Viewing the Rings of a Tree: Minimum Distortion Embeddings into Trees

Transcription

1 Viewing the Rings of a Tree: Minimum Distortion Embeddings into Trees Amir Nayyeri Benjamin Raihel Abstrat We desribe a 1+ε) approximation algorithm for finding the minimum distortion embedding of an n-point metri spae, X, d X ), into a tree with vertex set X. The running time of our algorithm is n 2 /ε) Oδopt/ε))2λ+1 parameterized with respet to the spread of X, denoted by, the minimum possible distortion for embedding X into any tree, denoted by δ opt, and the doubling dimension of X, denoted by λ. Hene we obtain a PTAS, provided δ opt is a onstant and X is a finite doubling metri spae with polynomially bounded spread, for example, a point set with polynomially bounded spread in onstant dimensional Eulidean spae. Our algorithm implies a onstant fator approximation with the same running time when Steiner verties are allowed. Moreover, we desribe a similar 1 + ε) approximation algorithm for finding a tree spanner of X, d X ) that minimizes the maximum streth. The running time of our algorithm stays the same, exept that δ opt must be interpreted as the minimum streth of any spanning tree of X. Finally, we generalize our tree spanner algorithm to a 1 + ε) approximation algorithm for omputing a minimum streth tree spanner of a weighted graph with a given upper bound deg on the maximum degree, where the running time is parameterized with respet to deg, in addition to the other parameters above. In partiular, we obtain a PTAS for omputing minimum streth tree spanners of weighted graphs, with polynomially bounded spread, onstant doubling dimension, and onstant maximum degree, when a tree spanner with onstant streth exists. The full version is also available online Shool of Eletrial Engineering and Computer Siene; Oregon State University; nayyeria@ees.oregonstate.edu; Work on this paper was partially supported by NSF CRII Award Department of Computer Siene; University of Texas at Dallas; Rihardson, TX 75080, USA; benjamin.raihel@utdallas.edu; Work on this paper was partially supported by NSF CRII Award and CAREER Award

2 1 Introdution Given a general metri spae X, d X ), onsider the problem of finding a host metri spae from within some lass of simple metri spaes that X, d X ) an be embedded into while preserving pairwise distanes as muh as possible. This is a entral problem in the algorithmi study of metri spaes, as naturally finding suh a simpler metri an unlok a set of effiient algorithmi tools whih may be less effetive on more omplex spaes. To quantify the extent to whih an embedding preserves distanes, we onsider the multipliative) distortion, whih is a widely used and studied measure, having many nie properties suh as sale invariane. Formally, given metri spaes X, d X ) and Y, d Y ), an embedding of X into Y is an injetive map f : X Y, with expansion e f and ontration f defined as e f = max x,x X x x d Y fx), fx )) d X x, x, f = max ) x,x X x x d X x, x ) d Y fx), fx )). The distortion of f is then defined as δ f = e f f. Low distortion embeddings have been extensively studied and have been used in a variety of omputer siene appliations see [IM04,Ind01,Mat13]). Among alternatives, one of the most widely studied lasses of simpler host metri spaes is the lass of weighted trees, whose struture is well understood and readily allows one to apply tools suh as dynami programming. Furthermore, suh embeddings have found natural appliations, for example, in estimating phylogeneti trees [KW99]. Closely related to minimum distortion embeddings into trees is the problem of finding tree spanners with minimum streth. Given a graph G, a tree spanner with streth δ is a spanning tree of G preserving distanes up to a multipliative fator of δ, i.e. a δ distortion embedding into a spanning tree of G. As minimal distane preserving strutures, tree spanners have for example found appliations in distributed systems [DH98,PR01]. In this paper, we provide parameterized approximation algorithms for minimum distortion embeddings into trees, and minimum streth tree spanners. In other words, we seek to answer the fundamental question, how well an a given metri spae or graph) be represented by a tree? Signifiane. Finding an approximate minimum distortion embedding into a tree is a provably hard problem, thus many previous works have foused on the simpler ase when the input is the shortest path metri of an unweighted graph as disussed in detail below). Here we onsider the far more general weighted ase, i.e. the input is any finite metri. In order to make suh a large jump we must parametrize our running times on ertain quantitative measures of the soure metri, in partiular, the doubling dimension and spread. 1 It is important to note that our running time depends only polynomially on the spread, and thus is designed to handle reasonably large ranges of distanes. Note for unweighted graphs the spread is trivially polynomially bounded.) Our running time is also parametrized on the optimal distortion, δ opt. This is natural beause when δ opt is large not only is the problem hard to approximate, but also a minimum distortion embedding beomes less informative. Note that more generally removing any of these parameterizations quikly either leads to an open problem or a known hard ase. Moreover, whenever these parameters are bounded we get a PTAS for finding the minimum distortion embedding into a tree or a PTAS for the minimum streth tree spanner). Thus as a natural example, given a point set in low dimensional Eulidean spae, with up to polynomially large spread, we an get a 1 + ε)δ opt embedding in polynomial time if δ opt is below some onstant threshold, and otherwise report that the input metri annot be well represented by a tree. 1 The spread is the ratio of the largest to smallest distane in the metri, sometimes referred to as the aspet ratio. 1

