## Sublinear-time algorithms (2006)

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Venue: | Bulletin of the EATCS |

Citations: | 11 - 2 self |

### BibTeX

@ARTICLE{Czumaj06sublinear-timealgorithms,

author = {Artur Czumaj and Christian Sohler},

title = {Sublinear-time algorithms},

journal = {Bulletin of the EATCS},

year = {2006},

volume = {89},

pages = {23--47}

}

### OpenURL

### Abstract

In this paper we survey recent advances in the area of sublinear-time algorithms. 1

### Citations

952 |
Approximation Algorithms
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Citation Context ... at least 3 4 . 4.1.1 Extensions: Sublinear-time (2 + ε)-approximation of metric TSP and Steiner trees Let us remark here one direct corollary of Theorem 7. By the well known relationship (see, e.g., =-=[51]-=-) between minimum spanning trees, travelling salesman tours, and minimum Steiner trees, the algorithm for estimating the weight of the minimum spanning tree from Theorem 7 immediately yields � O(n/ε O... |

430 | Property testing and its connection to learning and approximation
- Goldreich, Ron
- 1996
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Citation Context ...uss a large body of algorithms for dense graphs represented in the adjacency matrix model. Still, we mention the results of approximating the size of the maximum cut in constant time for dense graphs =-=[28, 32]-=-, and the more general results about approximating all dense problems in Max-SNP in constant time [2, 8, 28]. Similarly, we also have to mention about the existence of a large body of property testing... |

404 | Data streams: Algorithms and applications - Muthukrishnan - 2003 |

244 | Local search heuristics for k-median and facility location problems
- Arya, Garg, et al.
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Citation Context ...α + β) med(P,k) + ε, instead of 2αmed(P,k) + ε as in [46]. However, since the polynomial-time algorithm with the best known approximation guarantee has α = 3 + 1 c for the running time of O(nc ) time =-=[9]-=-, this significantly improves the running time of [46] for all realistic choices of the input parameters while achieving the same approximation guarantee. As a highlight, Theorem 10 yields a sublinear... |

181 | Fast Monte-Carlo Algorithms for Finding Low-Rank Approximations - Frieze, Kannan, et al. - 1998 |

172 | Approximation schemes for dense instances of np-hard problems
- Arora, Karpinski
- 1995
(Show Context)
Citation Context ...on the results of approximating the size of the maximum cut in constant time for dense graphs [28, 32], and the more general results about approximating all dense problems in Max-SNP in constant time =-=[2, 8, 28]-=-. Similarly, we also have to mention about the existence of a large body of property testing algorithms for graphs, which in many situations can lead to sublinear-time algorithms for graph problems. T... |

163 | Efficient testing of large graphs
- Alon, Fischer, et al.
(Show Context)
Citation Context ... give representative references, in addition to the excellent survey expositions [26, 30, 31, 40, 49], we want to mention the recent results on testability of graph properties, as described, e.g., in =-=[3, 4, 5, 6, 11, 21, 33, 43]-=-. 4 Sublinear Time Approximation Algorithms for Problems in Metric Spaces One of the most widely considered models in the area of sublinear time approximation algorithms is the distance oracle model f... |

132 | The art of uninformed decisions: A primer to property testing
- Fischer
(Show Context)
Citation Context ...geometry, algebraic computations, and computer graphics. Initially, the main research focus has been on designing efficient algorithms in the framework of property testing (for excellent surveys, see =-=[26, 30, 31, 40, 49]-=-), which is an alternative notion of approximation for decision problems. But more recently, we see some major progress in sublinear-time algorithms in the classical model of randomized and approximat... |

122 | Quick approximations to matrices and applications
- Frieze, Kannan
- 1997
(Show Context)
Citation Context ...uss a large body of algorithms for dense graphs represented in the adjacency matrix model. Still, we mention the results of approximating the size of the maximum cut in constant time for dense graphs =-=[28, 32]-=-, and the more general results about approximating all dense problems in Max-SNP in constant time [2, 8, 28]. Similarly, we also have to mention about the existence of a large body of property testing... |

