Compact Routing with Slack in Low Doubling Dimension Goran Konjevod, Andr é a W. Richa, Donglin Xia, Hai Yu CSE Dept., Arizona State University {goran, aricha, CS Dept., Duke University

Doubling Dimension The least value  s.t. any ball can be covered by at most 2  balls with half radius Euclidean plane:  = log 7

Related Work: Name-independent compact routing schemes ReferenceWith SlackStretchRouting TableHeaders [KRX’07] 9+  [Dinitz’07]  slack  This paper (1-  )n nodes 1+   n nodes This paper (1-  )n nodes1+   n nodes9+   Lower Bound [KRX’06]: GraphDoubling DimensionDiameterRouting tableStretch Tree  : Doubling Dimension;  1/polylog(n)

Overview Basic Idea Slack on Stretch Conclusion

Basic Idea Using underlying labeled routing scheme [KRX’07] (1+  ) stretch (log n)-bit label Mapping original names to routing labels Hierarchically storing (name, label) pairs Search procedure to retrieve routing label

r-Nets An r-net is a subset Y of node set V s.t.  x, y in Y, d(x,y)  r  u  V,  x  Y s.t. d(u,x)  r r-net nodes:

Hierarchy of r-nets r-nets: Y i : 2 i -net for i=0, …, log   : normalized diameter Zooming Sequence: u(0)=u u(i) is the nearest node in Y i to u(i-1)

Ball Packing s-size Ball Packing B Greedily select disjoint balls B u (r u (s)) in an ascending order of their radii r u (s) (where r u (s) is the radius s.t. |B u (r u (s)))|=s ) B j : 2 j -size ball packing, for j=0, …, log n B(u,j)  B j : the nearest one to u c(u,j): the center of B(u,j)

Counting Lemma D ij : the set of u  Y i s.t. c=c(u,j) Counting Lemma

Overview Basic Idea Slack on Stretch Conclusion

(1+  )-stretch B u(i) (2 i /  ) contains info of B u(i) (2 i /  2 ) Not found at u(t-1) Routing Cost:

Data Structure (1) A search tree on any B in B j, stores info of B c (r c (2 j g 1 )) where g 1 =log 2 n/(  14  )

Data Structure (2) For each u(i) If  B in B j s.t. B  B u(i) (2 i /  ) B u(i) (2 i /  2 )  B c (r c (2 j g 1 )) If not, search tree on B u(i) (2 i /  ) stores info of B u(i) (2 i /  3 )\B c (2 i+2 ), if u(i)  D ij where c=c(u,j), j=log (|B u (2 i /  )|g 2 ), and g 2 =log 2 n/(  10  ) B u(i) (2 i /  )

Searching at u(i) Go to c, and search on B cost: 2 i+1 /  info: B u(i) (2 i /  2 ) next level: i+1 Search on B u(i) (2 i /  ); if u(i)  D ij, go to c and search on B c (2 i+2 ) cost: 2 i+1 /  2 Info: B u(i) (2 i /  3 ) next level: i+log(1/  )+1

Slack on Stretch Counting lemma 9+  Stretch Not at level t-1 cost

Conclusion (1+  )-stretch compact name- independent routing schemes with slack either on storage, or on stretch, in networks of low doubling dimension. Dinitz provided 19-stretch  -slack compact name-independent routing scheme in general graphs Can we do better than 19 stretch in general graphs?

Thanks & Questions