Doubling Metrics Research Articles

MapReduce algorithms for robust center-based clustering in doubling metrics

Clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is the (k,ℓ)-clustering problem, where, given a pointset P from a metric space, one must determine a subset S of k centers minimizing the sum of the ℓ-th powers of the distances of points in P from their closest centers. This formulation covers the well-studied k-median (ℓ=1) and k-means (ℓ=2) clustering problems. A more general variant, introduced to deal with noisy pointsets, features a further parameter z and allows up to z points of P (outliers) to be disregarded when computing the sum. We present a distributed coreset-based 3-round approximation algorithm for the (k,ℓ)-clustering problem with z outliers, using MapReduce as a computational model. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by its doubling dimension D. Remarkably, for D=O(1), our algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term O(γ) away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where γ can be made arbitrarily small. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for metrics with constant doubling dimension.

Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Approximation Algorithms for Fair k-median Problem without Fairness Violation

The fair k-median problem is one of the important clustering problems. The current best approximation ratio is 4.675 for this problem with 1-fairness violation, which was proposed by Bercea et al. [APPROX-RANDOM'2019]. To our best knowledge, there is no available approximation algorithm for this problem without any fairness violation in doubling metrics. In this paper, we consider the fair k-median problem in doubling metrics and general metrics. We provide the first QPTAS for the fair k-median problem based on the hierarchical decomposition and dynamic programming process in doubling metrics. For applying a dynamic programming process to solve this problem, the distances from portals to facilities cannot be directly enumerated since each client may not be assigned to its closest open facility. To overcome the difficulties caused by the fairness constraints, we construct an auxiliary graph and use minimum weighted perfect matching to get the cost between the portals of each block and the ones in its children. In order to satisfy the fairness constraints, we bound the fairness constraints of each open facility in the leaves of the split-tree based on the relation between the subproblem and the subproblems of its children. To obtain the assignment of the given instance and remove the fairness violation, we construct a new b-value min-cost max-flow model based on the set of open facilities. For the fair k-median problem in general metrics, we present a polynomial-time approximation algorithm with ratio O(log⁡k). Our approximation algorithm for the fair k-median problem in doubling metrics is the first result for the corresponding problem without any fairness violation in doubling metrics.

Massively parallel and streaming algorithms for balanced clustering

Clustering is a fundamental problem with a wide range of applications. In this paper, we study a balanced version of the well-known k-center clustering problem, where the objective is to select k centers from a given set of points, and assign to each center at most L of the input points, so as to minimize the maximum distance from a point to the center to which it is assigned. We assume soft capacities, where points are allowed to be selected as center more than once. Motivated by applications involving massive datasets, we study the problem in the massively parallel computation (MPC) and streaming models. In particular, we present a two-round MPC algorithm for the balanced k-center problem, achieving an approximation factor of 5α+2, where α is the approximation factor of the corresponding sequential algorithm. This substantially improves the currently best approximation factor of 32α, available for the problem. We show that the approximation factor of our algorithm can be further improved to 5+ε in all spaces with bounded doubling dimension, including the Euclidean space. We also consider the balanced k-center problem in the streaming model, and present a constant-factor streaming algorithm in any metric space using O(knε) memory, and a factor 5+ε streaming algorithm using O(kpolylogn) memory in doubling metrics.

Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [28] showed that the greedy spanner in \(\mathbb {R}^2 \) admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^2 \) has a separator of size \(O(\sqrt {k}) \) . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in \(\mathbb {R}^d \) for any constant d ≥ 3 as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh [28] by showing that any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^d \) has a separator of size O ( k 1 − 1/ d ). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ -lanky : a geometric graph is τ -lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ -lanky geometric graph of n vertices in \(\mathbb {R}^d \) has a separator of size O ( τn 1 − 1/ d ). We then derive our main result by showing that the greedy spanner is O (1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in \(\mathbb {R}^d \) . Our technique naturally extends to doubling metrics. We use the τ -lanky criterion to show that there exists a (1 + ϵ)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size \(O(n^{1-\frac{1}{d}}) \) ; this result resolves an open problem posed by Abam and Har-Peled [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n -point metric space of doubling dimension d has a separator of size \(O((n^{1-\frac{1}{d}}) + \log \Delta) \) where Δ is the spread of the metric; the factor log ( Δ ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree . Finally, we discuss algorithmic implications of our results.