121 | Property Testing in Bounded Degree Graphs
- Goldreich, Ron
- 2002
(Show Context)
Citation Context ...mponents in each of these subgraphs. The algorithm to estimate the number of connected components is based on a property tester for connectivity in the bounded degree graph model by Goldreich and Ron =-=[35]-=-. To start with basic intuitions, let us assume that W = 2, i.e., the graph has only edges of weight 1 or 2. Let G (1) = (V, E (1) ) denote the subgraph that contains all edges of weight (at most) 1 a... |

93 | A Characterization of the (natural) Graph Properties Testable with One-Sided Error. preprint, available online at http://www.math.tau.ac.il/˜nogaa/PDFS/heredit2.pdf. 18
- Alon, Shapira
- 2008
(Show Context)
Citation Context ... give representative references, in addition to the excellent survey expositions [26, 30, 31, 40, 49], we want to mention the recent results on testability of graph properties, as described, e.g., in =-=[3, 4, 5, 6, 11, 21, 33, 43]-=-. 4 Sublinear Time Approximation Algorithms for Problems in Metric Spaces One of the most widely considered models in the area of sublinear time approximation algorithms is the distance oracle model f... |

82 | Sublinear time algorithms for metric space problems
- Indyk
- 1999
(Show Context)
Citation Context ...ices, no sublinear algorithm for sparse graphs can exists. It is also know that no constant factor approximation algorithm with o(n 2 ) query complexity in dense graphs (even in metric spaces) exists =-=[37]-=-. Given these facts, it is somewhat surprising that it is possible to approximate the cost of a minimum spanning tree in sparse graphs [15] as well as in metric spaces [19] to within a factor of (1 + ... |

77 | Better streaming algorithms for clustering problems
- Charikar, O’Callaghan, et al.
- 2003
(Show Context)
Citation Context ...). These sublinear-time results have been extended in many different ways, e.g., to efficient data streaming algorithms and very fast algorithms for Euclidean k-median and also to k-means, see, e.g., =-=[9, 12, 16, 27, 35, 36, 41, 42, 45]-=-. For another clustering problem, the min-sum k-clustering problem (which is complement to the Max-k-Cut), for the basic case of k = 2, Indyk [39] (see also [38]) gave a (1 + ǫ)-approximation algorith... |

76 | A Sublinear Bipartitness Tester for Bounded Degree Graphs
- Goldreich, Ron
- 1999
(Show Context)
Citation Context ... give representative references, in addition to the excellent survey expositions [26, 30, 31, 40, 49], we want to mention the recent results on testability of graph properties, as described, e.g., in =-=[3, 4, 5, 6, 11, 21, 33, 43]-=-. 4 Sublinear Time Approximation Algorithms for Problems in Metric Spaces One of the most widely considered models in the area of sublinear time approximation algorithms is the distance oracle model f... |

73 | Property testing
- Ron
- 2001
(Show Context)
Citation Context ...geometry, algebraic computations, and computer graphics. Initially, the main research focus has been on designing efficient algorithms in the framework of property testing (for excellent surveys, see =-=[26, 30, 31, 40, 49]-=-), which is an alternative notion of approximation for decision problems. But more recently, we see some major progress in sublinear-time algorithms in the classical model of randomized and approximat... |

70 | A combinatorial characterization of the testable graph properties: it’s all about regularity
- Alon, Fischer, et al.
- 2006
(Show Context)
Citation Context |

58 | Testing of clustering
- Alon, Dar, et al.
- 2000
(Show Context)
Citation Context ...oblems in either Euclidean spaces or metric spaces, when the number of clusters is small. For radius (k-center) and diameter clustering in Euclidean spaces, sublinear-time property testing algorithms =-=[1, 21]-=- and tolerant testing algorithms [48] have been developed. The first sublinear algorithm for the kmedian problem was a bicriteria approximation algorithm [37]. This algorithm computes in � O(nk) 16s1 ... |

54 |
High-dimensional computational geometry
- Indyk
- 2000
(Show Context)
Citation Context ...to k-means, see, e.g., [9, 12, 16, 27, 35, 36, 41, 42, 45]. For another clustering problem, the min-sum k-clustering problem (which is complement to the Max-k-Cut), for the basic case of k = 2, Indyk =-=[39]-=- (see also [38]) gave a (1 + ǫ)-approximation algorithm that runs in time O(21/ǫO(1) n (log n) O(1) ), which is sublinear in the full input description size. No such results are known for k ≥ 3, but r... |

