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## Compact Routing with Slack in Low Doubling Dimension

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### Compact Routing with Slack. in Low Doubling Dimension ... (1 )-stretch compact name-independent routing schemes with slack either on ... – PowerPoint PPT presentation

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Title: Compact Routing with Slack in Low Doubling Dimension

1
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_at_asu.eduCS Dept., Duke
University
• fishhai_at_cs.duke.edu

2
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

3
Related WorkName-independent compact routing
schemes
? Doubling Dimension ??1/polylog(n)
• Lower Bound KRX06

4
Overview
• Basic Idea
• Slack on Stretch
• Conclusion

5
Basic Idea
• Using underlying labeled routing scheme KRX07
• (1?) stretch
• (log n)-bit label
• Mapping original names to routing labels
• Hierarchically storing (name, label) pairs
• Search procedure to retrieve routing label

6
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
7
Hierarchy of r-nets
• r-nets
• Yi 2i-netfor i0,, log ?
• ? normalized diameter
• Zooming Sequence
• u(0)u
• u(i) is the nearest node in Yi to u(i-1)

8
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 j0,, log n
• B(u,j) ? Bj the nearest one to u
• c(u,j) the center of B(u,j)

9
Counting Lemma
• Dij the set of u?Yi s.t.
• cc(u,j)
• Counting Lemma

10
Overview
• Basic Idea
• Slack on Stretch
• Conclusion

11
(1e)-stretch
• Bu(i)(2i/e) contains info of Bu(i)(2i/e2)
• Routing Cost

12
Data Structure (1)
• A search tree on any B in Bj, stores info of
Bc(rc(2jg1))
• where g1log2 n/(??14?)

13
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(2i2), if u(i) ? Dij
• where cc(u,j), jlog (Bu(2i/?)g2), and g2log2
n/(??10?)
• Bu(i)(2i/?)

14
Searching at u(i)
• Go to c, and search on B
• cost 2i1/?
• info Bu(i)(2i/?2)
• next level i1
• Search on Bu(i)(2i/?) if u(i) ? Dij, go to c and
search on Bc(2i2)
• cost 2i1/?2
• Info Bu(i)(2i/?3)
• next level ilog(1/?)1

15
Slack on Stretch
• Counting lemma
• 9e Stretch
• Not at level t-1
• cost

16
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?

17
Thanks Questions