Light Spanners for High Dimensional Norms via Stochastic Decompositions

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har–Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with \(\tilde{O}(n^{1+1/t^2})\) edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of \(\ell _p\) with \(1<p\le 2\). Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of \(\ell _p\) for \(1<p\le 2\) has an O(t)-spanner with \(n^{1+\tilde{O}(1/t^p)}\) edges and lightness \(n^{\tilde{O}(1/t^p)}\). In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness \(O(n^{1/t})\). We exhibit the following tradeoff: metrics with decomposability parameter \(\nu =\nu (t)\) admit an O(t)-spanner with lightness \(\tilde{O}(\nu ^{1/t})\). For example, metrics with doubling constant \(\lambda \), graphs of genus g, and graphs of treewidth k, all have spanners with stretch O(t) and lightness \(\tilde{O}(\lambda ^{1/t})\), \(\tilde{O}(g^{1/t})\), \(\tilde{O}(k^{1/t})\) respectively. While these families do admit a (\(1+\epsilon \))-spanner, its lightness depend exponentially on the dimension (resp. \(\log g\), k). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

Dimensionality and Coordination in Voting: The Distortion of STV

We study the performance of voting mechanisms from a utilitarian standpoint, under the recently introduced framework of metric-distortion, offering new insights along two main lines. First, if d represents the doubling dimension of the metric space, we show that the distortion of STV is O(d log log m), where m represents the number of candidates. For doubling metrics this implies an exponential improvement over the lower bound for general metrics, and as a special case it effectively answers a question left open by Skowron and Elkind (AAAI '17) regarding the distortion of STV under low-dimensional Euclidean spaces. More broadly, this constitutes the first nexus between the performance of any voting rule and the ``intrinsic dimensionality'' of the underlying metric space. We also establish a nearly-matching lower bound, refining the construction of Skowron and Elkind. Moreover, motivated by the efficiency of STV, we investigate whether natural learning rules can lead to low-distortion outcomes. Specifically, we introduce simple, deterministic and decentralized exploration/exploitation dynamics, and we show that they converge to a candidate with O(1) distortion.

Covering metric spaces by few trees

A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y∈X has a low distortion path in one of the trees. If it has the stronger property that every point x∈X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

Double Metric Resolvability in Convex Polytopes

Nowadays, the study of source localization in complex networks is a critical issue. Localization of the source has been investigated using a variety of feasible models. To identify the source of a network’s diffusion, it is necessary to find a vertex from which the observed diffusion spreads. Detecting the source of a virus in a network is equivalent to finding the minimal doubly resolving set (MDRS) in a network. This paper calculates the doubly resolving sets (DRSs) for certain convex polytope structures to calculate their double metric dimension (DMD). It is concluded that the cardinality of MDRSs for these convex polytopes is finite and constant.

Bounded-degree light approximate shortest-path trees in doubling metrics

We study the Bounded-Degree Light Approximate Shortest-path Tree problem: Given a metric space (V,d) on n points, a root r∈V and a degree bound b, the goal is to find a tree T spanning V with maximum degree at most b, total weight comparable to the weight of a Minimum Spanning Tree of (V,d), and low stretch of tree distances from r to every vertex compared to input distances.Our main result is an algorithm that given a metric (V,d) with doubling dimension k, returns a tree T rooted at r spanning V with degree b and stretch max{O(k)/logb,O(1)} having weight within a constant factor the weight of a MST of (V,d).On the negative side, we show that the trade-off between degree and stretch is unavoidable (up to constant factors in the Big-O) even when the weight of the tree is not taken into account.

