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Metric Embeddings with Relaxed Guarantees Hubert Chan Joint work with Kedar Dhamdhere, Anupam Gupta, Jon Kleinberg, Aleksandrs Slivkins

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Embedding & Distortion Central Idea: Given finite metric (V,d), embed into simpler metric (V’,d’), (e.g. Euclidean space l 2 ) via mapping Application in approximation algorithms, e.g. sparsest cut Quality measured by distortion: The distortion of a mapping is D if for all pairs of nodes (x,y).

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Application in Networking Each node in the network gets virtual coordinates. Latencies between points can be approximated. Suppose you want to: find a “near” server in a network with replicated services find a “near” copy of file in P2P system Employ embedding techniques: Point-to-point latencies treated as a metric (V,d) Embed into Euclidean space f : (V,d) → l 2

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Practical Issues Some metrics embed into l 2 with (log n) dimensions. e.g. uniform metrics It is sufficient in some applications to obtain good approximation for most pairs of nodes. For example, one might need a server that is among the nearest 1% of all nodes. We can do better in this case! Distortion may be too restrictive: Some metrics embed into l 2 with (log n) distortion. e.g. metrics induced by constant degree expanders

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Roadmap Motivation Define Embeddings with Slack Example & Results - Upper Bounds - Lower Bounds Gracefully Degrading Embeddings Open Questions

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Definition Recall: An embedding f : V → V’ has distortion D if for all pairs of nodes (x,y). Define: An embedding f : V → V’ has distortion D with -slack if for all but n 2 pairs of nodes (x,y).

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Problem Formally, Are there functions D( ) and L( ) such that for every > 0, every finite metric space can be embedded into Euclidean space with L( ) dimensions and distortion D( ) with -slack? A question posed by Kleinberg et al in FOCS ’04: Can every finite metric space can be embedded into Euclidean space with constant number of dimensions and constant distortion with constant slack?

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Does such a result make sense? Suppose f : U n → l 2 L is an embedding into Euclidean space in L dimensions with (no slack) distortion D. Claim. # dimensions L = (log D n) 1. The images are contained in some ball of radius at most D. 2. Balls of radius 0.5 around the image of each point are pairwise disjoint. 3. Simple volume argument shows (D+0.5) L /(0.5) L is at least n. Consider a uniform metric U n on n points. D 4. Hence, L = (log D n)

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With slack comes more power… Consider the following embedding f : U n → R. 3124… 1/ Each cluster has n points. nn Each node ignores at most n other nodes. Pairs within a cluster are ignored. How many pairs are ignored? At most n 2 pairs are ignored. Distortion (with -slack) is 1/ .

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One of Our Results Theorem For every > 0, every finite metric space can be embedded into Euclidean space with O(log 2 1/ ) dimensions and O(log 1/ ) distortion with -slack. Theorem [Bourgain ’85] Every finite metric space of size n can be embedded into Euclidean space with O(log 2 n) dimensions and O(log n) distortion. With = 1/2n 2, our result reduces to Bourgain’s theorem.

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Corresponding Lower Bounds Theorem For every > 0, there exists a finite metric such that any - slack embedding into Euclidean space incurs distortion at least (log 1/ ). Theorem [Matousek ’97] There exists a finite metric space of size n such that any embedding into Euclidean space incurs distortion at least (log n). It seems we can just replace n with 1/ to get a lower bound for slack embeddings!

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General Principle for Translating Lower Bounds Theorem Suppose there exists a family of metrics for which any embedding into l 2 with at most L(n) dimensions incurs (no slack) distortion at least D(n). Then, for all > 0, there exists a family of metrics for which any embedding into l 2 with at most L(1/ 3 sqrt( ) dimensions incurs (no slack) distortion at least D(1/ 3 sqrt( )). ExampleSlackless distortion slack distortion Expanders into l 2 (log n) (log 1/ ) l 2 2m+1 into l 2 2m (n) 1/m (1/sqrt( )) 1/m

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Roadmap Motivation Define Embeddings with Slack Example & Results - Upper Bounds - Lower Bounds Gracefully Degrading Embeddings Open Questions

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One mapping does it all… We have shown: Given finite metric (V,d) and > 0, we can construct an -slack embedding with good guarantees. Question Given a finite metric (V,d), can we construct a single embedding f such that for every > 0, the mapping f is an -slack embedding with good guarantees?

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Gracefully Degrading Embedding Definition An embedding f : (V,d) → (V’,d’) has gracefully degrading distortion D( ) if for every > 0, f is an embedding with - slack distortion D( ).

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Some results for GD Embeddings Abraham, Bartal, and Neiman obtained similar results independently. Embedding into l 2 slack GD General metric O(log 2 1/ ) dim. O(log 1/ ) dist. ? Metric with doubling dim sameO(log 2 n) dim. O( ) (log 1/ ) 1/2 dist.

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Open Questions 2. Given a metric with doubling dimension , is there a gracefully degrading embedding into l 2 with the number of dimensions depending only on ? 1. Is there a function D( ) depending only on , such that every finite metric can be embedded into l 2 with gracefully degrading distortion D( ) and polylog(n) dimensions? Theorem Any finite metric can be embedded into l 1 with gracefully degrading distortion O(log 1/ ) in poly(n) dimensions.

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Questions?

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