## Fast construction of nets in low dimensional metrics, and their applications (2005)

### Cached

### Download Links

- [valis.cs.uiuc.edu]
- [arxiv.org]
- [arxiv.org]
- [valis.cs.uiuc.edu]
- [valis.cs.uiuc.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | SIAM J. Comput |

Citations: | 98 - 11 self |

### BibTeX

@INPROCEEDINGS{Har-peled05fastconstruction,

author = {Sariel Har-peled and Manor Mendel},

title = {Fast construction of nets in low dimensional metrics, and their applications},

booktitle = {SIAM J. Comput},

year = {2005},

pages = {150--158}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms for the following problems: approximate nearest neighbor search, well-separated pair decomposition, spanner construction, compact representation scheme, doubling measure, and computation of the (approximate) Lipschitz constant of a function. In all cases, the running (preprocessing) time is near linear and the space being used is linear. 1

### Citations

8835 |
Introduction to algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ... the lca of two vertices in T [5]. 2. A ε −1 -WSPD W on the net-tree T, with support for fast membership queries. For each pair we also store the distance between their representatives. Using hashing =-=[14]-=-, membership queries can be answered in constant time. 3. The (3n 2 )-approximation HST H of Section 3.2. The HST H should be augmented with the following features: (a) A constant time access to least... |

801 | An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions
- Arya, Mount, et al.
- 1998
(Show Context)
Citation Context ...l since there are examples of point sets in which the query time is 2 Ω(dim) log n [34], and examples in which the query time is ε −Ω(dim) . 1 Our result also matches the known results of Arya et al. =-=[1]-=- in Euclidean settings. Furthermore, our result improves over the recent work of Krauthgamer and Lee, which either assumes bounded spread [34], or requires quadratic space [33]. The algorithms in [1, ... |

746 | Approximate Nearest Neighbors: Towards Removing the
- Indyk, Motwani
- 1998
(Show Context)
Citation Context ...t-tree. This algorithm allows to boost an A-ANN solution to (1 + ε)-ANN solution in O(log n+log(A/ε)) query time. In Section 4.2 we present a fast construction of a variant of the ring separator tree =-=[29, 33]-=-, which support fast 2n-ANN queries. We conclude in Section 4.3 with the general scheme which is a combination of the previous two. 14s4.1 The Low Spread Case Lemma 4.1 Given a net-tree T of P, a quer... |

391 | Applications of random sampling in computational geometry
- Clarkson
- 1988
(Show Context)
Citation Context ...change O(log n) times in expectation. Thus, this would lead to O(n log 3 n) expected running time. Moreover, a slightly more careful analysis shows that the expected running time is O(n log 2 n). See =-=[16]-=- for details of such analysis. 31s8.2 Constant doubling dimension to arbitrary metric Theorem 8.5 Given a metric (P,ν) of n points having doubling dimension d, and a mapping f : P → (M,ρ), where M is ... |

292 |
Clustering to minimize the maximum inter-cluster distance
- GONZALEZ
- 1985
(Show Context)
Citation Context ...2). Currently, the known algorithms for constructing those nets require running time which is quadratic in n. An alternative way for constructing those nets is by the clustering algorithm of Gonzalez =-=[20]-=-. The algorithm of Gonzalez computes 2-approximate k-center clustering by repeatedly picking the point furthest away from the current set of centers. Setting k = n, this results in a permutation of th... |

262 | The Art of Computer Programming: Seminumerical Algorithms - Knuth - 1981 |

242 | A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields
- Callahan, Kosaraju
- 1995
(Show Context)
Citation Context ...from fixed dimensional Computational Geometry and extend them to finite metric spaces. In particular, Talwar [44] showed that one can extend the notion of well-separated pairs decomposition (WSPD) of =-=[11]-=- to spaces with low doubling dimension. Specifically, he shows that for every set P of n points having doubling dimension dim, and every ε > 0, there exists WSPD, with separation 1/ε and O(nε −O(dim) ... |

