Multi-Embedding and Path Approximation of Metric Spaces

Transcription

1 Multi-Embedding and Path Approximation of Metric Spaces Yair Bartal and Manor Mendel The Hebrew University Multi-Embedding and Path Approximation of Metric Spaces p1/4

2 Metric Embedding : a finite metric space An embedding of in is is called non-contractive if : a host" space for any The distortion of non-contractive embedding is "! # Multi-Embedding and Path Approximation of Metric Spaces p2/4

3 $ $ & & $ * + * $ * + / Algorithmic Paradigm : An algorithmic minimization problem : Instance of Contains a metric : a feasible solution of is a linear combination of the distances ) : a class of metric spaces for which Suppose Suppose Algorithm On input Embeds Apply - - an algorithm to solve for metric : : on 0 with is easy" for metrics in Multi-Embedding and Path Approximation of Metric Spaces p3/4

4 * Algorithmic Paradigm 0 and Prop: Suppose for any 0 5 ) )1 Then for any and 5 ) /243 ) Then 0 Proof Let 0 - )1 ) 0 5 ) ) / Multi-Embedding and Path Approximation of Metric Spaces p4/4

5 : : : 9 # : : : > ; B :! Host Spaces > ;=< > ;=? Tree metrics Ultrametrics Caveat: The -point cycle has distortion when embedded into a tree metric 0 > ;=@ 0 0 > ;=< >A ;=? >A ;=@ > ;=? #DC Example of an ultrametric Multi-Embedding and Path Approximation of Metric Spaces p5/4

6 * E H G F E I Probabilistic Embedding Def [APW B] A probabilistic embedding is an embedding into a metric where * E E /E is non-contractive E /E 0 E /E Theorem [B] Any -point metric has a probabilistic embedding into ultrametrics with distortion Furthermore the distribution can be sampled efficiently It has many algorithmic applications J ( Multi-Embedding and Path Approximation of Metric Spaces p6/4

7 * / Paradigm for Prob Embedding for metric : A randomized Algorithm : On input * E E Sample an embedding E 0 Lon Apply 0 0 and * 0 M Prop: Suppose that 0 0 (N ) )1 probabilistic + M and Then (N ) 2 3 / Q ) P O Multi-Embedding and Path Approximation of Metric Spaces p7/4

8 Applications of Prob Embedding Probabilistic embedding has found many applications for approximation algorithms online algorithms and distributed algorithms Examples: Group Steiner tree [Garg onjevod Ravi] Metrical task systems [Bartal Blum Burch Tomkins] [Fiat M] Metric labeling [leinberg Tardos] Clustering [Bartal Charikar Raz] Multi-Embedding and Path Approximation of Metric Spaces p/4

9 In This Talk Stronger notion special metrics Probabilistic Embedding Improved embedding Weaker Notion A weaker notion of embedding Useful for some algorithmic applications Sometimes has a better distortion" Multi-Embedding and Path Approximation of Metric Spaces p9/4

10 Motivation Weaker notions of embeddings may be of interest when: 1 There are algorithmic problems for which they make for feasible reductions Examples: group Steiner problem metrical task systems 2 They provide reduced overhead" (distortion) at least for some interesting metrics Examples: expanders low diameter graphs 3 The constructions are much simpler than those for probabilistic approximations 4 They are entertaining Multi-Embedding and Path Approximation of Metric Spaces p10/4

11 V \ W W Multi Embedding maps a point into a subset The inverse mapping is a function SR V UT The non-contraction property: M V T T M H The blow-up: Y Y Y[Z YX J^] ( N We require Multi-Embedding and Path Approximation of Metric Spaces p11/4

12 _ d a c e _ V E / + M V E a a _ e e _ Path Approximation Path": a sequence of points in `a ab V Its length: Ef a E a For any path p There exists path p A multi-embedding called -path approximation if st and d E E `a UT 0 E 0 is 0 d E E `a M H 0 / 3 Multi-Embedding and Path Approximation of Metric Spaces p12/4

