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12
Sublinear time algorithms
 SIGACT News
, 2003
"... Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algo ..."
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Cited by 22 (2 self)
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Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algorithmic research is to design efficient algorithms, where efficiency is typicallymeasured as a function of the length of the input. For instance, the elementary school algorithm for multiplying two n digit integers takes roughly n2 steps, while more sophisticated algorithmshave been devised which run in less than n log2 n steps. It is still not known whether a linear time algorithm is achievable for integer multiplication. Obviously any algorithm for this task, as for anyother nontrivial task, would need to take at least linear time in n, since this is what it would take to read the entire input and write the output. Thus, showing the existence of a linear time algorithmfor a problem was traditionally considered to be the gold standard of achievement. Nevertheless, due to the recent tremendous increase in computational power that is inundatingus with a multitude of data, we are now encountering a paradigm shift from traditional computational models. The scale of these data sets, coupled with the typical situation in which there is verylittle time to perform our computations, raises the issue of whether there is time to consider any more than a miniscule fraction of the data in our computations? Analogous to the reasoning thatwe used for multiplication, for most natural problems, an algorithm which runs in sublinear time must necessarily use randomization and must give an answer which is in some sense imprecise.Nevertheless, there are many situations in which a fast approximate solution is more useful than a slower exact solution.
Sublinear geometric algorithms
 In Proc. of the 35th Annual ACM Symp. on Theory of Computing
, 2003
"... Abstract. We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in twodimensional triangulations and Voronoi diagrams, and ray shooting in convex ..."
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Cited by 19 (2 self)
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Abstract. We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in twodimensional triangulations and Voronoi diagrams, and ray shooting in convex polyhedra, all of which run in expected time O ( √ n), where n is the size of the input. We also provide sublinear solutions for the approximate evaluation of the volume of a convex polytope and the length of the shortest path between two points on the boundary. Key words. sublinear algorithms, approximate shortest paths, polyhedral intersection
Estimating the weight of metric minimum spanning trees in sublineartime
 in Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC
"... In this paper we present a sublinear time (1+ ɛ)approximation randomized algorithm to estimate the weight of the minimum spanning tree of an npoint metric space. The running time of the algorithm is Õ(n/ɛO(1)). Since the full description of an npoint metric space is of size Θ(n 2),the complexity ..."
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Cited by 16 (5 self)
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In this paper we present a sublinear time (1+ ɛ)approximation randomized algorithm to estimate the weight of the minimum spanning tree of an npoint metric space. The running time of the algorithm is Õ(n/ɛO(1)). Since the full description of an npoint metric space is of size Θ(n 2),the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. Furthermore,it has been previously shown that no o(n 2) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum.
Abstract combinatorial programs and efficient property testers
, 2005
"... Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms. Our fr ..."
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Cited by 15 (6 self)
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Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms. Our framework is based on a connection of property testing and a new class of problems which we call abstract combinatorial programs. We show that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester. We apply our framework to a variety of problems. We present efficient property testing algorithms for geometric clustering problems, for the reversal distance problem, and for graph and hypergraph coloring problems. We also prove that, informally, any hereditary graph property can be efficiently tested if and only if it can be reduced to an abstract combinatorial program of small size. Our framework allows us to analyze all our testers in a unified way, and the obtained complexity bounds either match or improve the previously known bounds. Furthermore, even if the asymptotic complexity of the testers is not improved, the obtained proofs are significantly simpler than the previous ones. We believe that our framework will help to understand the structure of efficiently testable properties.
Sublineartime algorithms
 In Oded Goldreich, editor, Property Testing, volume 6390 of Lecture Notes in Computer Science
, 2010
"... In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1 ..."
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Cited by 10 (2 self)
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In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1
Online geometric reconstruction
 Proc. of 22nd SOCG
, 2006
"... We investigate a new class of geometric problems based on the idea of online error correction. Suppose one is given access to a large geometric dataset though a query mechanism; for example, the dataset could be a terrain and a query might ask for the coordinates of a particular vertex or for the ed ..."
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Cited by 7 (2 self)
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We investigate a new class of geometric problems based on the idea of online error correction. Suppose one is given access to a large geometric dataset though a query mechanism; for example, the dataset could be a terrain and a query might ask for the coordinates of a particular vertex or for the edges incident to it. Suppose, in addition, that the dataset satisfies some known structural property P (eg, monotonicity or convexity) but that, because of errors and noise, the queries occasionally provide answers that violate P. Can one design a filter that modifies the query’s answers so that (i) the output satisfies P; (ii) the amount of data modification is minimized? We provide upper and lower bounds on the complexity of online reconstruction for convexity in 2D and 3D. 1
Hierarchical PeertoPeer Networks using Lightweight Superpeer Topologies
 Proc. 10th IEEE Symp. Comp. and Commun. (ISCC 2005), La Manga del Mar Menor
, 2005
"... The use of SuperPeers has been proposed to improve the performance of both Structured and Unstructured PeertoPeer (P2P) Networks. In this paper, we study the performance of YaoGraph based SuperPeer Topologies for Hierarchical P2P networks. Since a YaoGraph is defined as a geometric structure, we ..."