214 | Geometric nonlinear functional analysis - Benyamini, Lindenstrauss - 2000 |

213 | Approximate distance oracles
- Thorup, Zwick
- 2005
(Show Context)
Citation Context ... log 2 n + ε −O(dim) n,S = ε −O(dim) n,Q = 2 O(dim) ,κ = 1 + ε)-CRS. (b) (P = 2 O(dim) · poly(n) + ε −O(dim) n,S = ε −O(dim) n,Q = O(dim),κ = 1 + ε)-CRS. For general n-point metrics, Thorup and Zwick =-=[45]-=- obtained a (kn 1+1/k ,kn 1+1/k ,O(k), 2k −1)- CRS, where k ∈ N is a prescribed parameter. The trade-off between the approximation and the space is essentially tight for general metrics. Closer in spi... |

187 | The LCA problem revisited
- Bender, Farach-Colton
- 2000
(Show Context)
Citation Context ...ss for ancestor of given x, when the level is at most 6log n levels below a given ancestor z. Again, Section 3.5 contains more information. (c) A constant time access for the lca of two vertices in T =-=[4]-=-. 3 Caveat: They use a weaker model of computation. 20s2. A ε −1 -WSPD W on the net-tree T, with support for fast membership queries. For each pair we also store the distance between their representat... |

172 |
Optimal algorithms for approximate clustering
- Feder, Greene
- 1988
(Show Context)
Citation Context ...r, the running time of Gonzalez algorithm in this case is still quadratic. Although, in fixed dimensional Euclidean space the algorithm of Gonzalez was improved to O(n log k) time by Feder and Greene =-=[17]-=-, and linear time by Har-Peled [25], those algorithms require specifying k in advance, and they do not generate the permutation of the points, and as such they cannot be used in this case. Our results... |

161 |
Leeuwen. Maintenance of configurations in the plane
- Overmars, van
- 1981
(Show Context)
Citation Context ...through p, then we can compute the Lipschitz constant of p in constant time (i.e., it is the slope of the tangent with largest slope). Here, one can use the data-structure of Overmars and van Leeuwen =-=[38]-=-, which supports the maintenance of convex-hull under insertions, deletions and tangent queries in O(log 2 n) per operation. Indeed, sort the points of P from left to right. Let p1,... ,pn be the sort... |

158 |
Graph spanners
- Peleg, Schaffer
- 1989
(Show Context)
Citation Context ...nner of a metric is a sparse weighted graph whose vertices are the metric’s points, and in which the graph metric is t-approximation to the original metric. Spanners were first defined and studied in =-=[40]-=-. Construction of (1 + ε)-spanners for points in low dimensional Euclidean space is considered in [31, 10]. Using Callahan’s technique [10], the WSPD construction also implies a near linear-time const... |

155 | Bounded geometries, fractals, and lowdistortion embeddings
- Krauthgamer, Gupta, et al.
- 2003
(Show Context)
Citation Context ...t got considerable attention recently is to define a notion of dimension on a finite metric space, and develop efficient algorithms for this case. One such concept is the notion of doubling dimension =-=[2, 27, 23]-=-. The doubling constant of metric space M is the maximum, over all balls b in the metric space M, of the minimum number of balls needed to cover b, using balls with half the radius of b. The logarithm... |

150 | Finding nearest neighbors in growth-restricted metrics
- Karger, Ruhl
- 2002
(Show Context)
Citation Context ... be thought as a generalization of the Euclidean dimension, as IR d has Θ(d) doubling dimension. Furthermore, the doubling dimension extends the notion of growth restricted metrics of Karger and Ruhl =-=[30]-=-. Understanding the structure of such spaces (or similar notions), and how to manipulate them efficiently received considerable attention in the last few years [14, 30, 23, 28, 34, 33, 44]. The low do... |

