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Metric Embedding with Relaxed Guarantees Ofer Neiman Ittai Abraham Yair Bartal

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Embedding metric spaces Representation of the metric in a simple and structured space. Representation of the metric in a simple and structured space. Common target spaces: l p, trees (ultra-metrics). Common target spaces: l p, trees (ultra-metrics). The price of simplicity: distortion, which is the multiplicative amount by which distances can change. The price of simplicity: distortion, which is the multiplicative amount by which distances can change. Goal: find low distortion embeddings. Goal: find low distortion embeddings. A tool for approximation algorithms A tool for approximation algorithms Useful for many practical applications. Useful for many practical applications.

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Metric embedding Let X,Y be metric spaces with metrics d x, d y respectively. Let X,Y be metric spaces with metrics d x, d y respectively. f : X→Y is an embedding of X into Y. f : X→Y is an embedding of X into Y. The distortion of f is the minimal α such that for some c: The distortion of f is the minimal α such that for some c:

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Every metric space on n points can be embedded into Euclidean space with distortion O(log n) and dimension O(log 2 n). [Bourgain/LLR] Every metric space on n points can be embedded into Euclidean space with distortion O(log n) and dimension O(log 2 n). [Bourgain/LLR] Basic results Every metric space on n points can be embedded into a tree metric with distortion Every metric space on n points can be embedded into a tree metric with distortion [Bartal/BLMN/RR]. [Bartal/BLMN/RR].

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problem The lower bounds on the distortion and the dimension are high, and grow with n. The lower bounds on the distortion and the dimension are high, and grow with n. In some cases, weaker guarantees are acceptable.. In some cases, weaker guarantees are acceptable..

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Some Alternative Schemes Probabilistic embedding: considering the expected distortion. [Bartal, FRT] Probabilistic embedding: considering the expected distortion. [Bartal, FRT] Ramsey theorems: embedding a large subspace of the original metric. [BFM, BLMN] Ramsey theorems: embedding a large subspace of the original metric. [BFM, BLMN] Partial embedding: embedding all but a fraction of the distances. [KSW, ABCDGKNS] Partial embedding: embedding all but a fraction of the distances. [KSW, ABCDGKNS]

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Motivation Estimating latencies (round-trip time) in the internet. Estimating latencies (round-trip time) in the internet. - the distance matrix is almost a metric. - the distance matrix is almost a metric. - embedding heuristics yield surprisingly good results... - embedding heuristics yield surprisingly good results... [Ng+Zhang ’ 02, ST ’ 03, DCKM ’ 04] [Ng+Zhang ’ 02, ST ’ 03, DCKM ’ 04] Practical network embedding requires: Practical network embedding requires: - Small number of dimensions. - Small number of dimensions. - No centralized co-ordination. - No centralized co-ordination. - Linear number of distances measurement. - Linear number of distances measurement. Finding nearest: copy of a file, service from some server, ect. Finding nearest: copy of a file, service from some server, ect.

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(1-ε) partial embedding X, Y are metric spaces. X, Y are metric spaces. f : X→Y has (1-ε) partial distortion at f : X→Y has (1-ε) partial distortion at most α if there exists a set of pairs G ε such that: most α if there exists a set of pairs G ε such that: For all pairs (u,v)єG ε.

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Scaling Embedding A stronger requirement is a map that will be good for all ε simultaneously. A stronger requirement is a map that will be good for all ε simultaneously. Definition: an embedding f has scaling distortion D(ε) if for any ε>0, it is an (1-ε) partial embedding with distortion D(ε). Definition: an embedding f has scaling distortion D(ε) if for any ε>0, it is an (1-ε) partial embedding with distortion D(ε).

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Scaling & Average Distortion Thm: every metric space has scaling probabilistic embedding with distortion O(log(1/ε)) into trees. Thm: every metric space has scaling probabilistic embedding with distortion O(log(1/ε)) into trees. Thm: every metric space has scaling embedding with distortion O(log(1/ε)) and dimension O(log n) into Euclidean space. Thm: every metric space has scaling embedding with distortion O(log(1/ε)) and dimension O(log n) into Euclidean space. implies constant average distortion! implies constant average distortion! Applications: weighted average problems Applications: weighted average problems sparsest cut, quadratic assignment, linear arrangement, ect. sparsest cut, quadratic assignment, linear arrangement, ect.

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Any λ-doubling metric space X can be embedded into l 2 with (1-ε) partial distortion [KSW]. Any λ-doubling metric space X can be embedded into l 2 with (1-ε) partial distortion [KSW]. Previous work Definition: a metric space X is called λ-doubling if for any r>0, any ball of radius r can be covered by λ balls of radius r/2. Definition: a metric space X is called λ-doubling if for any r>0, any ball of radius r can be covered by λ balls of radius r/2.