3 1.1 Previous Work Embedding into trees. Nearly a half entury ago, Buneman studied the problem of reonstruting trees from distane measures [Bun71]. He showed that an embedding with distortion one an be found in On 4 ) time if it exists. Later, Agarwala et al. [ABF + 99] showed that in the absene of a perfet embedding, finding a minimum distortion embedding is not only NP-omplete, but atually APX-hard. 2 Moreover, in ertain ases muh stronger hardness results are known. For example, finding the minimum distortion embedding into the real line, that is a tree of maximum degree two, is hard to approximate within a polynomial fator even when embedding from weighted tree metris with polynomial spread [BCIS05] note the problem is muh easier for additive distortion, as there is a 2-approximation [HIL98]). Thus it is natural to onsider restritions on the soure metri spae. In partiular, Bădoiu et al. [BIS07] showed the minimum distortion embedding for unweighted graphs into trees an be approximated within a onstant fator in polynomial time. Their result leads to the state of the art 6-approximation after a ouple of improvements [BDH + 08, CDN + 10]. In ontrast to unweighted graphs, far less is known about embedding general metris into trees. In fat, the only non-trivial approximation algorithm, found by Bădoiu et al. [BIS07], gives an embedding with distortion δ opt log n) log, where δ opt is the minimum distortion. 3 Tree spanners. The history of tree spanner algorithms is somewhat similar. Cai and Corneil initiated the study of tree spanners [CC95], and showed that the 1-spanner of a weighted graph, if it exists, oinides with the minimum spanning tree, and therefore an be omputed effiiently. Nevertheless, omputing t-spanners is NP-omplete for t > 1. For unweighted graphs, in the same paper it was shown that the situation is slightly better: there are polynomial time algorithms to find 1-spanners or 2-spanners, if they exist, while finding t-spanners is NP-omplete for t > 4. For unweighted graphs, Emek and Peleg [EP08] and Dragan and Köhler [DK14] show Olog n)- approximation algorithms for finding minimum streth tree spanners. More reently, Fomin et al. [FGvL11] showed that for onstants t and w, t-spanners of treewidth w for bounded degree graphs an be found in polynomial time if they exist [FGvL11] also see [Pap15]). To the best of the authors knowledge no approximation algorithm is known for general metri spaes for t > 1. Geometri tree spanners are an interesting speial ase, where the input is a weighted graph representing the distanes between points from a metri spae. Not muh is known even for this speial ase. Note the signifiane of requiring that the spanner is a tree, as there are many results when other sparse graphs are allowed.) Eppstein [Epp00] asked whether one an ompute the minimum streth geometri tree spanner or the minimum streth hamiltonian path for a planar point set, either exatly or approximately, in polynomial time. Cheon et al. [CHL07] partially answers this question by showing NP-hardness for the deision problem. Eppstein and Wortman [EW05] give a nearly linear time algorithm to find the minimum streth star for a planar point set. As for approximation algorithms, prior to our work, no non-trivial approximation was known even for the ase when the input is a planar point set. Our results imply a PTAS for omputing the minimum streth spanning tree and the minimum streth hamiltonian path of a planar point set provided polynomially bounded spread and onstant streth. Here we seek tree spanners minimizing the maximum multipliative streth, although different variants have been studied before. We refer the reader to [LW08] for a list of different tree spanner problems and a survey of orresponding results.) 2 Note [ABF + 99] states the additive distortion ase is APX-hard, however, Chepoi et al. [CDN + 10] noted that the proof also implies the same for the multipliative distortion for a smaller onstant. 3 There is a different line of researh for embedding a graph into a given tree or graph), see for example [KRS04, FFL + 13, NR17]. We emphasize that the goal of this paper is different as here we look for the best possible tree to embed into. Also, note we fous on multipliative distortion. See [HIL98,ABF + 99] for results on additive distortion. 2

4 1.2 Our results In this paper, we onsider the problems of embedding a general metri spae into a tree, and finding the minimum streth tree spanner of a metri spae. We give approximation algorithms whose running times are parameterized with respet to: δ opt, the minimum distortion or streth);, the spread of X; and λ, the doubling dimension of X. Our main result is an algorithm to embed a general metri spae X, d X ) into a tree with vertex set X. 4 Theorem 1.1. Let X be an n-point metri spae, with doubling dimension λ and spread. Also, let δ opt be the minimum distortion of any embedding of X into any tree with vertex set X. For any 0 < ε < 1, there is an n 2 /ε) Oδopt/ε))2λ+1 time algorithm to ompute a 1 + ε)δ opt distortion embedding of X into a tree with vertex set X. To obtain the above result we first show how to ompute a 1+ε)-approximation to the minimum distortion embedding into a tree on vertex set X with bounded degree whih may be of independent interest). Then it is argued our bounded doubling dimension assumption implies that a tree with arbitrary degree an be embedded into a tree with bounded degree with distortion at most 1 + ε. The result of Gupta [Gup01], whih shows that Steiner verties an help only up to a fator of eight in the distortion see Lemma 2.3), implies that the output of the algorithm of Theorem 1.1 is also a onstant fator approximation for embedding into a tree when Steiner verties are allowed. Corollary 1.2. Let X be an n-point metri spae, with doubling dimension λ and spread. Let δ opt be the minimum distortion of any embedding of X into any tree. For any 0 < ε < 1, in n 2 /ε) Oδopt/ε))2λ+1 time one an ompute an 8 + ε)δ opt distortion embedding of X into a tree. Our approah an be adapted to ompute tree spanners of finite metri spaes. Here, the input is a finite metri spae, and the tree spanner must be hosen from the set of all spanning trees of the omplete graph representing the metri spae. Note this theorem is different from Theorem 1.1 as the weight of an edge x, x ) in the tree spanner, for any x, x X, must be equal to d X x, x ). Theorem 1.3. Let X be a metri spae with doubling dimension λ and spread. Let δ opt be the minimum possible streth of any spanning tree of X. For any 0 < ε < 1, there is an n 2 /ε) Oδopt/ε))2λ+1 time algorithm to ompute a 1 + ε)δ opt -tree-spanner. Note the above theorem only onsiders spanning trees from metri omplete graphs. We an strengthen this result to an algorithm for omputing tree spanners for general weighted graphs, however, the running time depends on the maximum allowable degree for the tree spanner. Theorem 1.4. Let G = X, E, w) be a weighted graph, and let X, d X ) be its shortest path metri spae. Let λ and denote the doubling dimension and spread of X, d X ), respetively, and let deg > 0 be some integer. Let δ opt be the minimum possible streth of any spanning tree of G of maximum degree at most deg. For any 0 < ε < 1, there is an n 2 /ε) logdeg)oδopt/ε))2λ+1 time algorithm to ompute a 1 + ε)δ opt -tree-spanner with maximum degree at most deg. Note that Theorem 1.1 gives a PTAS for the minimum distortion embedding of a finite metri spae X, d X ) into a tree on vertex set X, provided that δ opt and λ are onstants, and that is polynomially bounded. Under the same set of onditions, Corollary 1.2 gives a onstant fator approximation algorithm for embedding X into any tree. Again, under the same onditions, Theorem 1.3 gives a PTAS for omputing the minimum streth geometri tree spanner, and Theorem 1.4 gives a PTAS for omputing the minimum streth bounded degree tree spanner of a weighted graph. 4 For all our results we atually prove a stronger running time bound. Namely the Oδ opt/ε)) 2λ+1 term in the exponent an instead be written as log1/ε)1/ε)oδ 2 opt/ε)) λ. In the theorem statements, however, we prefer a less luttered form, as it allows one to more learly see the rough dependene on eah parameter. 3