145 | Cover trees for nearest neighbor
- Beygelzimer, Kakade, et al.
- 2006
(Show Context)
Citation Context ...l() data-structure is not needed since Rel(root) = {root}. Therefore the storage for this ANN scheme is O(n), with no dependency on the dimension. A similar construction was obtained independently in =-=[8]-=-. However, their construction time is O(n 2 ). 4.2 Low Quality Ring Separator Tree Lemma 4.2 One can construct a data-structure which supports 2n-ANN queries in 2 O(dim) log n time. The construction t... |

123 | Navigating nets: simple algorithms for proximity search - Krauthgamer, Lee - 2004 |

112 | Nearest neighbor queries in metric spaces
- Clarkson
- 1999
(Show Context)
Citation Context ...stricted metrics of Karger and Ruhl [30]. Understanding the structure of such spaces (or similar notions), and how to manipulate them efficiently received considerable attention in the last few years =-=[14, 30, 23, 28, 34, 33, 44]-=-. The low doubling metric approach can be justified in two levels. 1. Arguably, non-Euclidean, low (doubling) dimensional metric data appears in practice, and deserves an efficient algorithmic treatme... |

105 |
Decomposable searching problems i: Static-to-dynamic transformation
- Bentley, Saxe
- 1980
(Show Context)
Citation Context ...n the plane. Such a diagram can be computed in O(n log n) time for n points, and a point-location query in it can be performed in O(log n) time. In fact, using the standard Bentley and Saxe technique =-=[6]-=-, one can build a data-structure, where one can insert such upper cones in O(log 2 n) amortized time, and given a query point q in the plane, decide in O(log 2 n) which of the cones inserted lies on t... |

103 |
Lectures on analysis on metric spaces,” Universitext
- Heinonen
- 2001
(Show Context)
Citation Context ...t got considerable attention recently is to define a notion of dimension on a finite metric space, and develop efficient algorithms for this case. One such concept is the notion of doubling dimension =-=[2, 27, 23]-=-. The doubling constant of metric space M is the maximum, over all balls b in the metric space M, of the minimum number of balls needed to cover b, using balls with half the radius of b. The logarithm... |

99 | Distance labeling in graphs
- Gavoille, Peleg, et al.
- 2004
(Show Context)
Citation Context ...air of points, it is possible to compute efficiently an approximation of the the pairwise distance. Thus, ADLS is a stricter notion of compact representation. 5 This notion was studied for example in =-=[19, 18, 45]-=-. In the constant doubling dimension setting Gupta et al.] [23] have shown an (1+ε)-embedding O(log n) of the metric in ℓ∞ . This implies (1 + ε)-ADLS with O(log n log Φ) bits for each label (the O no... |

91 |
An O(n log n) algorithm for the all-nearest-neighbors problem
- Vaidya
- 1989
(Show Context)
Citation Context ... is to compute for a set P of n points the (exact) nearest neighbor for each point of p ∈ P in the set P \ {p}. It is known that in low dimensional Euclidean space this can be done in O(n log n) time =-=[13, 46, 11]-=-. One can ask if a similar result can be attained for finite metric spaces with low doubling dimensions. Below we show that this is impossible. Consider the points p1,... ,pn, where the distance betwe... |

89 | A replacement for Voronoi diagrams of near linear size - Har-Peled - 2001 |

87 | Nearest-neighbor searching and metric space dimensions. NearestNeighbor Methods for Learning and Vision: Theory and Practice
- Clarkson
- 2006
(Show Context)
Citation Context ...g properties easily imply that each vertex has at most λ O(1) children. Net-trees are roughly equivalent to compressed quadtrees [1]. The Net-tree is also similar to the sb data-structure of Clarkson =-=[15]-=-, but our analysis and guaranteed performance is new. 2.2 The Computational Model. The model of computation we use is the “unit cost floating-point word RAM model”. More precisely, for a given input c... |

84 | Measured descent: A new embedding method for finite metrics
- Krauthgamer, Lee, et al.
(Show Context)
Citation Context ...stricted metrics of Karger and Ruhl [31]. Understanding the structure of such spaces (or similar notions), and how to manipulate them efficiently received considerable attention in the last few years =-=[12, 31, 22, 27, 34, 35, 33, 46]-=-. It would have been nice if algorithmic results developed for low dimensional Euclidean space were directly applicable to low dimensional metrics. However, this is impossible in general, as there are... |