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Partial embedding into trees with distortion. Partial embedding into trees with distortion. Distortion & Dimension don ’ t depend on the size of X! Distortion & Dimension don ’ t depend on the size of X! Our Results Partial embedding into l 2 with distortion and dimension O(log(1/ε)). Partial embedding into l 2 with distortion and dimension O(log(1/ε)). General theorem converting classical l p embeddings into the partial model. General theorem converting classical l p embeddings into the partial model. Tight lower bounds. Tight lower bounds. Appeared in FOCS05 together with CDGKS

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Thm: Any subset-closed family of metric spaces X, that has Thm: Any subset-closed family of metric spaces X, that has for any Xє X on n points, an embedding φ:X→ l p with for any Xє X on n points, an embedding φ:X→ l p with - distortion α(n). - distortion α(n). - dimension β(n). - dimension β(n). φ can be converted into (1-ε) partial embedding of X with φ can be converted into (1-ε) partial embedding of X with - distortion - distortion - dimension - dimension Embedding into l p In practice..

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(1-ε) partial embedding of any metric space into l p with distortion (1-ε) partial embedding of any metric space into l p with distortion and dimension [ Bourgain,Matousek,Bartal] and dimension [ Bourgain,Matousek,Bartal] (1-ε) partial embedding of any negative type metric (l 1 metrics) into l 2 with distortion and dimension (1-ε) partial embedding of any negative type metric (l 1 metrics) into l 2 with distortion and dimension [ARV, ALN] [ARV, ALN] (1-ε) partial embedding of any doubling metric into l p with distortion and dimension [KLMN] (1-ε) partial embedding of any doubling metric into l p with distortion and dimension [KLMN] (1-ε) partial embedding of any tree metric into l 2 with distortion (1-ε) partial embedding of any tree metric into l 2 with distortion and dimension [Matousek] and dimension [Matousek] Main Results

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Definitions Let r ε (u) be the minimal radius such that |B(u,r ε (u))| ≥ εn. Let r ε (u) be the minimal radius such that |B(u,r ε (u))| ≥ εn. A pair (u,v), w.l.o.g r ε (u) ≥ r ε (v): has short distance if d(u,v) < r ε (u) has short distance if d(u,v) < r ε (u) has medium distance if r ε (u) ≤ d(u,v) < 4∙r ε (u). has medium distance if r ε (u) ≤ d(u,v) < 4∙r ε (u). has long distance if 4·r ε (u) ≤ d(u,v). has long distance if 4·r ε (u) ≤ d(u,v).

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Close Distances (u,v) is a short pair. (u,v) is a short pair. Short pairs are ignored - at most εn 2. Short pairs are ignored - at most εn 2. u r ε (u) v r ε (v) r ε (w) w

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Beacon Based Embedding Randomly choose beacons = B. Randomly choose beacons = B. Each point attached to nearest beacon. Each point attached to nearest beacon.

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Some More Bad Points If d(u,B) > r ε (u) then If d(u,B) > r ε (u) then is bad. is bad. For each uєX : For each uєX : With probability ½ at most 2εn 2 bad pairs. With probability ½ at most 2εn 2 bad pairs. u r ε (u) v r ε (v) r ε (w) w

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Partial Embedding Use the embedding φ:B→l p. Use the embedding φ:B→l p. φ has distortion guarantee of. φ has distortion guarantee of. The partial embedding is: The partial embedding is: φ(b)φ(b)φ(b)φ(b) h(u)h(u)h(u)h(u) f (u) u attached to beacon b

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bvbv bubu Upper Bound We assume for the pair (u,v): - Each point has a beacon in its ball. - Each point has a beacon in its ball. - Both u,v are outside each other ’ s ball. - Both u,v are outside each other ’ s ball. - The mapping φ is a contraction. - The mapping φ is a contraction. u r ε (u) r ε (v) v

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bvbv d(u,v) ≥ 4·max{r ε (u), r ε (v)} Lower Bound - Long Distances r ε (u) u r ε (v) v bubu d(b u,b v ) ≥ d(u,v)/2

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u v Medium Distances?? There is a problem in this case: There is a problem in this case: r ε (u) r ε (v) u,v are attached to the same beacon!! The additional coordinates h will guarantee enough contribution.. The additional coordinates h will guarantee enough contribution..

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Medium Distances With probability < ε the pair (u,v) will be smaller than half its expectation. Pairs satisfying: r ε (u) ≤d(u,v) ≤ 4r ε (u) r ε (u) ≤ d(u,v) ≤ 4r ε (u) [w.l.o.g r ε (u) ≥ r ε (v) ] [w.l.o.g r ε (u) ≥ r ε (v) ] r ε (u),0 r ε (v),0 rε(u)rε(u)rε(u)rε(u)0 rε(u)rε(u)rε(u)rε(u) In expectation ¼ of the coordinates will be r ε (u). With probability ¼ we get r ε (u) h(u)-h(v)

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Medium Distances With probability ½, 2εn 2 medium pairs failed, but for the others: With probability ½, 2εn 2 medium pairs failed, but for the others: End of proof! End of proof!