5 Outline. After overing basi bakground in Setion 2, we give an overview of our approah in Setion 3. In Setion 4 we present our main result for approximating minimum distortion embeddings of metri spaes into bounded degree trees. We then show how to remove the bounded degree assumption in Setion 5, by proving that with 1 + ε) distortion, any tree an be embedded into a tree whose degree is bounded by a funtion only depending on ε and the doubling dimension. Finally, in Setion 6, we show that our algorithm an be adapted to find tree spanners by formulating the problem in a more general setting. 2 Preliminaries Graphs and metris. We use G = V, E, w) to denote an undireted graph with vertex set V, edge set E, and positive edge weight funtion w : E R +. The shortest path metri V, d G ) of a graph G is defined by the distane funtion d G : V V R 0, where d G u, v) is the length of the shortest u-to-v path in G. For eah u V, we define adj G u) to be a list of all inident edges to u in G. Throughout the paper we use to denote the spread of the given finite metri spae X, d X ), that is = max x y X d X x, y))/min x y X d X x, y)). Given a metri spae X, d X ), a point x X, and a radius r R + {0}, the ball Bx, r) is the subset of all points of X whose distane to x is at most r. The doubling dimension of a metri spae X, d X ) is the smallest λ R + suh that for any r R +, eah ball of radius r an be overed by at most 2 λ balls of radius r/2. A metri spae is alled doubling if λ is bounded by a onstant independent of the size of the problem). The following lemma of Gupta et al. [GKL03] is helpful in the analysis in this paper. Lemma 2.1 [GKL03], Proposition 1.1). Let X, d X ) be a metri with doubling dimension λ, and let X X. If all pairwise distanes in X are at least l, then any ball of radius R in X ontains at most ) 2R λ l points of X. 5 Embeddings and distortion. An embedding of a metri spae X, d X ) to a metri spae Y, d Y ) is an injetive map f : X Y. The ontration f and the expansion e f of f are defined as f = max x,y X,x y d X x, y) d Y fx), fy)), and e f = max x,y X,x y d Y fx), fy)). d X x, y) An embedding is alled non-ontrating if its ontration is at most one. The distortion of f is defined as δ = f e f. Often in this paper we onsider the identity map as an embedding from a metri spae X, d X ) to the shortest path metri X, d T ) of a tree T = X, E T, w T ). To simplify notation, in these ases, we drop f and ompare x-to-y distane in X, denoted by d X x, y), with the x-to-y distane in T, denoted by d T x, y). Also to simplify, we refer to the identity map X, d X ) to X, d T ) as the embedding defined by T. We use δ opt X) to refer to the smallest possible distortion for embedding X into any tree. When it is lear from the ontext we use the same notation, δ opt X), to refer to the smallest possible distortion for embedding X into any tree with vertex set X. We use δ opt X, deg) to refer to the smallest possible distortion for embedding X into any tree of maximum degree at most deg. Sine distortion is sale invariant a non-ontrating embedding of expansion δ opt X) always exists, and throughout the text we assume we are looking for a suh an embedding. We found the following lemma helpful when working with embeddings between shortest path metris of graphs. 5 Note that λ in their paper is the doubling onstant, whereas in this paper it denotes the doubling dimension. 4

