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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)
  • Not found at u(t-1)
  • 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
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