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A Note on Finding the Nearest Neighbor in Growth-Restricted Metrics Kirsten Hildrum John Kubiatowicz Sean Ma Satish Rao
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Peer-to-Peer Networks A B C D
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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
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Overlay Routing Source Destination
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One Way (prefix routing) Level 1 Level 2 Level 3
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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)
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Metric space Basic technique distance-halving Need to have some point at half-distance Points roughly evenly-spaced good query bad query
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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 r 3r
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Algorithm Idea, in pictures new
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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 3r
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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. r 3r
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Counting Messages Only O(log n) per level needed. Total is O(log 2 n). –(proved previously) We show: Only O(log n) total need to be contacted.
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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)
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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( log n) r 3r
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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 r 3r
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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 r 3r
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(Some) Related Work Karger and Ruhl, STOC02, find nearest neighbor with O(log n) queries HKRZ, fixed k technique, O(log 2 n) Krauthgamer and Lee, SODA04 –deterministic, wider class of metric spaces –not peer-to-peer
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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?
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