6 Lemma 2.2 [KRS04], Proposition 2.3). Let G = V G, E G ) and H = V H, E H ) be two positively weighted undireted graphs, and let d G and d H be their shortest path metris, respetively. Let f : V G V H be a bijetion. Then the expansion of f is ahieved by an adjaent pair u, v V G, and the ontration of f or the expansion of f 1 ) is ahieved by an adjaent pair x, y V H. In this paper, we onsider embedding into trees both when Steiner verties are allowed and when they are not allowed. The following Lemma of Gupta, ensures that the optimal tree metris for these two problems differ up to a fator of at most eight. Lemma 2.3 [Gup01], Theorem 1.1). Given a tree T = V, E, w ) with shortest path metri d T, and a set of required verties V V, there exists a tree T = V, E, w) with shortest path metri d T suh that for all x, y V, 1 d T x,y) d 8. Moreover, T an be omputed in polynomial time. T x,y) Tree spanners. Let G = V, E G, w G ) be a graph, and let T = V, E T, w T ) be a spanning tree of G, where w T is the restrition of w G to E T. Let e = u, v) E G. The streth or dilation) of the edge e is defined as str T e) = d T u, v)/d G u, v). Note that d T u, v) d G u, v). The streth or dilation) of T is then defined as the maximum streth of the edges in E G, str T = max e EG str T e). By Lemma 2.2, the identity map is a map of distortion str T from V, d G ) to V, d T ). If str T = t, we say that T is a t-tree spanner of G. Hene, finding the minimum streth spanning tree is equivalent to finding the spanning tree into whih the identity map has the lowest distortion. Let X, d X ) be a metri spae, and let T = X, E T, w T ), where w T is the restrition of d X to E T. Let x, y X X. The geometri streth or dilation) of the pair x, y) is defined as str T x, y) = d T x, y)/d X x, y). Again, note d T x, y) d X x, y). The geometri streth of T is then defined as str T = max x,y X & x y str T x, y). If str T = t, we say that T is a geometri t-tree spanner of X. Alternatively, let G X = X, E X, d X ) be the omplete graph over X, where for eah x, x X, the weight of edge x, x ) is d X x, x ). Then, for any tree T with vertex set X, the geometri streth of T with respet to X is equal to the streth of T as a spanning tree of G X.) When it is lear from the ontext, we sometimes drop the adjetive geometri. 3 Overview Here we sketh our algorithm and its analysis for embedding an n-point metri spae X, d X ) into a tree with vertex set X, whih by Lemma 2.3 will also serve as a sketh for the ase when Steiner verties are allowed. Our algorithms for the tree spanner ases follow a similar high level approah as skethed here, but require enforing a set of additional onstraints on edge weights). The details of these onstraints and their enforement an be found in Setion 6. Throughout, for any x X the term point is used when referring to x in the metri spae X, d X ), and the term vertex when referring to x in the tree. Here we assume we are given a value δ suh that δ δ opt, where δ opt is the minimum distortion of any embedding of X, d X ) into a tree with vertex set X). Ultimately our atual algorithm performs an exponential searh to approximately find δ opt, where the proedure skethed below an be seen to fail and hene return false if δ < δ opt. Moreover, assume the spread of X, d X ), denoted by, is polynomially bounded, that δ is a onstant, and X is doubling. Under these onditions, we desribe a polynomial time algorithm to ompute an embedding of X into a tree with Oδ) distortion. Our atual algorithm ahieves 1 + ε)δ distortion, though for simpliity here we are satisfied with this weaker guarantee. As distortion is sale invariant, as remarked in the previous setion, we an restrit our attention to non-ontrating embeddings where the expansion is at most δ. Moreover, sale invariane also implies that we an assume the smallest distane in X, d X ) is 1, and hene the largest distane 5

7 is. As the expansion is at most δ, this implies we an restrit our attention to trees with edge weights in the interval [1, δ ]. Finally, to make things simpler we assume all edge weights are integers whih is valid sine we are only seeking a onstant fator approximation). We start by desribing a more omprehendible version of the algorithm ontaining many of the key ideas, though with an exponential running time. Then we modify the algorithm step by step to obtain a polynomial running time. Stithing loal views. For eah point x X, our algorithm tries to enumerate all possible loal views of what a distortion at most δ embedding ould look like when standing at the vertex x i.e. the image of point x). Then, our algorithm tries to stith together these views eah ontaining only partial information of the tree) into a tree T on X with Oδ) distortion. As a first attempt, we define a loal view at a vertex x to ontain preise information about the loation of all other verties relative to the vertex x. Speifially, a view V x at a vertex x inludes the following information from the tree of an at most δ distortion embedding): 1) The degree of x. 2) For eah y X: a) the branh of x leading to y, and b) the distane of x to y. Figure 1-left shows a possible view at a vertex x and a possible view at vertex y on a tree with vertex set {a, b,..., h, i, x, y}. Note any embedding of X into a tree T implies a view V x at eah vertex x. In this ase, we also say V x extends to T. Note that a view an extend to more than one tree, as the distane/branh information at one vertex is not suffiient to uniquely reonstrut a tree. Figure 1-right shows a tree that is an extension of both of the views at x and y) on the left. Figure 1: Left: views at x and y, right: a tree that is an extension of both views. To keep the figure readable, unweighted distane are used, though in general weights are allowed. We now formalize the notion of stithing. For a given view V x at a vertex x, for every z X define i) b x z) to be the branh of x that leads to z aording to V x, and ii) d x z) to be the x, z-distane aording to V x. For a given view V y at another vertex y, similarly define b y z) and d y z). Let b = b x y) denote the branh label of y in V x, and let b = b y x) denote the branh label of x in V y. Intuitively, we say that V x and V y are stithable if when we identify the labels b and b, all piees of information in V x and V y look onsistent. Speifially, 1) d x y) = d y x). Call this value l i.e. the length of the edge x, y)). 2) For any z X, a) b x z) = b if and only if b y z) b, and b) if b x z) = b then d x z) = d y z) + l, otherwise d x z) = d y z) l. For example, the views in Figure 1 are stithable, and the stithed result is shown in Figure 2-left. The stithing operation tells us how to build one edge of our desired tree. Next, by stithing another view to this edge one obtains a larger subtree see Figure 2-right). By ontinually stithing together more and more views, our ultimate goal is to obtain a full tree T on vertex set X. So suppose we suessfully stithed together views into suh a tree. What an be said about the resulting tree this stithing produes? First, it is not hard to see that requiring onsisteny of 6