51 | Intersection of convex objects in two and three dimensions
- Chazelle, Dobkin
- 1987
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Citation Context ...ning time. The complexity of this problem depends on the input representation. In the most powerful model, if the vertices of both polygons are stored in an array in cyclic order, Chazelle and Dobkin =-=[13]-=- showed that the intersection of the polygons can be determined in logarithmic time. However, a standard geometric representation assumes that the input is not stored in an array but rather A has key ... |

48 |
Tolerant property testing and distance approximation
- Parnas, Ron, et al.
(Show Context)
Citation Context ...ric spaces, when the number of clusters is small. For radius (k-center) and diameter clustering in Euclidean spaces, sublinear-time property testing algorithms [1, 21] and tolerant testing algorithms =-=[48]-=- have been developed. The first sublinear algorithm for the kmedian problem was a bicriteria approximation algorithm [37]. This algorithm computes in � O(nk) 16s1 1 1 1 d(e) = 1 1 1 (a) L R 1 1 1 1 d(... |

45 | Approximating the Minimum Spanning Tree Weight in Sublinear Time
- Chazelle, Rubinfeld, et al.
- 2001
(Show Context)
Citation Context ...ional geometry [14]. Next, we present recent sublineartime algorithms for basic graph problems: approximating the average degree in a graph [25, 34] and estimating the cost of a minimum spanning tree =-=[15]-=-. Then, we discuss sublinear-time algorithms for optimization problems in metric spaces. We present the main ideas behind recent algorithms for estimating the cost of minimum spanning tree [19] and fa... |

45 | Sublinear time approximate clustering
- Mishra, Oblinger, et al.
- 2001
(Show Context)
Citation Context ...rithms for estimating the cost of minimum spanning tree [19] and facility location [10], and then we discuss the quality of random sampling to obtain sublinear-time algorithms for clustering problems =-=[20, 46]-=-. We finish with some conclusions. 2 Basic Sublinear Algorithms The concept of sublinear-time algorithms is known for a very long time, but initially it has been used to denote “pseudo-sublinear-time”... |

44 | Every monotone graph property is testable
- Alon, Shapira
- 2005
(Show Context)
Citation Context |

42 | Combinatorial property testing - a survey
- Goldreich
- 1998
(Show Context)
Citation Context ...geometry, algebraic computations, and computer graphics. Initially, the main research focus has been on designing efficient algorithms in the framework of property testing (for excellent surveys, see =-=[26, 30, 31, 40, 49]-=-), which is an alternative notion of approximation for decision problems. But more recently, we see some major progress in sublinear-time algorithms in the classical model of randomized and approximat... |

39 | A sublinear time approximation scheme for clustering in metric spaces
- Indyk
- 1999
(Show Context)
Citation Context ..., e.g., [9, 12, 16, 27, 35, 36, 41, 42, 45]. For another clustering problem, the min-sum k-clustering problem (which is complement to the Max-k-Cut), for the basic case of k = 2, Indyk [39] (see also =-=[38]-=-) gave a (1 + ǫ)-approximation algorithm that runs in time O(21/ǫO(1) n (log n) O(1) ), which is sublinear in the full input description size. No such results are known for k ≥ 3, but recently, [22] g... |

34 | Graph limits and parameter testing
- Borgs, Chayes, et al.
(Show Context)
Citation Context ...representative references, in addition to the excellent survey expositions [28, 32, 33, 43, 53], we would like to mention the recent results on testability of graph properties, as described, e.g., in =-=[3, 4, 5, 6, 11, 12, 19, 23, 36, 46]-=-. 4 Sublinear Time Approximation Algorithms for Problems in Metric Spaces One of the most widely considered models in the area of sublinear time approximation algorithms is the distance oracle model f... |

32 | Optimal time bounds for approximate clustering
- Mettu, Plaxton
- 2002
(Show Context)
Citation Context ...et of O(k) centers that are a constant factor approximation to the k-median objective function. Later, standard constant factor approximation algorithms were given that run in time �O(nk) (see, e.g., =-=[44, 50]-=-). These sublinear-time results have been extended in many different ways, e.g., to efficient data streaming algorithms and very fast algorithms for Euclidean k-median and also to k-means, see, e.g., ... |