79 |
Plongements Lipschitziens dans
- Assouad
- 1983
(Show Context)
Citation Context ...t got considerable attention recently is to define a notion of dimension on a finite metric space, and develop efficient algorithms for this case. One such concept is the notion of doubling dimension =-=[2, 27, 23]-=-. The doubling constant of metric space M is the maximum, over all balls b in the metric space M, of the minimum number of balls needed to cover b, using balls with half the radius of b. The logarithm... |

79 | Sublinear time algorithms for metric space problems
- Indyk
- 1999
(Show Context)
Citation Context ...nce between any two points in the metric space in constant time. Since the matrix of all ( ) n 2 distances has quadratic size, this means that in some sense our algorithms have sublinear running time =-=[28, 29]-=-. This is not entirely surprising since o(n2 ) time algorithms exist for those problems in low dimensional Euclidean space. Thus, our paper can be interpreted as further strengthening the perceived co... |

67 | On metric ramsey-type phenomena
- Bartal, Linial, et al.
- 2003
(Show Context)
Citation Context ...lp later in eliminating the dependence on the spread of the running time for constructing net-tree and in distance queries. We remark that similar use of low quality approximation by HST were used in =-=[3]-=- for fast computation of subsets of metric spaces which are well approximated by HSTs. We begin by constructing a sparse graph that approximates the original metric (this is sometimes called spanner).... |

65 | Bypassing the embedding: algorithms for low dimensional metrics
- Talwar
- 2004
(Show Context)
Citation Context ...stricted metrics of Karger and Ruhl [30]. Understanding the structure of such spaces (or similar notions), and how to manipulate them efficiently received considerable attention in the last few years =-=[14, 30, 23, 28, 34, 33, 44]-=-. The low doubling metric approach can be justified in two levels. 1. Arguably, non-Euclidean, low (doubling) dimensional metric data appears in practice, and deserves an efficient algorithmic treatme... |

64 | Distance estimation and object location via rings of neighbors
- Slivkins
- 2005
(Show Context)
Citation Context ... log log Φ) on the average label’s length for 1.9-ADLS for this family of HSTs. Since these HSTs are binary their doubling dimension is 1. After a preliminary version of this paper appeared, Slivkins =-=[42]-=- managed to produce an ADLS with labels length of ε −O(dim) log n log log Φ, which improves upon our construction in the range n loglog n ≪ Φ ≪ 2 n . 7 Doubling Measure A measure µ on a metric space M... |

56 |
Fast algorithms for the all nearest neighbors problem
- Clarkson
- 1983
(Show Context)
Citation Context ... is to compute for a set P of n points the (exact) nearest neighbor for each point of p ∈ P in the set P \ {p}. It is known that in low dimensional Euclidean space this can be done in O(n log n) time =-=[13, 46, 11]-=-. One can ask if a similar result can be attained for finite metric spaces with low doubling dimensions. Below we show that this is impossible. Consider the points p1,... ,pn, where the distance betwe... |

48 |
Informative labeling schemes for graphs
- Peleg
- 2000
(Show Context)
Citation Context ..., given two points x 1 ,x 2 , using the identifiers I(x 1 1 ), I(x2 1 ), I(x1 2 ), I(x2 2 ), I(u1), I(u2), we can compute 1 + ε approximation of dM(x1,x2). We now use (the proof of) a result of Peleg =-=[39]-=-: Given an n-vertex rooted tree with identifiers I(v) of maximum length s on the vertices, it is possible to efficiently compute labels L(v) of length O(log n(log n + s)) to the vertices, such that gi... |