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Cited by 3 (2 self)
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The use of SuperPeers has been proposed to improve the performance of both Structured and Unstructured PeertoPeer (P2P) Networks. In this paper, we study the performance of YaoGraph based SuperPeer Topologies for Hierarchical P2P networks. Since a YaoGraph is defined as a geometric structure, we are using the ”Highways ” proximity clustering and placement scheme to assign geometric coordinates to SuperPeers and Peers with respect to the underlying network conditions. Because of the lightweight structure of YaoGraphs, the resulting hierarchical P2P networks have promising properties with regard to scalability and performance, while still offering the benefits of the P2P approach with regard to resiliency. 1
A case for lightweight superpeer topologies
 In KiVS Kurzbeiträge und Workshop
, 2005
"... Abstract: The usage of SuperPeers has been proposed to improve the performance of both Structured and Unstructured PeertoPeer (P2P) networks. In this paper we explore a networkaware class of Lightweight SuperPeer Topologies (LSTs). The proposed LST is based on the geometric principle of YaoGraph ..."
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Cited by 2 (1 self)
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Abstract: The usage of SuperPeers has been proposed to improve the performance of both Structured and Unstructured PeertoPeer (P2P) networks. In this paper we explore a networkaware class of Lightweight SuperPeer Topologies (LSTs). The proposed LST is based on the geometric principle of YaoGraphs, a class of graphs allowing the development of simple and efficient broadcast algorithms. The prerequisite of the LST approach is a function for mapping nodes in a network into a geometric space. In this paper we use the ”Highways ” proximity clustering and geometric placement model, introduced by one of the authors for this purpose. LST is evaluated based on PlanetLab measurements. 1
On derandomizing probabilistic sublineartime algorithms
 IN PROCEEDINGS OF THE 22TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2007
"... There exists a positive constant α < 1 such that for any function T (n) ≤ n α and for any problem L ∈ BPTIME(T (n)), there exists a deterministic algorithm running in poly(T (n)) time which decides L, except for at most a 2 −Ω(T (n) log T (n)) fraction of inputs of length n. The proof uses a nove ..."
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Cited by 2 (0 self)
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There exists a positive constant α < 1 such that for any function T (n) ≤ n α and for any problem L ∈ BPTIME(T (n)), there exists a deterministic algorithm running in poly(T (n)) time which decides L, except for at most a 2 −Ω(T (n) log T (n)) fraction of inputs of length n. The proof uses a novel derandomization technique based on a new type of randomness extractors, called exposureresilient extractors. An exposureresilient extractor is an efficient procedure that, from a random variable with imperfect minentropy, produces randomness that passes all statistical tests including those that have bounded access to the random variable, with adaptive queries that can depend on the string being tested. More precisely, EXT: {0, 1} n × {0, 1} d → {0, 1} m is a (k, ɛ)exposure resilient extractor resistant to q queries if, when the minentropy of x is at least k and y is random, EXT(x, y) looks ɛrandom to all statistical tests modeled by oracle circuits of unbounded complexity that can query q bits of x. Besides the extractor that is needed for the above derandomization (whose parameters are tailored for this application), we construct, for any δ < 1, a (k, ɛ)exposure resilient extractor with query resistance n δ, k = n − n Ω(1) , ɛ = n −Ω(1) , m = n Ω(1) and d = O(log n).
Testing Euclidean minimum spanning trees in the plane
 ACM Transactions on Algorithms
, 2007
"... Given a Euclidean graph G over a set P of n points in the plane, we are interested in verifying whether G is a Euclidean minimum spanning tree (EMST) of P or G differs from it in more than ǫn edges. We assume that G is given in adjacency list representation and the point/vertex set P is given in an ..."
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Cited by 2 (2 self)
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Given a Euclidean graph G over a set P of n points in the plane, we are interested in verifying whether G is a Euclidean minimum spanning tree (EMST) of P or G differs from it in more than ǫn edges. We assume that G is given in adjacency list representation and the point/vertex set P is given in an array. We present a property testing algorithm that accepts graph G if it is an EMST of P and that rejects with probability at least 2 3 if G differs from every EMST of P in more than ǫn edges. Our algorithm runs in O ( � n/ǫ · log2 (n/ǫ)) time and has a query complexity of O ( � n/ǫ · log(n/ǫ)).