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Coarse Partial Embedding Another version: ignoring only the short distances (i.e., from each point to its nearest εn neighbors). the dimension increases to O(log(n)·β(1/ε)). Another version: ignoring only the short distances (i.e., from each point to its nearest εn neighbors). the dimension increases to O(log(n)·β(1/ε)).

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Partial Embedding into Trees Thm: every metric space has (1-ε) partial embedding with distortion into a tree (ultra-metric). Thm: every metric space has (1-ε) partial embedding with distortion into a tree (ultra-metric).

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Ultra-metrics Metric on leaves of rooted labeled tree. Metric on leaves of rooted labeled tree. 0 ≤ Δ(D) ≤ Δ(B) ≤ Δ(A). 0 ≤ Δ(D) ≤ Δ(B) ≤ Δ(A). d(x,y) = Δ(lca(x,y)). d(x,y) = Δ(lca(x,y)). d(x,y) = Δ(D). d(x,w) = Δ(B). d(w,z) = Δ(A). xyz Δ(A)Δ(A) Δ(B)Δ(B) Δ(C)Δ(C) Δ(D)Δ(D) w

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Embedding into Ultra-metric Partition X into 2 sets X 1, X 2 Partition X into 2 sets X 1, X 2 Create a root labeled Δ = diam(X). Create a root labeled Δ = diam(X). The children of the root are created recursively on X 1, X 2 The children of the root are created recursively on X 1, X 2 Using induction the number of distances we ignore is Using induction the number of distances we ignore is B – bad distances for current level. B – bad distances for current level. X X1X1X1X1 X2X2X2X2 Δ X1X1X1X1 X2X2X2X2 |B|≤ ε|X 1 ||X 2 |

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Take a point u such that |B(u,Δ/2)| ≤ n/2. Take a point u such that |B(u,Δ/2)| ≤ n/2. Let Let i=1,…,1/ε i=1,…,1/ε Let S i =A i+1 -A i Let S i =A i+1 -A i We need a “slim” shell… We need a “slim” shell… only distances inside the shell are distorted by more than only distances inside the shell are distorted by more than Where to Cut? A1A1A1A1 AiAiAiAi u A i+1

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Case 1: |A 1 |< εn. X 1 = u, X 2 = X\{u} X 1 = u, X 2 = X\{u} Where to Cut? A1A1A1A1 AiAiAiAi Δ X\u u

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Case 2: Case 2: Choose an i such that: Choose an i such that: Let X 1 =A i+½, X 2 = X\X 1 Let X 1 =A i+½, X 2 = X\X 1 A1A1A1A1 AiAiAiAi u A i+1 X1X1X1X1 X2X2X2X2 Δ

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Assume by contradiction for all Assume by contradiction for all |S i | 2 > εn|A i | |S i | 2 > εn|A i | Then by induction |A i | ≥ εn·i 2. which implies |A t | ≥ n. Then by induction |A i | ≥ εn·i 2. which implies |A t | ≥ n. End of proof! End of proof! Finding Shell S i

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Lower Bounds General method to obtain partial lower bounds from known classical ones. General method to obtain partial lower bounds from known classical ones. Thm: given a lower bound α for embedding a family X into a family Y : i.e. for any n there is Xє X on n points and any embedding of X requires distortion at least α(n). Thm: given a lower bound α for embedding a family X into a family Y : i.e. for any n there is Xє X on n points and any embedding of X requires distortion at least α(n). Then there is X’є X for which any (1-ε) partial embedding requires distortion Then there is X’є X for which any (1-ε) partial embedding requires distortion The family X must be nearly closed under composition!

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distortion for partial embedding into trees. [Bartal/BLMN/RR]. distortion for partial embedding of doubling or l 1 metrics into l 2. [NR] distortion for probabilistic partial embedding into trees. [Bartal] Main corollaries distortion for partial embedding into l p. [LLR, Mat]

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Choose Xє X such that Choose Xє X such that For each xєX create a metric C x such that For each xєX create a metric C x such that - C x є X. - C x є X. - X’ contain many “copies” of X. X’ contain many “copies” of X. Let f be a (1-ε) partial embedding that ignores the set of edges I. By definition. Let f be a (1-ε) partial embedding that ignores the set of edges I. By definition. General idea d δ X X’X’X’X’

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T: vertices intersecting less than edges in I. T: vertices intersecting less than edges in I. For each xєX, choose some For each xєX, choose some v x єC x ∩T. v x єC x ∩T. For each pair (v x,v y ) find t єC y such that: For each pair (v x,v y ) find t єC y such that: Finding a copy of X in T vxvx vyvy t CxCx CyCy

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Distortion of the Copy vxvxvxvx vyvyvyvy t f has distortion guarantees for both these distances Its distortion must be at least d(t,v y ) is negligible

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