Compact Routing with Slack in Low Doubling Dimension
Goran Konjevod, Andréa W. Richa, Donglin Xia, Hai Yu
CSE Dept., Arizona State University{goran, aricha, dxia}@asu.edu
CS Dept., Duke [email protected]
Doubling Dimension 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
Reference
With Slack
Stretch
Routing Table Headers
[KRX’07] 9+
[Dinitz’07]
slack
This paper
(1-)n nodes 1+n nodes
This paper
(1-)n nodes
1+
n nodes 9+
3 ( )log / On
4 4log log /( log log )O n n n2log
log log
nO
n
4 ( )1/ log log / On n
( log )O n n
4
( )
logO
n
Lower Bound [KRX’06]:
Graph
Doubling Dimension
Diameter Routing table
Stretch
Tree6 log 1/(2 )O n
2( / 60)( )o n 9
: 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
r
xy
u
An r-net is a subset Y of node set V s.t. x, y in Y,
d(x,y) r uV, x Y s.t. d(u,x) r
r-net nodes:
Hierarchy of r-nets r-nets:
Yi: 2i-netfor i=0,…, log
: normalized diameter
Zooming Sequence: u(0)=u u(i) is the nearest
node in Yi to u(i-1)21-net2i-1-net
2i-net
u
u(1)
u(i-1)
u(i)
Ball Packing s-size Ball Packing B
Greedily select disjoint balls Bu(ru(s)) in an ascending order of their radii ru(s)
(where ru(s) is the radius s.t. |Bu(ru(s)))|=s )
Bj: 2j-size ball packing, for j=0,…, log n B(u,j) Bj : the nearest one to u c(u,j): the center of B(u,j)
Counting Lemma
Dij: the set of uYi s.t.
c=c(u,j) Counting Lemma
( )log
,0
1 log
2
O
i j ji
n nD
3 2 3| (2 / ) \ (2 ) | (4 / ) 2i i ju cB B
u
Bc(2i+2)
Bu(2i/3)
c
Overview
Basic Idea Slack on Stretch Conclusion
(1+)-stretch
Bu(i)(2i/) contains info of Bu(i)(2i/2)
Not found at u(t-1)
Routing Cost:
1 2( ( 1), ) 2 /td u t v
1 2
0
2 / ( ( ), ) ( , ) 2 /
(1 ( )) ( , )
ti t
i
d u t v d u v
O d u v
Data Structure (1)
A search tree on any B in Bj, stores info of Bc(rc(2jg1)) where g1=log2 n/(14)
Data Structure (2)For each u(i)
If B in Bj s.t. B Bu(i)(2i/) Bu(i)(2i/2)Bc(rc(2jg1))
If not, search tree on Bu(i)(2i/) stores info of Bu(i)(2i/3)\Bc(2i+2), if u(i) Dij
where c=c(u,j), j=log (|Bu(2i/)|g2), and g2=log2 n/(10)
Bu(i)(2i/)
u(i)
Bu(i)(2i/2)
Bc(rc2ig1)
cB
2i
u(i)
Bu(i)(2i/)
Bu(i)(2i/)
c
2i2
Bc(2i+2)
Searching at u(i) Go to c, and search on B
cost: 2i+1/ info: Bu(i)(2i/2) next level: i+1
Search on Bu(i)(2i/); if u(i) Dij, go to c and search on Bc(2i+2) cost: 2i+1/2 Info: Bu(i)(2i/3) next level: i+log(1/)+1
u(i)
Bu(i)(2i/2)
Bc(rc2ig1)
cB
2i
u(i)
Bu(i)(2i/)
Bu(i)(2i/)
c
2i2
Bc(2i+2)
Slack on Stretch
Counting lemma
9+ Stretch Not at level t-1
cost
, ( ) iju i u i D n u(i)
Bu(i)(2i/)
Bu(i)(2i/)
Bc(2i+2)
1( ( 1), ) 2 /td u t v
1 2
0
2 / ( ( ), ) ( , ) 2 /
(9 ( )) ( , )
ti t
i
d u t v d u v
O d u v
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