8 Figure 2: Left: stithing together a view at x and a view at y to build the edge x, y)), right: stithing a view at e to the view at x to build the subtree of x, y, and e). the branh information implies the resulting tree defines a valid embedding i.e. a point annot be mapped to two different verties). Seond, observe that a view entered at some vertex x reords the distane from x to the image of any other z X under this embedding, and so requiring onsisteny of distanes an be shown to imply that the view at z must also reord the same distane to x. This implies that if for eah x X, if loally at the view of x no distane from x was expanded by more than δ, then globally the resulting tree defines an embedding with expansion at most δ, and hene distortion at most δ by our non-ontrating assumption. Thus we restrit our attention to plausible views, where a view at a vertex x is plausible if for all z X, the x-z distane in the view has a value between d X x, z) and δd X x, z). There are many potential views at a vertex x whih are plausible. Though as desribed below, by a number of areful summarization steps we an make the view desriptions ompat enough suh that we an enumerate all possible plausible views. However, deiding whih views to stith together from these lists is still a daunting task. Fortunately, dynami programming an be used to give an algorithm whose running time is polynomial in the number of views. Interestingly, while this dynami programming ultimately works beause our goal is to stith together a tree a struture amenable to dynami programming), the dynami programming we now desribe is not atually done over a tree. Assembling the tree. Provided the set of all plausible views at every vertex, we now desribe a dynami program whih builds a tree T on vertex set X by stithing together appropriate views. To failitate our dynami program, we fix an arbitrary point r X, and root all trees with vertex set X at r. Fixing r allows us to uniquely define the set of desendants for eah view V x at a vertex x). Namely, y X is a desendant of x in V x if the branh of x leading to y aording to V x ) is different from the branh of x leading to r aording to V x ). In other words, y is a desendant in V x if for all extensions of V x to a tree T with root r, y is a desendant of x in T. We denote the set of desendants of x aording to V x by desx, V x ). We emphasize it is possible a vertex y is a desendant of x aording to a view V x and not a desendant of x aording to another view V x. Now we are ready to define our subproblems. For any plausible view V x at any x X, we say Subtreex, V x ) is true if and only if there is a set of views V, one per vertex of desx, V x ), suh that V {V x } an be stithed together to build a tree with vertex set desx, V x ) {x} and root x. The definition of Subtree, ) implies the following reursive algorithm to hek if Subtreex, V x ) is true. Let b 0, b 1, b 2,..., b t be all the branhes of x aording to V x, and let b 0 be the branh that 7

9 leads to r. Subtreex, V x ) is true if and only if there are y 1,..., y t X and views V y1,..., V yt suh that for every i {1,..., t} we have: 1) b i is the branh of x that leads to y i aording to V x ), 2) V x an be stithed to V yi, and 3) Subtreey i, V yi ) is true. Using this reursive relation we an build our dynami programming table to hek if there exists a plausible view V r at the root r suh that Subtreer, V r ) is true. If so, by traing bak through the dynami programming table, we an stith together a set of plausible views to build a tree T with distortion at most δ. Note it is now easy to see that if we had allowed δ < δ opt, in this ase the dynami program would fail, and hene we would know to return false.) Observe that the running time of our dynami program is learly polynomial in terms of n = X and the maximum number of plausible views at any vertex. Unfortunately however, with the urrent definition of a view, the total number of plausible views at a vertex an be exponentially large. A trivial bound on the number of suh views at a vertex x X is, n n) n δ ) n, as there are at most n hoies for the degree of x, and for any other y in X \ {x} there are at most n hoies for its branh, and δ hoies for its distane. Reall the δ bound on the number of distanes follows sine the maximum distane in X, d X ) was, and we are looking for an embedding of distortion at most δ and we assumed integral distanes). In Setion 4 we prove that a doubling tree metri embeds into a bounded degree tree metri with onstant distortion. Therefore, if our goal is to ahieve a onstant fator approximation, then we an assume that the degree of our target tree is bounded by a onstant. This redues our bound on the number of plausible views at any vertex to O1) O1)) n δ ) n = O )) n, as we assumed δ is a onstant. At this point the number of views, and therefore the running time of our dynami program, is still exponential in n. Note however that up until the point we assumed the tree degree was onstant, our algorithm had atually been exat. Thus we an now take advantage of the extra slak of moving to an approximation, to drastially improve the running time. Hierarhial nets. To redue the above running time we have no hoie but to make the views more onise. To that end, for eah point x X, we hoose a subset I x X, and inlude branh/distane information only for the points of I x instead of all of X) in any view at x. We say a vertex y is visible from x if y I x. We now argue that if one selets the visible verties for eah view arefully, then only a logarithmi number is suffiient to guarantee that the resulting assembled tree of plausible views has Oδ) distortion. 6 Now lets figure out how to onstrut I x. This subset of X will still somehow need to approximately apture the distane information from all y / I x to x. Suppose that for any y / I x, we guarantee that there is some z I x suh that d X y, z) d X x, y)/ for some onstant > 1. Then in this ase up to a onstant fator d X x, z) d X x, y), and so potentially the information reorded for z an be used as a proxy for that of y. For example, if d X y, z) d X x, y)/2, then by applying the triangle inequality twie) we have d X x, z) [1/2)d X x, y), 3/2)d X x, y)].) Ultimately, however, we need to approximate the distane from y to x in the tree, not in the input metri spae. 6 For our dynami program to work, it is ruial these views determine the branh information for all verties. However, we now fous only on preserving distanes, as we an prove this implies we an determine branhes exatly. 8

