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) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

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

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

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

Distances

All-Pairs Shortest Paths in weighted directed graphs Dynamic vertex updates Authors Update time Demetrescu-Italiano (2004) n2 (amortized) Thorup (2005) n2.75 (worst case) Random edge updates Authors Update time Friedrich-Hebbinghaus (2008) n (expected) * (2010) log2 n (expected)

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

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

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

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

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

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

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

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

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

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

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

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

LSPs of length  3

High probability bound on #LSPs

Short and long SPs and LSPs

SPs and LSPs

Long LSPs New Lemma:

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

Short SPs passing through e … … New Lemma 2:

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

Large (under) deviations theorem [Maurer 2003]

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)