47 | Coresets for k-means and k-median clustering and their applications
- Har-Peled, Mazumdar
- 2004
(Show Context)
Citation Context ... d, one can answer such queries in O(log log(Φ/ε)) time, using near linear space, see [24] and references therein (in fact, it is possible to achieve constant query time using slightly larger storage =-=[26]-=-). Note however, that the latter results strongly use the Euclidean structure. Recently, Krauthgamer and Lee [33] overcame the restriction on the spread, presenting a data-structure with nearly quadra... |

46 | Approximate distance labeling schemes
- Gavoille, Pérennès
- 1996
(Show Context)
Citation Context ...air of points, it is possible to compute efficiently an approximation of the the pairwise distance. Thus, ADLS is a stricter notion of compact representation. 5 This notion was studied for example in =-=[19, 18, 45]-=-. In the constant doubling dimension setting Gupta et al.] [23] have shown an (1+ε)-embedding O(log n) of the metric in ℓ∞ . This implies (1 + ε)-ADLS with O(log n log Φ) bits for each label (the O no... |

45 | Extending Lipschitz functions via random metric partitions
- Lee, Naor
(Show Context)
Citation Context ... metrics (and in fact for complete metrics [39]) the existence of doubling measure is quantitively equivalent to the metric being doubling. This measure has found some recent algorithmic applications =-=[38, 45]-=-, and we anticipate more applications. Following the proof of Wu [50], we present in Section 7 a near linear time algorithm for constructing doubling measure. 2Lipschitz Constant of a Mapping. In Sec... |

40 | Farach-Colton. The Level Ancestor Problem simplified
- Bender, Martin
- 2004
(Show Context)
Citation Context ...ccessed in O(λ 4 ) time. The second data-structure enables the following seek operation: Given a leaf x, and a level l, find the ancestor y of x such that ℓ(p(y)) > l ≥ ℓ(y). Bender and Farach-Colton =-=[5]-=- present a data-structure D that can be constructed in linear time over a tree T, such that given a node x, and depth d, it outputs the ancestor of x at depth d at x. This takes constant time per quer... |

40 | A sublinear time approximation scheme for clustering in metric spaces
- Indyk
- 1999
(Show Context)
Citation Context ...nce between any two points in the metric space in constant time. Since the matrix of all ( ) n 2 distances has quadratic size, this means that in some sense our algorithms have sublinear running time =-=[28, 29]-=-. This is not entirely surprising since o(n2 ) time algorithms exist for those problems in low dimensional Euclidean space. Thus, our paper can be interpreted as further strengthening the perceived co... |

34 | Approximate distance oracles for geometric graphs
- Gudmundsson, Levcopoulos, et al.
- 2002
(Show Context)
Citation Context ...swer approximate distance queries between pairs of points, in essentially constant time. CRS were coined “approximate distance oracles” in [45]. Our result extends recent results of Gudmunsson et al. =-=[21, 22]-=- who showed the existence of CRS with similar parameters for metrics that are “nearly” fixed-dimensional Euclidean (which are sub-class of fixed doubling dimension metrics). We also mention in passing... |

33 |
On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A∞-weights
- Semmes
- 1996
(Show Context)
Citation Context ...lgorithmic results developed for fixed dimensional Euclidean space to doubling metrics, since there exists doubling metrics that can not embedded in Hilbert space with low distortion of the distances =-=[41, 35]-=-. Hence, some of the aforementioned works apply notions and techniques from fixed dimensional Computational Geometry and extend them to finite metric spaces. In particular, Talwar [44] showed that one... |

32 | A Course in Convexity, Volume 54 - Barvinok - 2002 |

31 | Distributed approaches to triangulation and embedding
- Slivkins
- 2005
(Show Context)
Citation Context ...nearly” fixed-dimensional Euclidean (which are sub-class of fixed doubling dimension metrics). We also mention in passing that our CRS technique can be applied to improve and unify two recent results =-=[44, 43]-=- on distance labeling. Doubling Measure. A doubling measure µ is a measure on the metric space with the property that for every x ∈ P and r > 0, the ratio µ(b(x,2r))/µ(b(x,r)) is bounded, where b(x,r)... |

