
n2 (amortized)
Thorup (2005)
n (worst case)
Random edge updates
Authors
Update time
Friedrich-Hebbinghaus (2008)
n (expected)
* (2010)
log2 n (expected)

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

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

9. Demetrescu-Italiano (2004)
(Static version)
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

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

11. Demetrescu-Italiano (2004)
New implementation and analysis
Running time:
Show that:
Implement a priority queue with

12. … Bucket based (approx.) priority queue
O(1) amortized cost per operation
Algorithm remains correct

13. Expected number of LSPs
LSPs of length 1:
LSPs of length 2:
LSPs of length  3:

14. The events are not independent !
LSPs of length 2
The events are not independent !

15. LSPs of length 2
Subgraph of size n/2
Subgraph of size n/2

16. But, the events are dependent …
LSPs of length  3
But, the events are dependent …

17. Distance from b to c avoiding a
LSPs of length  3
Distance from b to c avoiding a

18. LSPs of length  3
Choose all edge weights except
Then choose

19. LSPs of length  3

20. High probability bound on #LSPs

21. Short and long SPs and LSPs

22. SPs and LSPs

23. Long LSPs
New Lemma:

24. Efron-Stein inequality (1981)
Number of short LSPs passing through ei

25. Short SPs passing through e… …
New Lemma 2:

26. Number of vertices at a given distance
New Lemma 2:
Use a large deviation theorem of Maurer (2003)

27. Large (under) deviations theorem [Maurer 2003]

28. Open problems
Improve high probability bound to O(nc) for every c>0
Obtain O(n2) expected running time for the more general endpoint independent model
Obtain O(n2) expected running time for unweighted directed G(n,p), for any 0<p<1
Improve expected random update time from O(log2n) to O(log n)