A Note on Finding the Nearest Neighbor in Growth-Restricted Metrics

1. A Note on Finding the Nearest Neighbor in Growth-Restricted Metrics Kirsten Hildrum John Kubiatowicz Sean Ma Satish Rao

2. A B C D Peer-to-Peer Networks

3. Challenge: Efficiency • Low number of links per node • Short paths • Hops • Network distance • Challenging • Nodes connect via the Internet • Can’t see the “big picture” when choosing overlay links

4. Destination Overlay Routing Source

5. One Way(prefix routing) Level 3 Level 2 Level 1

6. Maintaining Structure • Make assumptions about network • Note: • Linear space search for sequential case • Really, tree for every destination This talk: Routing structure can be used to find nearest neighbor (or nearest-at-level)

7. Metric space • Basic technique distance-halving • Need to have some point at half-distance  Points roughly evenly-spaced good query bad query

8. 3r r Growth Restriction • Growth Restriction: • Ball of radius 3r contains only a factor of c more nodes than ball of radius r. • Stronger than “polynomial growth” • Average tree degree more than c • Sample less than 1/c

9. new Algorithm Idea, in pictures

10. 3r The Trimming Lemma • Circle of radius r around new with at least one level i node. • To find level (i-1) must find its parent. • Level (i-1) node in little circle must point to a level i node in 3r of new <2r r

11. How Many? • Keep fixed k. • Expect about c nodes in big circle, O(log n) with high probability • Closest O(log n) level i nodes, have all within circle, w.h.p. 3r r

12. Counting Messages • Only O(log n) per level needed. •  Total is O(log2 n). • (proved previously) We show:Only O(log n) total need to be contacted.

13. Certificate of “neighborliness” • A “proof” that the end node found really is the closest (all the nodes that had to be contacted) • Recursive, use level i to verify level (i-1) • For every level, keep nodes required by the trimming lemma and no more • Then bound overall size of certificate • (Algorithm is closely connected)

14. Adaptive k • Before, saved closest k • Only need ones required by trimming lemma • c, in expectation • log n levels, each constant expectation, total is O( logn) 3r r

15. My little lie • Trimming lemma fails if level (i-1) node was outside circle • Must expand circle & keep more level i nodes • More level i means more level i+1… • if sample rate < 1/c, level (i-1)’s overall effect is still constant 3r r

16. Algorithm • Keep what you think you need for trimming lemma • Get more later if needed • Touch only nodes in certificate • Return nodes on a level in order of distance from query node • Use get-level-i-in-order to implement get-level-(i-1)-in-order 3r r

17. (Some) Related Work • Karger and Ruhl, STOC02, find nearest neighbor with O(log n) queries • HKRZ, fixed k technique, O(log2 n) • Krauthgamer and Lee, SODA04 • deterministic, wider class of metric spaces • not peer-to-peer

18. Final Notes • O(log n) neighbor search w/ applications to peer-to-peer networks • Prefix routing: PRR97, Pastry, Tapestry, LAND • Question Is stronger metric assumption necessary for load balance?