Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion Ittai Abraham, Yair Bartal, Ofer Neiman The Hebrew University

Embedding Metric Spaces Metric spaces M X =(X,d X ), M Y =(Y,d y ) Metric spaces M X =(X,d X ), M Y =(Y,d y ) Embedding is a function f : X→Y Embedding is a function f : X→Y For u,v in X, non-contracting embedding f : dist f (u,v)= d y (f(u),f(v)) / d x (u,v) For u,v in X, non-contracting embedding f : dist f (u,v)= d y (f(u),f(v)) / d x (u,v) Distortion : dist(f)= max {u,v  X} dist f (u,v) Distortion : dist(f)= max {u,v  X} dist f (u,v)

Two Schemes 1. Embedding a graph into a spanning tree of the graph. 2. Embedding a metric into an ultrametric xyz Δ(A)Δ(A) Δ(B)Δ(B) Δ(C)Δ(C) Δ(D)Δ(D) w 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). Given a weighted graph, the distance between 2 points is the length of the shortest path between them

Motivation Simple and compact representation of a metric space. Simple and compact representation of a metric space. Ultrametric embedding provides approximation algorithms to numerous NP-hard problems. Ultrametric embedding provides approximation algorithms to numerous NP-hard problems. Constructing a spanning tree is a well studied network design objective. Constructing a spanning tree is a well studied network design objective.

Previous Results For embedding n point metric into ultrametrics: For embedding n point metric into ultrametrics: A single ultrametric/tree requires Θ(n) distortion. [Bartal 96/BLMN 03/HM 05/RR 95]. A single ultrametric/tree requires Θ(n) distortion. [Bartal 96/BLMN 03/HM 05/RR 95]. Probabilistic embedding with Θ(log n) expected distortion. [Bartal 96,98,04, FRT 03] Probabilistic embedding with Θ(log n) expected distortion. [Bartal 96,98,04, FRT 03] Embedding into spanning trees: Embedding into spanning trees: Minimum Spanning Tree: n-1 distortion. Minimum Spanning Tree: n-1 distortion. Probabilistic embedding with Õ(log 2 n) expected distortion. [EEST 05] Probabilistic embedding with Õ(log 2 n) expected distortion. [EEST 05]

Average Distortion Average distortion : Average distortion : l q -distortion : l q -distortion : Any metric embeds into Hilbert space with constant average distortion [ABN 06]. Any metric embeds into Hilbert space with constant average distortion [ABN 06]. Any metric probabilistically embeds into ultrametrics with constant average distortion [ABN 05/06, CDGKS 05]. Any metric probabilistically embeds into ultrametrics with constant average distortion [ABN 05/06, CDGKS 05]. Also: Simultaneously tight l q -distortion for all q. Also: Simultaneously tight l q -distortion for all q. l ∞ -dist = distortion l 1 -dist = average distortion.

Our Results An embedding of any n point metric into a single ultrametric. An embedding of any n point metric into a single ultrametric. An embedding of any graph on n vertices into a spanning tree of the graph. An embedding of any graph on n vertices into a spanning tree of the graph. Average distortion = O(1). Average distortion = O(1). l 2 -distortion = l 2 -distortion = l q -distortion = Θ(n 1-2/q ), for 2<q≤∞ l q -distortion = Θ(n 1-2/q ), for 2<q≤∞

Embeddings with scaling distortion Definition: f has scaling distortion α, if for every ε there exist at least pairs (u,v) such that dist f (u,v) ≤ α(ε). Definition: f has scaling distortion α, if for every ε there exist at least pairs (u,v) such that dist f (u,v) ≤ α(ε). Thm: Every metric space embeds into an ultrametric and every graph has a spanning tree with scaling distortion For ε=¼, ¾ of pairs have distortion < c·2 For ε=1/16, 15/16 of pairs have distortion < c·4 … For ε=1/n 2, all pairs have distortion < c·n

Additional Result Thm: Any graph probabilistically embeds into a distribution of spanning trees with expected scaling distortion Õ(log 2 (1/ε)). Thm: Any graph probabilistically embeds into a distribution of spanning trees with expected scaling distortion Õ(log 2 (1/ε)). Implies that the l q -distortion is bounded by O(1) for any fixed 1≤q<∞. Implies that the l q -distortion is bounded by O(1) for any fixed 1≤q<∞. For q=∞ matches the [EEST 05] result. For q=∞ matches the [EEST 05] result.

