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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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In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1
New Sublinear Methods in the Struggle against Classical Problems
, 2010
"... We study the time and query complexity of approximation algorithms that access only a minuscule fraction of the input, focusing on two classical sources of problems: combinatorial graph optimization and manipulation of strings. The tools we develop find applications outside of the area of sublinear ..."
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We study the time and query complexity of approximation algorithms that access only a minuscule fraction of the input, focusing on two classical sources of problems: combinatorial graph optimization and manipulation of strings. The tools we develop find applications outside of the area of sublinear algorithms. For instance, we obtain a more efficient approximation algorithm for edit distance and distributed algorithms for combinatorial problems on graphs that run in a constant number of communication rounds.
(1 + ε)Approximation for Facility Location in Data Streams∗
"... We consider the Euclidean facility location problem with uniform opening cost. In this problem, we are given a set of n points P ⊆ R2 and an opening cost f ∈ R+, and we want to find a set of facilities F ⊆ R2 that minimizes f · F + p∈P min q∈F d(p, q), where d(p, q) is the Euclidean distance betw ..."
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We consider the Euclidean facility location problem with uniform opening cost. In this problem, we are given a set of n points P ⊆ R2 and an opening cost f ∈ R+, and we want to find a set of facilities F ⊆ R2 that minimizes f · F + p∈P min q∈F d(p, q), where d(p, q) is the Euclidean distance between p and q. We obtain two main results: • A (1 + ε)approximation algorithm with running time O(n log2 n log log n) for constant ε, • The first (1 + ε)approximation algorithm for the cost of the facility location problem for dynamic geometric data streams, i.e., when the stream consists of insert and delete operations of points from a discrete space {1,...,∆}2. The streaming algorithm uses log ∆ ε)O(1) space. Our PTAS is significantly faster than any previously known (1 + ε)approximation algorithm for the problem, and is also relatively simple. Our algorithm for dynamic geometric data streams is the first (1 + ε)approximation algorithm for the cost of the facility location problem with polylogarithmic space, and it resolves an open problem in the streaming area. Both algorithms are based on a novel and simple decomposition of an input point set P into small subsets Pi, such that: • the cost of solving the facility location problem for each Pi is small (which means that one needs to open only a small, polylogarithmic number of facilities), • ∑i OPT(Pi) ≤ (1 + ε) ·OPT(P), where for a point set P, OPT(P) denotes the cost of an optimal solution for P. ∗Research partially supported by the EU within the 7th Framework
A SublinearTime Approximation Scheme for Bin Packing
, 2008
"... The bin packing problem is defined as follows: given a set of n items with sizes 0 < w1, w2,...,wn ≤ 1, find a packing of these items into minimum number of unitsize bins possible. We present a sublineartime asymptotic approximation scheme for the bin packing problem; that is, for any ɛ> 0, ..."
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The bin packing problem is defined as follows: given a set of n items with sizes 0 < w1, w2,...,wn ≤ 1, find a packing of these items into minimum number of unitsize bins possible. We present a sublineartime asymptotic approximation scheme for the bin packing problem; that is, for any ɛ> 0, we present an algorithm Aɛ that has sampling access to the input instance and outputs a value k such that Copt ≤ k ≤ (1+ɛ)·Copt+1, where Copt is the cost of an optimal solution. It is clear that uniform sampling by itself will not allow a sublineartime algorithm in this setting; a small number of items might constitute most of the total weight and uniform samples will not hit them. In this work we use weighted samples, where item i is sampled with probability proportional to its weight: that is, with probability wi / ∑ i wi. In the presence of weighted samples, the approximation algorithm runs in Õ(√n · poly(1/ɛ)) + g(1/ɛ) time, where g(x) is an exponential function of x. When both weighted and uniform sampling are allowed, Õ(n1/3 · poly(1/ɛ)) + g(1/ɛ) time suffices. In addition to an approximate value to Copt, our algorithm can also output a constantsize “template ” of a packing that can later be used to find a nearoptimal packing in linear time.
ConstantTime Approximation . . .