A QPTAS for ɛ-Envy-Free Profit-Maximizing Pricing on Line Graphs

We consider the problem of pricing edges of a line graph so as to maximize the profit made from selling intervals to single-minded customers. An instance is given by a set E of n edges with a limited supply for each edge, and a set of m clients, where each client specifies one interval of E she is interested in and a budget B j which is the maximum price she is willing to pay for that interval. An envy-free pricing is one in which every customer is allocated an (possibly empty) interval maximizing her utility. Grandoni and Rothvoss (SIAM J. Comput. 2016) proposed a polynomial-time approximation scheme ( PTAS ) for the unlimited supply case with running time ( nm ) O ((1/ɛ) 1/ɛ ) , which was extended to the limited supply case by Grandoni and Wiese (ESA 2019). By utilizing the known hierarchical decomposition of doubling metrics , we give a PTAS with running time ( nm ) O (1/ ɛ 2 ) for the unlimited supply case. We then consider the limited supply case, and the notion of ɛ-envy-free pricing in which a customer gets an allocation maximizing her utility within an additive error of ɛ. For this case, we develop an approximation scheme with running time ( nm ) O (log 5/2 max e H e /ɛ 3 ) , where H e = B max ( e )/ B min ( e ) is the maximum ratio of the budgets of any two customers demanding edge e . This yields a PTAS in the uniform budget case, and a quasi-PTAS for the general case. The best approximation known, in both cases, for the exact envy-free pricing version is O (log c max ), where c max is the maximum item supply. Our method is based on the known hierarchical decomposition of doubling metrics, and can be applied to other problems, such as the maximum feasible subsystem problem with interval matrices.

Near Isometric Terminal Embeddings for Doubling Metrics

Given a metric space (X, d), a set of terminals $$K\subseteq X$$ , and a parameter $$0<\epsilon <1$$ , we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in $$K\times X$$ up to a factor of $$1+\epsilon$$ , and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion $$1+\epsilon$$ and space s(|X|) has its terminal counterpart, with distortion $$1+O(\epsilon )$$ and space $$s(|K|)+n$$ . In particular, for any doubling metric on n points, a set of k terminals, and constant $$0<\epsilon <1$$ , there exists Moreover, surprisingly, the last two results apply if only the metric space on K is doubling, while the metric on X can be arbitrary.

Local routing in a tree metric 1-spanner

Solomon and Elkin (SIAM J Discret Math 28(3):1173–1198, 2014) constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight \((1+\epsilon )\)-spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after \(O(\log n)\) hops and requires \(O(\varDelta \log n)\) bits of storage per vertex where \(\varDelta \) is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.

A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in Talwar (STOC 2004) and Bartal, Gottlieb, Krauthgamer (STOC 2012). However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

The Parameterized Hardness of the k-Center Problem in Transportation Networks

In this paper we study the hardness of the $$k$$ -Center problem on inputs that model transportation networks. For the problem, a graph $$G=(V,E)$$ with edge lengths and an integer k are given and a center set $$C\subseteq V$$ needs to be chosen such that $$|C|\le k$$ . The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the $$k$$ -Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and the pathwidth p. Moreover, under the exponential time hypothesis there is no $$f(k,p,h)\cdot n^{o(p+\sqrt{k+h})}$$ time algorithm for any computable function f. Thus it is unlikely that the optimum solution to $$k$$ -Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized $$(1+{\varepsilon })$$ -approximation algorithm for inputs of doubling dimension d with runtime $$(k^k/{\varepsilon }^{O(kd)})\cdot n^{O(1)}$$ . This generalizes a previous result, which considered inputs in D-dimensional $$L_q$$ metrics.