Embedding into an ultrametric 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 Plan : show for all ε, at most ε fraction of distances are distorted “too much”. Plan : show for all ε, at most ε fraction of distances are distorted “too much”. Using induction, for all 0<ε≤1 simultaneously: Using induction, for all 0<ε≤1 simultaneously: B ε – distorted distances for current level and ε. B ε – distorted distances for current level and ε. X X1X1X1X1 X2X2X2X2 Δ X1X1X1X1 X2X2X2X2 | B ε |≤ ε|X 1 ||X 2 | A separated pair (x,y) is distorted “ too much ” if

Partition Algorithm Fix some point u, such that |B(u,Δ/2)|<n/2 fix a constant c = 1/150. Fix some point u, such that |B(u,Δ/2)|<n/2 fix a constant c = 1/150. Goal: find r>0, define X 1 =B(u,r), X 2 =X\X 1. Goal: find r>0, define X 1 =B(u,r), X 2 =X\X 1. Such that for all ε>0 : Such that for all ε>0 : (the set of possible “bad” pairs) u r X1X1X1X1 S1S1S1S1 S2S2S2S2 X2X2X2X2 A separated pair (x,y) is distorted if

Partition Algorithm Let Let Choose r from the interval Choose r from the interval Claim 1: The interval is “sparse”, contains at most points. Claim 1: The interval is “sparse”, contains at most points. Claim 2: Any r in the interval is good for all Claim 2: Any r in the interval is good for all Proof: Proof: By the maximality of, By the maximality of, Clearly |S 1 |≤|X 1 |. Clearly |S 1 |≤|X 1 |.

Small values of ε Claim 3: There exists some r in the interval which is good for all simultaneously. Claim 3: There exists some r in the interval which is good for all simultaneously. While there exists uncolored r in the interval which is “bad” for some : While there exists uncolored r in the interval which is “bad” for some : Take uncolored r i with largest bad. Take uncolored r i with largest bad. Color the segment of length around r i. Color the segment of length around r i. u r1r1r1r1 r2r2r2r2 r is bad for ε if letting X 1 =B(u,r) will imply |B ε |>ε|X 1 |·|X 2 | r3r3r3r3

Small values of ε T = number of points in all bad segments. T = number of points in all bad segments. u r1r1r1r1 r2r2r2r2 A bad segment contains at least points Otherwise |B ε | is bounded by By claim 1 the interval contains at most points Bound on the length of all the bad segments S1S1S1S1 S2S2S2S2 Every point can be at most at 2 bad segments

Embedding into a Spanning Tree The spanning tree is created by a hierarchical star decomposition that uses ideas from [EEST 05]. The spanning tree is created by a hierarchical star decomposition that uses ideas from [EEST 05]. The decomposition for ultrametrics is in the heart of the star decomposition. The decomposition for ultrametrics is in the heart of the star decomposition. Furthermore, the spanning tree construction requires some additional ideas. Furthermore, the spanning tree construction requires some additional ideas.

y2y2 x1x1 y1y1 Star Decomposition Let R be the radius for x 0. Let R be the radius for x 0. Cut a central ball X 0 with radius ≈R/2. Cut a central ball X 0 with radius ≈R/2. While un-assigned points exist: While un-assigned points exist: Let x i with a neighbor y i. Let x i with a neighbor y i. Apply decompose algorithm with cone-radius α k R. Apply decompose algorithm with cone-radius α k R. ( k =level of recursion). ( k =level of recursion). Add edges (x i,y i ) to the tree. Add edges (x i,y i ) to the tree. Continue recursively inside each cluster. Continue recursively inside each cluster. x0x0 x2x2 A point z is in the cone with radius r if d(z,x 1 )+d(x 1,x 0 )-d(z,x 0 )≤r

y2y2 x1x1 y1y1 Cone-radius Cone-radius α k R = loss of 1/α k in distortion. Cone-radius α k R = loss of 1/α k in distortion. Tree radius blow-up = Tree radius blow-up = EEST chose α=1/log n EEST chose α=1/log n To ensure small blow-up and scaling distortion take To ensure small blow-up and scaling distortion take as long as as long as rad(X) decreases geometrically. rad(X) decreases geometrically. Work for all ε<ε lim Work for all ε<ε lim x0x0 x2x2 n = size of original metric Δ = radius of original metric Reset the parameters and k when this fails If u,v are separated then d T (u,v)<2rad(T[X])

Conclusion An scaling approximation of An scaling approximation of Metrics by ultrametrics. Metrics by ultrametrics. Graphs by spanning trees. Graphs by spanning trees. Implies constant approximation on average. Implies constant approximation on average. Implies l 2 -distortion. Implies l 2 -distortion. A Õ(log 2 (1/ε)) scaling probabilistic approximation of graphs by a random spanning tree. A Õ(log 2 (1/ε)) scaling probabilistic approximation of graphs by a random spanning tree. Implies constant l q -distortion for all fixed q<∞. Implies constant l q -distortion for all fixed q<∞.