Compact Routing with Slack in Low Doubling Dimension

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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

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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

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Related WorkName-independent compact routing
schemes
? Doubling Dimension ??1/polylog(n)

• Lower Bound KRX06

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Overview

• Basic Idea
• Slack on Stretch
• Conclusion

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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

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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
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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)

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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)

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Counting Lemma

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

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Overview

• Basic Idea
• Slack on Stretch
• Conclusion

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(1e)-stretch

• Bu(i)(2i/e) contains info of Bu(i)(2i/e2)
• Not found at u(t-1)
• Routing Cost

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Data Structure (1)

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

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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/?)

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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

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Slack on Stretch

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

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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?

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Thanks Questions