10 Assuming that the x-z and y-z distanes are not ontrated and expand by at most δ when going from X, d X ) to the tree, then by simply hanging our requirement to d X y, z) d X x, y)/δ), we assure only a onstant fator distane error in the tree. So how do we ensure that the x-z and y-z distanes are not ontrated and expand by at most δ? Well, sine z is visible to x, the plausibility of our urrent view at x ensures this for the x-z distane. For the y-z distane we then should ensure z is also visible from the view at y. In short, for any y / I x, we need to guarantee there is a z I x that is i) suffiiently lose to y, and ii) is visible from y. We now define I x sets whih ahieve this goal while being onise. Figure 3: Left: hierarhial nets around x net points are red, x is blue), right: z is used to estimate the distane to y in x s view. To onstrut the I x we use r-nets, a standard geometri tool, where an r-net is any subset of X suh that i) pairs of net points are at least r apart and ii) every point in X has distane at most r to its nearest net point. The above disussion then implies that I x should be onstruted suh that for any y X it ontains y s nearest net point from an r-net of X where up to a onstant) r = d X x, y)/δ. Now we want I x to be small, so we annot afford to build a ustom radius net for every possible distane to x. Thus instead we bin distane by fators of δ. Speifially, we onstrut a set of nested nets: X = X 0 X 1... X logδ, where X s is a δ s -net. Note δ log δ = is the largest radius we need to onsider as is the largest distane in X, d X ). Also, suh a nested set of nets an be easily omputed with the standard greedy k-enter algorithm.) So onsider any y X, where d X y, x) lies somewhere in the interval [δ s+1, δ s+2 ], for some integer s. Then I x should be onstruted so that it inludes y s nearest net point in X s as d X y, x) may be as small as δ s+1 ). All points whose distane to x lie in this range are ontained in the ball Bx, δ s+2 ), and hene their nearest X s net points are ontained in Bx, 2δ s+2 ). Thus in general I x is onstruted by inluding all net points from X s ontained in Bx, 2δ s+2 ), for all values of s. Intuitively, I x is thus a net of points whose density exponentially dereases with respet to the distane from x see Figure 3). Construt the I x sets as desribed above for all x X. Now fix some I x, and for any y / I x, onsider its nearest neighbor in all different sale nets. Speifially, let z s be the nearest neighbor of y in X s. By onstrution all these nearest neighbors are visible from y i.e. are in I y ). Let t be the smallest index suh that z t is also visible from x. Note t is well defined as the points in X logδ are visible to everyone.) It an be shown that for this hoie of z t in partiular, beause z t 1 is not visible from x), that z t is suffiiently lose to y relative to the distane to x), and thus z t is the point we sought above visible to both x and y and) whih guarantees our desired properties. The only question that remains, is how big is I x? Sine X is doubling and δ is a onstant, there are O1) points from X s inside eah ball Bx, 2δ s+2 ). This follows from Lemma 2.1 and the paking property of nets, i.e. property i) above.) Therefore, the total size of I x is bounded by Olog δ ), the number of onentri balls. Note that log δ = Olog ) if δ > 2, whih we an assume as a onstant fator approximation suffies for the overview.) With these oniser views, 9

11 the number of plausible views per vertex goes down to O )) Olog ) = Olog ), whih readily implies an algorithm whose running is polynomial in n and quasi-polynomial in. Anhors and mile markers. We managed to redue the number of visible verties in eah view to Olog ), thus obtaining an algorithm with quasi-polynomial dependene on the spread. Getting a polynomial dependene on the spread by reduing the number of visible verties in eah view to a onstant seems impossible. Thus alternatively, we now seek to improve this dependene by storing less information about the distanes to eah visible vertex. At first blush, the solution may seem obvious. Just reord distanes approximately rather than exatly, sine our solution is already approximate beause instead of mapping eah point we only mapped its losest net point at an appropriate sale. Speifially, if a view V x is mapping a sale s net point y, then reord the distane from x to y in the image up to a fator of roughly δ s. This approah however has a fatal flaw, as deiding whether views an stith together beomes ambiguous, espeially over relatively short edges. Suppose the view at x laims the distane to y in the tree is in between 10δ s and 11δ s. As we walk from x to y in the tree, at some point our estimate of the distane to y in the urrent view will have to be dereased otherwise we never reah y). The issue is that in our dynami program as we stith views, sine we don t atually know the tree struture, there is no way to know when this update should happen. Speifially, our dynami program must try both long and short edges on this path. If it tries an edge that is longer than δ s, well then it knows the estimate must be dereased at the next view. However, if the next edge is muh smaller than δ s then knowing whether to update or not means knowing where in the range [10δ s, 11δ s ] the distane to y lies, i.e. we are bak to needing to know the distane exatly. In other words, if we walk down a long path with short edges, the views aross eah edge look onsistent, but by the time we reah y something will have gone wrong. a Figure 4: Anhor yellow square), beaon rings blue irles), and x-to-y mile marks red triangles). To resolve this issue, rather than reording the exat distane to the image of eah net point, instead we fix an arbitrary vertex a X, alled the anhor, and for any given view V x entered at a vertex x we only reord the distane from x to a exatly. Note that to hek if two views aross a given edge in the tree are onsistent with respet to the anhor, we just verify that their laimed distanes from the view enters to the anhor differ by exatly the length of the edge. Note whether the distane should go up or down, depends on whether we are walking towards or away from the anhor, and hene we also reord the branh of the anhor.) Now onsider a tree T, and for a given integer s log δ, imagine plaing a set of onentri rings around the anhor a, with radii iδ s for all integers i 0 see Figure 4). Call these beaon rings of sale s and onsider the loations where these rings ross T. For any verties x, y we define their sale s approximate distane to be the number sale s beaon rings on the unique x, y-path in T. As 10

