1 / 17

Compact Routing with Slack in Low Doubling Dimension

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, dxia}@asu.edu CS Dept., Duke University fishhai@cs.duke.edu. Doubling Dimension. Doubling Dimension

sveta
Download Presentation

Compact Routing with Slack in Low Doubling Dimension

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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}@asu.eduCS Dept., Duke University fishhai@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 Work:Name-independent compact routing schemes  : Doubling Dimension; 1/polylog(n) • Lower Bound [KRX’06]:

  4. Overview • Basic Idea • Slack on Stretch • Conclusion

  5. 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

  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 i=0,…, 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 j=0,…, 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. • c=c(u,j) • Counting Lemma

  10. Overview • Basic Idea • Slack on Stretch • Conclusion

  11. (1+e)-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 g1=log2 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(2i+2), if u(i) Dij • wherec=c(u,j), j=log (|Bu(2i/)|g2), and g2=log2 n/(10) • Bu(i)(2i/)

  14. 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 onBc(2i+2) • cost: 2i+1/2 • Info: Bu(i)(2i/3) • next level: i+log(1/)+1

  15. Slack on Stretch • Counting lemma • 9+e 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

More Related