30 |
Approximating the complete Euclidean graph
- Keil
- 1988
(Show Context)
Citation Context ...raph metric is t-approximation to the original metric. Spanners were first defined and studied in [40]. Construction of (1 + ε)-spanners for points in low dimensional Euclidean space is considered in =-=[31, 10]-=-. Using Callahan’s technique [10], the WSPD construction also implies a near linear-time construction of (1 + ε)-spanners having linear number of edges for such metrics. Independently of our work, Cha... |

30 | The black-box complexity of nearest neighbor search
- Krauthgamer, Lee
- 2004
(Show Context)
Citation Context ...stricted metrics of Karger and Ruhl [30]. Understanding the structure of such spaces (or similar notions), and how to manipulate them efficiently received considerable attention in the last few years =-=[14, 30, 23, 28, 34, 33, 44]-=-. The low doubling metric approach can be justified in two levels. 1. Arguably, non-Euclidean, low (doubling) dimensional metric data appears in practice, and deserves an efficient algorithmic treatme... |

30 | Clustering motion - Har-Peled |

28 |
Plane with A∞-weighted metric not bi-Lipschitz embeddable to
- Laakso
(Show Context)
Citation Context ...lgorithmic results developed for fixed dimensional Euclidean space to doubling metrics, since there exists doubling metrics that can not embedded in Hilbert space with low distortion of the distances =-=[41, 35]-=-. Hence, some of the aforementioned works apply notions and techniques from fixed dimensional Computational Geometry and extend them to finite metric spaces. In particular, Talwar [44] showed that one... |

28 |
Using the Borsuk-Ulam theorem. Universitext
- Matouˇsek
- 2003
(Show Context)
Citation Context ...hitz. By Kirszbraun Theorem without (see [7, Ch. 1]), the embedding φ ′ can be extended to the whole sphere � φ ′ : S d2 �·� 2 → R d2 �·� 2 increasing the Lipschitz constant. Borsuk-Ulam theorem (cf. =-=[37]-=-) states that there exists x ∈ S k such that � φ ′ (x) = � φ ′ (−x). Note that ∃y,z ∈ Pη such that�x − y� 2 ≤ η, and �(−x) − z� 2 ≤ η. Since �φ ′ is 1-Lipschitz, we have � � � � ′ ′ φ (y) − φ (z) �2 �... |

16 | A note on the nearest neighbor in growth-restricted metrics
- Hildrum, Kubiatowicz, et al.
- 2004
(Show Context)
Citation Context |

12 |
Clustering motion. Discrete Comput
- Har-Peled
(Show Context)
Citation Context ...orithm in this case is still quadratic. Although, in fixed dimensional Euclidean space the algorithm of Gonzalez was improved to O(n log k) time by Feder and Greene [17], and linear time by Har-Peled =-=[25]-=-, those algorithms require specifying k in advance, and they do not generate the permutation of the points, and as such they cannot be used in this case. Our results. In this paper, we present improve... |

12 |
Every complete doubling metric space carries a doubling measure
- Luukkainen, Saksman
- 1998
(Show Context)
Citation Context ...at for every x ∈ P and r > 0, the ratio µ(b(x,2r))/µ(b(x,r)) is bounded, where b(x,r) = {y : d(x,y) ≤ r}. Vol ′ berg and Konyagin [47] proved that for finite metrics (and in fact for complete metrics =-=[36]-=-) the existence of doubling measure is quantitatively equivalent to the metric being doubling. This measure has found some recent algorithmic applications [43], and we anticipate more applications. Fo... |

12 |
A new theory of dimension
- Larman
- 1967
(Show Context)
Citation Context ... got considerable attention recently, is to define a notion of dimension on a finite metric space, and develop efficient algorithms for this case. One such concept is the notion of doubling dimension =-=[37, 2, 26, 22]-=-. The doubling constant of metric space M is the maximum, over all balls b in the metric space M, of minimum number of balls needed to cover b, using balls with half the radius of b. The logarithm of ... |