12 a simple analogy, when driving from point A to point B on the highway, if one reords the number of mile markers that get passed, then one will know the distane from A to B, at the resolution of a mile. Of ourse, our algorithm does not know T a priori, but when stithing together two views V x and V y, the number of rings that ross the resulting edge x, y) an be omputed from the exat distane of x and y to the anhor without knowing T ). Thus, we an ahieve an approximate version of our stithing definition. In a view V x at x, we therefore register the exat distane to the anhor point, and for eah visible sale s net point we register its distane from x with only δ s auray by reording the number of beaon ring of sale s rossings on its path from x). Sine any visible sale s net point has distane Oδ s+2 ) from x, as δ is a onstant, there are O1) hoies for its distane estimate from x. As there are Olog ) visible points from x, there are O1)) Olog ) hoies for the branh/distane information of all visible points from x. Moreover, there are O ) hoies for the branh/distane of the anhor point, and O1) hoies for the degree of x. Overall, the number of plausible views at vertex s is bounded by, O1)) Olog ) O ) O1) = Opoly )). Reall that ultimately we list the set of all possible plausible views at eah of the n verties in X, and then run a dynami programming algorithm whose running time is polynomial in the total number of views. Thus, overall our running time is Opolyn )). Tehniques of this paper. The idea of enumerating seleted piees of information about the embedding and ombining these piees using dynami programming over an amenable struture suh as a tree or line, has a long tradition in the embedding ommunity. See for example [KRS04, BDG + 05, FFL + 13, NR15, NR17], whih inludes previous works by the authors. However, whih piees of information to onsider and how to apply the dynami programming is problem speifi, and is what distinguishes these result from one another. Thus it is important to note that while for onsisteny we adopt the terminology of views previously used by the authors in [NR15, NR17], the information ontained in these views differs substantially. Moreover, the main idea of defining approximate distane relative to anhor points is new, and has the potential for future appliations as well as potentially improving/simplifying previous results). Additionally, in these listed previous works the target struture a tree or line) was known and fixed though whih points map to whih verties was not), and so the dynami programming was more natural. Interestingly in our ase, as the tree struture is not fixed in advane, our dynami programming is not done over the tree, though still manages to ompute it in the end. 4 Bijetive embedding into trees In this setion, we onsider the problem of embedding a metri spae X, d X ), with doubling dimension λ and spread into a weighted tree T = X, E T, w T ) with vertex set X i.e. defining a bijetion), and maximum degree deg. We use δ opt X, deg) to denote the minimum ahievable distortion of suh an embedding. We show a 1 + ε)-approximation algorithm for finding this optimal embedding Theorem 4.13). Theorem 1.1 is then immediately implied from Theorem 4.13 and Corollary Setup During this setion assume that the smallest distane in X is one, so the largest distane is. This an be ensured by saling X. Let δ δ opt X, deg), and let a be an arbitrary fixed point of X, 11

13 alled the anhor. Let {x 0 } = X S+1 X S... X 0 = X, where X s is a maximal subset of points with mutual distanes at least δ s, S = log δ ), and x 0 X is an arbitrary point. For tehnial reasons we extend these sets to be defined for negative values of s, where X s = X for any negative value s. A point x X is alled a sale s point if X s is the sparsest net in the sequene that ontains x. The sale s nearest neighbor of a point x X is denoted by nn s x), and is defined to be the losest point of X s to x. Note that by the maximality of X s, we have that d X x, nn s x)) < δ s. Therefore, for example, nn s x) = x for all x X and for any s 0. We build a tree into whih X an be embedded with nearly optimal distortion in this setion. Part of the proess is to find the edge weights of this tree. The following lemma limits the range of the searh spae for the weight values, while ensuring a bounded approximation fator. Lemma 4.1. Let G = V, E, w) be a graph with minimum edge weight one, and let d G be the shortest path metri of G. For any 0 < σ 1, G an be embedded to a graph G = V, E, w ) whose edge weights are multiples of σ with distortion at most 1 + σ. Proof. Let G = V, E, w ) be the graph obtained from G by setting the weight of eah edge e to w e) = we)/σ σ. We bound the distortion of the identity map from V, d G ) to V, d G ). Sine w e) we) the map is non-ontrating. Furthermore, for eah e = x, y) E, we have w e) we) + σ we) 1 + σ ) we) 1 + σ), we) as we) is at least one. Hene, by Lemma 2.2, the distortion is at most 1 + σ. 4.2 Views The key building bloks used in our algorithm are views, whih are olletions of relevant information about what the embedding looks like around the images of points in X. This information is limited in sope so that it an be guessed by our algorithm. Speifially the view at a vertex x vertex signifying it is the image of the point x), speifies the degree of the image of x, the loation of the anhor vertex relative to the image of x, and approximate relative loations of the images of all sale s points that are at distane Oδ s+2 ) from x in the preimage. To desribe the loation of the anhor vertex we speify the branh edge) adjaent to vertex x whih leads to the anhor as well as the exat distane to the anhor. Similarly, for eah vertex y whih is the image of one of these sale s points, we speify the branh, but rather than speifying the distane to y exatly we just reord the number of beaon ring rossings as desribed in the overview) of the x-to-y path in the image. Formally, a view V x = deg x, A b x, A d x), {b s x, r s x)} L s S ) at x X is a tuple with parameters deg, σ,, δ) where eah omponent is defined as follows. 1) An integer, deg x {1, 2,..., deg}. 2) a) A b x {1, 2,..., deg x } {null}. b) A d x {0, σ, 2σ,..., δ /σ σ}. 3) For eah L s S a) b s x : X s Bx, δ s+2 ) {1,..., deg x } {null}. b) r s x : X s Bx, δ s+2 ) {0, 1, 2,..., δ 3 }. 12