Reducing Curse of Dimensionality

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find the shortest tour that visits each of a given collection of subsets (regions or neighborhoods) in the underlying metric space. We give a randomized polynomial-time approximation scheme (PTAS) when the regions are fat weakly disjoint. This notion of regions was first defined when a QPTAS was given for the problem in SODA 2010 (Chan and Elbassioni 2010). The regions are partitioned into a constant number of groups, where in each group, regions should have a common upper bound on their diameters and each region designates one point within it such that these points are far away from one another. We combine the techniques in the previous work, together with the recent PTAS for TSP (STOC 2012: Bartal, Gottlieb, and Krauthgamer 2012) to achieve a PTAS for TSPN. However, several nontrivial technical hurdles need to be overcome for applying the PTAS framework to TSPN: (1) Heuristic to detect sparse instances. In the STOC 2012 paper, a minimum spanning tree heuristic is used to estimate the portion of an optimal tour within some ball. However, for TSPN, it is not known if an optimal tour would use points inside the ball to visit regions that intersect the ball. (2) Partially cut regions in the recursion. After a sparse ball is identified by the heuristic, the PTAS framework for TSP uses dynamic programming to solve the instance restricted to the sparse ball and recurse on the remaining instance. However, for TSPN, it is an important issue to decide whether each region partially intersecting the sparse ball should be solved in the sparse instance or considered in the remaining instance. Surprisingly, we show that both issues can be resolved by conservatively making the ball in question responsible for all intersecting regions. In particular, a sophisticated charging argument is needed to bound the cost of combining tours in the recursion. Moreover, more refined procedures are used to improve the dependence of the running time on the doubling dimension k from the previous exp[( O (1)) k 2 ] (even for just TSP) to exp[2 O ( k log k ) ].

A PTAS for the Steiner Forest Problem in Doubling Metrics

We achieve a (randomized) polynomial-time approximation scheme (PTAS) for the Steiner forest problem in doubling metrics. Before our work, a PTAS was given only for the Euclidean plane in [G. Borradaile, P. N. Klein, and C. Mathieu, in FOCS, IEEE Computer Society, 2008, pp. 115--124]. Our PTAS also shares similarities with the dynamic programming for sparse instances used in [Y. Bartal, L. Gottlieb, and R. Krauthgamer, in STOC, ACM, 2012, pp. 663--672] and [T-H. H. Chan and S.-H. Jiang, in SODA, SIAM, 2016, pp. 754--765]. However, extending previous approaches requires overcoming several nontrivial hurdles, and we make the following technical contributions. (1) We prove a technical lemma showing that Steiner points have to be “near” the terminals in an optimal Steiner tree. This enables us to define a heuristic to estimate the local behavior of the optimal solution, even though the Steiner points are unknown in advance. This lemma also generalizes previous results in the Euclidean plane and may be of independent interest for related problems involving Steiner points. (2) We develop a novel algorithmic technique known as “adaptive cells” to overcome the difficulty of keeping track of multiple components in a solution. Our idea is based on but significantly different from the previously proposed “uniform cells” in [G. Borradaile, P. N. Klein, and C. Mathieu, in FOCS, IEEE Computer Society, 2008, pp. 115--124], where techniques cannot be readily applied to doubling metrics.

A $(1+\varepsilon)$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.

Doubling metric Diophantine approximation in the dynamical system of continued fractions

This paper is concerned with the Diophantine properties of the orbits of real numbers in continued fraction system under the doubling metric. More precisely, let φ be a positive function defined on N. We determine the Lebesgue measure and Hausdorff dimension of the setE(φ)={(x,y)∈[0,1)×[0,1):|Tnx−y|<φ(n)fori.m.n},where T is the Gauss map and “i.m.” stands for “infinitely many”.

On Hierarchical Routing in Doubling Metrics

We study the problem of routing in doubling metrics and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with a small amount of routing information stored at each vertex). We say that a metric ( X , d ) has doubling dimension dim(&lt;i&lt;X&lt;/i&lt;) at most α if every ball can be covered by 2 α balls of half its radius. (A doubling metric is one whose doubling dimension dim(&lt;i&lt;X&lt;/i&lt;) is a constant.) We consider the metric space induced by the shortest-path distance in an underlying undirected graph G . We show how to perform (1 + τ)-stretch routing on such a metric for any 0 &lt; τ ≤ 1 with routing tables of size at most (α/τ) O (α) log Δlog δ bits with only (α/τ) O (α) log Δ entries , where Δ is the diameter of the graph, and δ is the maximum degree of the graph G ; hence, the number of routing table entries is just τ − O (1) log Δ for doubling metrics. These results extend and improve on those of Talwar (2004). We also give better constructions of sparse spanners for doubling metrics than those obtained from the routing tables earlier; for τ &gt; 0, we give algorithms to construct (1 + τ)-stretch spanners for a metric ( X , d ) with maximum degree at most (2 + 1/τ) O(dim(X)) , matching the results of Das et al. for Euclidean metrics.