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

Slides:



Advertisements
Similar presentations
Routing Complexity of Faulty Networks Omer Angel Itai Benjamini Eran Ofek Udi Wieder The Weizmann Institute of Science.
Advertisements

Lower Bounds for Additive Spanners, Emulators, and More David P. Woodruff MIT and Tsinghua University To appear in FOCS, 2006.
Embedding Metric Spaces in Their Intrinsic Dimension Ittai Abraham, Yair Bartal*, Ofer Neiman The Hebrew University * also Caltech.
1 Discrete Structures & Algorithms Graphs and Trees: III EECE 320.
Trees and Markov convexity James R. Lee Institute for Advanced Study [ with Assaf Naor and Yuval Peres ] RdRd x y.
Metric Embeddings with Relaxed Guarantees Hubert Chan Joint work with Kedar Dhamdhere, Anupam Gupta, Jon Kleinberg, Aleksandrs Slivkins.
Metric Embedding with Relaxed Guarantees Ofer Neiman Ittai Abraham Yair Bartal.
Cse 521: design and analysis of algorithms Time & place T, Th pm in CSE 203 People Prof: James Lee TA: Thach Nguyen Book.
Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs Chenyu Yan, Yang Xiang, and Feodor F. Dragan (WADS 2009) Kent State University, Kent,
Metric embeddings, graph expansion, and high-dimensional convex geometry James R. Lee Institute for Advanced Study.
Geometric embeddings and graph expansion James R. Lee Institute for Advanced Study (Princeton) University of Washington (Seattle)
Approximation Algorithms for Unique Games Luca Trevisan Slides by Avi Eyal.
Interchanging distance and capacity in probabilistic mappings Uriel Feige Weizmann Institute.
Navigating Nets: Simple algorithms for proximity search Robert Krauthgamer (IBM Almaden) Joint work with James R. Lee (UC Berkeley)
1 Discrete Structures & Algorithms Graphs and Trees: II EECE 320.
Approximation Algorithms: Combinatorial Approaches Lecture 13: March 2.
Balanced Graph Partitioning Konstantin Andreev Harald Räcke.
1 University of Freiburg Computer Networks and Telematics Prof. Christian Schindelhauer Wireless Sensor Networks 19th Lecture Christian Schindelhauer.
Network Design Adam Meyerson Carnegie-Mellon University.
L16: Micro-array analysis Dimension reduction Unsupervised clustering.
Approximating metrics by tree metrics Kunal Talwar Microsoft Research Silicon Valley Joint work with Jittat Fakcharoenphol Kasetsart University Thailand.
Advances in Metric Embedding Theory Ofer Neiman Ittai Abraham Yair Bartal Hebrew University.
Sublinear Algorithms for Approximating Graph Parameters Dana Ron Tel-Aviv University.
Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs Y. Rabinovich R. Raz DCG 19 (1998) Iris Reinbacher COMP 670P
Chapter 4: Straight Line Drawing Ronald Kieft. Contents Introduction Algorithm 1: Shift Method Algorithm 2: Realizer Method Other parts of chapter 4 Questions?
Sublinear Algorithms for Approximating Graph Parameters Dana Ron Tel-Aviv University.
Constant Factor Approximation of Vertex Cuts in Planar Graphs Eyal Amir, Robert Krauthgamer, Satish Rao Presented by Elif Kolotoglu.
The Art Gallery Problem
The Art Gallery Problem
Algorithms on negatively curved spaces James R. Lee University of Washington Robert Krauthgamer IBM Research (Almaden) TexPoint fonts used in EMF. Read.
Domain decomposition in parallel computing Ashok Srinivasan Florida State University COT 5410 – Spring 2004.
1 Introduction to Approximation Algorithms. 2 NP-completeness Do your best then.
1 Introduction to Approximation Algorithms. 2 NP-completeness Do your best then.
Trees and Distance. 2.1 Basic properties Acyclic : a graph with no cycle Forest : acyclic graph Tree : connected acyclic graph Leaf : a vertex of degree.
Expanders via Random Spanning Trees R 許榮財 R 黃佳婷 R 黃怡嘉.
Embeddings, flow, and cuts: an introduction University of Washington James R. Lee.
Topics in Algorithms 2007 Ramesh Hariharan. Tree Embeddings.
Multicommodity flow, well-linked terminals and routing problems Chandra Chekuri Lucent Bell Labs Joint work with Sanjeev Khanna and Bruce Shepherd Mostly.
Doubling Dimension: a short survey Anupam Gupta Carnegie Mellon University Barriers in Computational Complexity II, CCI, Princeton.
Lower Bounds for Embedding Edit Distance into Normed Spaces A. Andoni, M. Deza, A. Gupta, P. Indyk, S. Raskhodnikova.
What is a metric embedding?Embedding ultrametrics into R d An embedding of an input metric space into a host metric space is a mapping that sends each.
On the Impossibility of Dimension Reduction for Doubling Subsets of L p Yair Bartal Lee-Ad Gottlieb Ofer Neiman.
Advances in Metric Embedding Theory Yair Bartal Hebrew University &Caltech UCLA IPAM 07.
1 Assignment #3 is posted: Due Thursday Nov. 15 at the beginning of class. Make sure you are also working on your projects. Come see me if you are unsure.
Graphs Definition: a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected.
1 Approximation Algorithms for Low- Distortion Embeddings into Low- Dimensional Spaces Badoiu et al. (SODA 2005) Presented by: Ethan Phelps-Goodman Atri.
Introduction Wireless Ad-Hoc Network  Set of transceivers communicating by radio.
Oct 23, 2005FOCS Metric Embeddings with Relaxed Guarantees Alex Slivkins Cornell University Joint work with Ittai Abraham, Yair Bartal, Hubert Chan,
1 Schnyder’s Method. 2 Motivation Given a planar graph, we want to embed it in a grid We want the grid to be relatively small And we want an efficient.
Succinct Routing Tables for Planar Graphs Compact Routing for Graphs Excluding a Fixed Minor Ittai Abraham (Hebrew Univ. of Jerusalem) Cyril Gavoille (LaBRI,
Theory of Computational Complexity Probability and Computing Chapter Hikaru Inada Iwama and Ito lab M1.
Coarse Differentiation and Planar Multiflows
Approximating Graphs by Trees Marcin Bieńkowski University of Wrocław
Proof technique (pigeonhole principle)
Efficient methods for finding low-stretch spanning trees
Advances in Metric Embedding Theory
COMP 6/4030 ALGORITHMS Prim’s Theorem 10/26/2000.
Algorithms and Complexity
Autumn 2016 Lecture 11 Minimum Spanning Trees (Part II)
The Art Gallery Problem
Near-Optimal (Euclidean) Metric Compression
Introduction Wireless Ad-Hoc Network
Metric Methods and Approximation Algorithms
Embedding Metrics into Geometric Spaces
Lecture 19 Linear Program
Clustering.
Winter 2019 Lecture 11 Minimum Spanning Trees (Part II)
Treewidth meets Planarity
Routing in Networks with Low Doubling Dimension
Autumn 2019 Lecture 11 Minimum Spanning Trees (Part II)
Presentation transcript:

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<∞.