14 In the definition above we set L = log δ + 3), as the minimum distane in X is one, and so Bx, δ s ) is empty for s L. Throughout the text is onsidered to be a suffiiently large value, whih will be speified later, and intuitively ats as a dial ontrolling the approximation quality of the embedding. Whenever, it is lear from the ontext, we drop the speifiation of the parameters to simplify the explanation. We say that a point y is visible under V x at sale s, if it is in the domain of b s x and r s x. We say that a point y is visible under V x if there exists an L s S suh that y is is visible under V x at sale s. The first step of our algorithm is to list the set of possible views at every point of X. The following lemma bounds the number of suh views. Lemma 4.2. There are at most 1 σ )Olog δ deg))2δ2 ) λ different views at any x X. Moreover, a list of these views an be onstruted in time linear in the list size. Proof. We enumerate all possibilities for a view V x = deg x, A b x, A d x), {b s x, rx)} s L s S ) at x. For the first parameter deg x in the tuple, there are at most deg possibilities. For A b x, A d x) there are at most deg + 1)δ /σ + 1) possibilities. So what remains is to bound the possibilities for the b s x and rx s funtions. For eah point y X s Bx, δ s+2 ) and for eah sale L s S, there are at most deg+1 possibilities for b s xy), and at most δ 3 +1 possibilities for rxy). s By Lemma 2.1, there are at most 2δ s+2 /δ s ) λ = 2δ 2 ) λ points in X s Bx, δ s+2 ). Therefore, for any L s S, there are at most deg + 1) δ 3 + 1)) 2δ2 ) λ number of hoies for b s x and rx. s There are S L + 1 sales overall, so the total number of views at x is at most: deg deg + 1)δ /σ + 1) deg + 1) δ 3 + 1)) 2δ2 ) λ) log δ ) log δ +3) +1 = σ deg δ) Olog δ ))2δ2 ) λ) = σ δ Olog δ deg) log δ ))2δ2 ) λ) = 1 σ )Olog δ deg))2δ2 ) λ. Feasibility/Plausibility. From a list of views generated by Lemma 4.2, we would like to only keep the views that an be ompleted into feasible trees on X, that is trees that define nonontrating embeddings of expansion at most δ. To formally desribe our desired properties for suh views, we define restrition of embeddings, and extensions of views as follows. Let T = X, E T, w T ) be a tree, and let x X. We define the restrited view of T around x to be the view V x = deg x, A b x, A d x), {b s x, rx)} s L s S ) speified as follows. 1) deg x = deg T x), where deg T x) denotes the degree of x in T. 2) A d x = d T a, x)/σ σ. 3) Fix a global ordering on the edges of T, and let l : adjx) {1,..., deg x } be the bijetion that for every 1 i deg x ) assigns the ith element of adjx) to i. We have: a) If x = a then A b x = null, otherwise A b x = le), where e is the first edge of the x-to-a path in T. b) For eah L s S and y X s Bx, δ s+2 ), i) If x = y then b s xy) = null, otherwise b s xy) = le), where e is the first edge of the x-to-y path in T. ii) r s xy) = d T x, a)/δ s + d T y, a)/δ s - 2 d T u, a)/δ s, where u is the vertex in T that is losest to a on the path from x to y, i.e. u is the lowest ommon anestor of x and y if the root is a. Roughly speaking, up to the δ s fators this is the distane between x and y as d T x, u) + d T y, u) = d T x, y)) 13

15 Note that the restrited view of T around x is uniquely defined for fixed values of a, deg, δ,, and σ. If V x is the restrited view of T around x, we say that T is an extension of V x. Note that a view an possibly be extended to several different trees. A view is alled feasible if it an be extended to a feasible tree. Suh an extension is alled a feasible extension of the view. Ideally, we would like to be able to disregard all non-feasible views from the lists omputed by Lemma 4.2. However, it seems impossible to determine feasibility by merely examining a view in isolation from other views. Fortunately, the following weaker ondition on views, whih an be tested quikly, suffies for our algorithm. We say that a view V x is plausible if for any L s S and any y Domr s x), we have d X x, y) 2δ s r s xy) δ s δd X x, y) + 2δ s. Note radii of suessive beaon rings differ by δ s, and hene the need for the additive fator of 2δ s, as this is longest a shortest path an be without rossing a beaon ring. Moreover, observe that at a suffiiently small sale the additive error in the above definition will beome a multipliative one. Intuitively a view is plausible if non-feasibility of the view annot be onluded by examining it in isolation from other views. The following lemma ensures that the plausibility of a view an be heked effiiently. Lemma 4.3. There is an O2δ 2 ) λ log δ )) time algorithm to hek the plausibility of a view. Proof. Let V x = deg x, A b x, A d x), {b s x, rx)} s L s S ) be a view at x X. For eah L s S, we have Domrx) s 2δ 2 ) λ by Lemma 2.1). For eah element in Domrx) s the plausibility ondition an be heked in onstant time. Therefore, the total running time for heking plausibility is O2δ 2 ) λ S L)) = O2δ 2 ) λ log δ )). The partition funtion. Although a view provides information only about the images of a relatively small subset of X, more an be dedued from it. Speifially, a view V x at x uniquely determines the onneted omponents of T \{x} for every feasible extension T of V x if any exists). Note that a priori it is not even lear that these onneted omponents must be the same in different feasible extensions of V x. Lemma 4.4. Let V x = deg x, A b x, A d x), {b s x, r s x)} L s S ) be any view at x X. There is an algorithm to ompute a partition P of X\{x} in On log δ )) time with the following property. For every feasible extension T of V x, and any y, z X\{x}, y and z belong the the same onneted omponent of T \{x} if and only if y and z belong to the same set of P. Proof. Let y X, and let s be the smallest sale suh that b s x and r s x at on nn s y), where s is well defined as b S x ats on all X S. We show that y and nn s y) must belong to the same onneted omponent of T \{x} in any feasible extension T of V x. Note that sine b s x ats on nn s y), the onneted omponent ontaining nn s y) is speified by V x, and so the lemma statement will then follow. By the definition of nn s y) and the feasibility of T, we have: d X y, nn s y)) δ s d T y, nn s y)) δ s+1. Suppose, to derive a ontradition, that y and nn s y) belong to different onneted omponents of T \{x}. That is, the path from y to nn s y) in T ontains x. Consequently, d T x, y) d T y, nn s y)) δ s+1. 14