29 |
Random sampling and approximation of MAX-CSPs
- Alon, Vega, et al.
(Show Context)
Citation Context ...on the results of approximating the size of the maximum cut in constant time for dense graphs [28, 32], and the more general results about approximating all dense problems in Max-SNP in constant time =-=[2, 8, 28]-=-. Similarly, we also have to mention about the existence of a large body of property testing algorithms for graphs, which in many situations can lead to sublinear-time algorithms for graph problems. T... |

28 | Smaller coresets for k-median and k-means clustering. http://www.uiuc.edu/˜sariel/papers/04/small coreset
- Har-Peled, Kushal
- 2004
(Show Context)
Citation Context ...). These sublinear-time results have been extended in many different ways, e.g., to efficient data streaming algorithms and very fast algorithms for Euclidean k-median and also to k-means, see, e.g., =-=[9, 12, 16, 27, 35, 36, 41, 42, 45]-=-. For another clustering problem, the min-sum k-clustering problem (which is complement to the Max-k-Cut), for the basic case of k = 2, Indyk [39] (see also [38]) gave a (1 + ǫ)-approximation algorith... |

27 |
Property testing in computational geometry
- Czumaj, Sohler, et al.
(Show Context)
Citation Context ...t the most natural characterization would be to remove at least ε n points in A and B, for an appropriate parameter ε (see [18] for a discussion about other geometric characterization). Czumaj et al. =-=[23]-=- gave a simple algorithm that for any ε > 0, can distinguish between the case when A and B do not intersect, and the case when at least ε n points has to be removed from A and B to make them intersect... |

26 |
Coresets in dynamic geometric data streams
- Frahling, Sohler
- 2005
(Show Context)
Citation Context ...). These sublinear-time results have been extended in many different ways, e.g., to efficient data streaming algorithms and very fast algorithms for Euclidean k-median and also to k-means, see, e.g., =-=[9, 12, 16, 27, 35, 36, 41, 42, 45]-=-. For another clustering problem, the min-sum k-clustering problem (which is complement to the Max-k-Cut), for the basic case of k = 2, Indyk [39] (see also [38]) gave a (1 + ǫ)-approximation algorith... |

25 | Every minorclosed property of sparse graphs is testable - Benjamini, Schramm, et al. |

24 | Sublinear time algorithms
- Kumar, Rubinfeld
(Show Context)
Citation Context |

22 | Sublinear geometric algorithms
- Chazelle, Liu, et al.
- 2003
(Show Context)
Citation Context ...related to the research presented in this survey. Organization. We begin with an introduction to the area and then we give some sublinear-time algorithms for a basic problem in computational geometry =-=[14]-=-. Next, we present recent sublineartime algorithms for basic graph problems: approximating the average degree in a graph [25, 34] and estimating the cost of a minimum spanning tree [15]. Then, we disc... |

21 | On k-median clustering in high dimensions
- Chen
- 2006
(Show Context)
Citation Context |

19 | Testing hereditary properties of nonexpanding bounded-degree graphs
- Czumaj, Shapira, et al.
- 2009
(Show Context)
Citation Context ...representative references, in addition to the excellent survey expositions [28, 32, 33, 43, 53], we would like to mention the recent results on testability of graph properties, as described, e.g., in =-=[3, 4, 5, 6, 11, 12, 19, 23, 36, 46]-=-. 4 Sublinear Time Approximation Algorithms for Problems in Metric Spaces One of the most widely considered models in the area of sublinear time approximation algorithms is the distance oracle model f... |

18 | Estimating the weight of metric minimum spanning trees in sublinear-time
- Czumaj, Sohler
- 2009
(Show Context)
Citation Context ...g tree [15]. Then, we discuss sublinear-time algorithms for optimization problems in metric spaces. We present the main ideas behind recent algorithms for estimating the cost of minimum spanning tree =-=[19]-=- and facility location [10], and then we discuss the quality of random sampling to obtain sublinear-time algorithms for clustering problems [20, 46]. We finish with some conclusions. 2 Basic Sublinear... |