13 i j H I i j I k I i j j Metrics of Expander graphs Constant degree expanders are badly embeddable in ehg Prop : -vertex unweighted graph maximal degree diameter Then has path-approx by a tree with blow-up of k Proof There are only paths of length in Put them all in one metric space with pairwise distance of 3 D/ 2 D D D D/ 2 D Multi-Embedding and Path Approximation of Metric Spaces p13/4

14 l o n q l R r p v u t s m p y x w I I Group Steiner Problem (GSP) Instance: A metric space subsets ( groups") of points Feasible sol : A graph q R is connected MDs Minimize: p m R z and a collection of satisfying Thm [GR]: GSP on -point tree metrics has poly-time approx alg m Y Y Using probab -approx : GSP on poly-time J ( b Ym Y -point metric space has approx alg Multi-Embedding and Path Approximation of Metric Spaces p14/4

15 * 2 / * m 2 / \ m s X m 0p { p V { p 0m p m Reduction via Path Approx Given - an We construct: An -approx alg Let Apply 0 Return -approx alg to solve GSP for metrics in Z s - T path-approx of for GSP instance 0 m : The definition of implies that is a feasible solution for Multi-Embedding and Path Approximation of Metric Spaces p15/4

16 5 67 n & _ 0 _ / _ n _ Analysis of the Reduction m & & 2 / p Let Claim Proof an Euler tour of Let in path approx of be an Let & 2 / 0 _ 2 0 p p / 2 Multi-Embedding and Path Approximation of Metric Spaces p16/4

17 I j j GSP on Expanders Prop GSP on metrics of the type: D/ 2 D/ 2 has Ym Y approximation algorithms D D D D Proof Two cases: The optimal solution is inside one -path: It is an interval in that path and therefore easy to find The optimal solution spans more than one path: Its cost is dominated by the inter-distances between paths These distances are all equal ( ) therefore It is an instance of the Hitting Set Problem Multi-Embedding and Path Approximation of Metric Spaces p17/4

18 I H I GSP on Expanders Corollary GSP on constant degree expander graphs has poly-time approx alg Ym Y This is almost optimal since expanders contains large subset with distortion from equilateral space Perspective: using probabilistic embedding it s unclear how to improve the approximation factor below b m Y Y Multi-Embedding and Path Approximation of Metric Spaces p1/4

19 } > } Œ I Œ Œ / Multi-embedding into Ultrametrics Def the aspect ratio of metric space: > #DC ; B^ ~ ƒ ; B Š ˆ Thm Any -point metric space with ar has multi-embedding into UM of size with path-approx at most VŽ #DC \ Œ J ( Œ J ( X Remarks: The dependence on is much better than in probab -embedding for which it is The construction and its analysis are much simpler than for probab -embedding There are lower bounds of and on Œ Multi-Embedding and Path Approximation of Metric Spaces p19/4

20 I 9 Probabilistic Multi Embedding It is possible to combine multi-embedding with probab -embedding Thm Any -point metric space has probabilistic multi embedding to spaces of size at most for which the path-approx is at most VŽ J ( J ( The reductions for MTS and GSP also hold for this type of embedding We thus obtain a slight improvement wrt approx factor for these problems There is a lower bound of of this type of embedding in the on the path-approx Multi-Embedding and Path Approximation of Metric Spaces p20/4

21 I š & W Multi-embedding into Ultrametrics Thm Let Any point metric space has path approximation by a UM with points and Proof: V X \ Œ X \ b Œ a A B S b Partition the diameter equal width shells Pick one shell into and duplicate it S= A (intersection) B Construct recursively UMs for the inner shells and for the outer shells Join them with a new root labelled with a A B S S" b Multi-Embedding and Path Approximation of Metric Spaces p21/4

22 Summary Definition of a metric multi-embedding Has very low distortion" for expanders Applicable to MTS and GSP Improves on probab -embedding into UM May have very low distortion" embedding into trees Multi-Embedding and Path Approximation of Metric Spaces p22/4

23 Open Problems What is the trade off between the blow-up and the path-approx in multi-embeddings into trees More applications Tight bounds on [probabilistic] path-approx into UM Is probab multi embedding really necessary? Other types of embeddings" or distortions" Multi-Embedding and Path Approximation of Metric Spaces p23/4