18 | A simple linear time (1 + ε)-approximation algorithm for k-means clustering in any dimensions
- Kumar, Sabharwal, et al.
- 2004
(Show Context)
Citation Context |

17 | Property testing with geometric queries
- Czumaj, Sohler
- 2001
(Show Context)
Citation Context ...he notion of “significantly modify” may depend on the application at hand, but the most natural characterization would be to remove at least ε n points in A and B, for an appropriate parameter ε (see =-=[18]-=- for a discussion about other geometric characterization). Czumaj et al. [23] gave a simple algorithm that for any ε > 0, can distinguish between the case when A and B do not intersect, and the case w... |

17 |
Krzysztof Onak. Constant-time approximation algorithms via local improvements
- Nguyen
- 2008
(Show Context)
Citation Context ...ar-time algorithms for basic graph problems: approximating the average degree in a graph [27, 37], estimating the cost of a minimum spanning tree [16] and approximating the size of a maximum matching =-=[51, 56]-=-. Then, we discuss sublinear-time algorithms for optimization problems in metric spaces. We present the main ideas behind recent algorithms for estimating the cost of minimum spanning tree [21] and fa... |

16 | Graph limits and testing hereditary graph properties, manuscript
- Lovász, Szegedy
- 2005
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Citation Context |

16 |
Quick k-median, k-center, and facility location for sparse graphs
- Thorup
- 2001
(Show Context)
Citation Context ...imized. Here, d(p,F) denote the distance from p to the nearest point in F . It is known that one cannot find a solution that approximates the optimal solution within a constant factor in o(n 2 ) time =-=[50]-=-. However, it is possible to approximate the cost of an optimal solution within a constant factor. The main idea is as follows. Let us denote by B(p,r) the set of points from P with distance at most r... |

15 | Abstract combinatorial programs and efficient property testers
- Czumaj, Sohler
(Show Context)
Citation Context ...oblems in either Euclidean spaces or metric spaces, when the number of clusters is small. For radius (k-center) and diameter clustering in Euclidean spaces, sublinear-time property testing algorithms =-=[1, 21]-=- and tolerant testing algorithms [48] have been developed. The first sublinear algorithm for the kmedian problem was a bicriteria approximation algorithm [37]. This algorithm computes in � O(nk) 16s1 ... |

15 | On sums of independent random variables with unbounded variance, and estimating the average degree in a graph
- Feige
- 2004
(Show Context)
Citation Context ...me sublinear-time algorithms for a basic problem in computational geometry [14]. Next, we present recent sublineartime algorithms for basic graph problems: approximating the average degree in a graph =-=[25, 34]-=- and estimating the cost of a minimum spanning tree [15]. Then, we discuss sublinear-time algorithms for optimization problems in metric spaces. We present the main ideas behind recent algorithms for ... |

13 | Sublinear-time approximation of Euclidean minimum spanning tree - Czumaj, Ergun, et al. - 2003 |

12 | On the complexity of linear programming
- Megiddo
- 1987
(Show Context)
Citation Context ... problem can be solved in O(n) time, for example, by observing that it can be described as a linear programming instance in 2-dimensions, a problem which is known to have a linear-time algorithm (cf. =-=[24]-=-). In fact, within the same time one can either find a point that is in the intersection of A and B, or find a line L that separates A from B (actually, one can even find a bitangent separating line L... |

10 | Homomorphisms in graph property testing - a survey - Shapira, Alon - 2005 |

10 |
Coresets for k-means and k-medians and their applications
- Har-Peled, Mazumdar
- 2004
(Show Context)
Citation Context |

9 | Facility Location in Sublinear Time
- Badoiu, Czumaj, et al.
- 2005
(Show Context)
Citation Context ...ss sublinear-time algorithms for optimization problems in metric spaces. We present the main ideas behind recent algorithms for estimating the cost of minimum spanning tree [19] and facility location =-=[10]-=-, and then we discuss the quality of random sampling to obtain sublinear-time algorithms for clustering problems [20, 46]. We finish with some conclusions. 2 Basic Sublinear Algorithms The concept of ... |

9 | Linear time algorithms for clustering problems in any dimension
- Kumar, Sabharwal, et al.
- 2005
(Show Context)
Citation Context |