"... We present a technique for transforming classical approximation algorithms into constanttime algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based o ..."
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We present a technique for transforming classical approximation algorithms into constanttime algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based on greedily considering local improvements in random order. The problems amenable to our technique include
PassEfficient Algorithms for Clustering
, 2006
"... The proliferation of computational problems involving massive data sets has necessitated the design of computational paradigms that model the extra constraints placed on systems processing very large inputs. Among these algorithmic paradigms, the passefficient model captures the constraints that th ..."
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The proliferation of computational problems involving massive data sets has necessitated the design of computational paradigms that model the extra constraints placed on systems processing very large inputs. Among these algorithmic paradigms, the passefficient model captures the constraints that the input may be much too large to fit in main memory for processing, and that system performance is optimized by sequential access to the data in storage. Thus, in the passefficient model of computation, an algorithm may make a constant number of sequential passes over readonly input while using a small amount of random access memory. The resources to be optimized are memory, number of passes, and per element processing time. We give passefficient algorithms for clustering and finding structure in large amounts of data. Our algorithms have the property that the number of passes allotted is an input parameter to the algorithm. We answer questions regarding the intrinsic tradeoffs between the number of passes used by a passefficient algorithm and the amount of random access memory required. Our algorithms use adaptive sampling techniques that are quite general and can be used to solve many massive data set problems. The first family of clustering problems that we consider is learning mixtures of distributions. In these problems, we are given samples drawn according to a probability distribution known as a mixture of distributions, and must reconstruct the density function of the original mixture. Our algorithms show tradeoffs between the number of passes and the amount of memory required: if the algorithm makes a few extra passes, the amount of memory required drops off sharply. We also prove lower bounds on the amount of memory needed by any `pass randomized algorithm, thus showing that our tradeoff is nearly tight. The second family of clustering problems that we consider is the combinatorial optimization problem of facility location and related problems. Our passefficient algorithms for this problem exhibit the same sharp tradeoffs as our algorithm for learning mixtures of distributions. We also give clustering algorithms that are not in the streaming model for partitioning a graph to approximately minimize certain natural objective functions.
Electronic Colloquium on Computational Complexity, Report No. 94 (2005) On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms
"... We consider the problem of estimating the size, V C(G), of a minimum vertex cover of a graph G, in sublinear time, by querying the incidence relation of the graph. We say that an algorithm is an (α, ɛ)approximation algorithm if it outputs with high probability an estimate V C such that V C(G) − ɛn ..."
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We consider the problem of estimating the size, V C(G), of a minimum vertex cover of a graph G, in sublinear time, by querying the incidence relation of the graph. We say that an algorithm is an (α, ɛ)approximation algorithm if it outputs with high probability an estimate V C such that V C(G) − ɛn ≤ V C ≤ α · V C(G) + ɛn, where n is the number of vertices of G. We show that the query complexity of such algorithms must grow at least linearly with the average degree ¯ d of the graph. In particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o(n). On the positive side we first describe a simple (O(log ( ¯ d/ɛ), ɛ)approximation algorithm, whose query complexity is ɛ −2 · ( ¯ d/ɛ) log ( ¯ d/ɛ)+O(1) We then show a reduction from local distributed approximation algorithms to sublinear approximation algorithms. Using this reduction and the distributed algorithm of Kuhn, Moscibroda, and Wattenhofer [KMW05] we can get an (O(1), ɛ)approximation algorithm, whose query complexity is ɛ −2 · ( ¯ d/ɛ) O(log ( ¯ d/ɛ) ISSN 14338092
Sequence Comparison
"... ensuring compliance with copyright. For more information, please contact scholarworks@uno.edu. Robust and Efficient Algorithms for Protein 3D Structure Alignment and Genome ..."
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ensuring compliance with copyright. For more information, please contact scholarworks@uno.edu. Robust and Efficient Algorithms for Protein 3D Structure Alignment and Genome
Approximating Average Parameters of Graphs In Memory of Shimon Even (1935{2004)
, 2005
"... Inspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph. Specically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph. Since our focus is on sublinear algorithms ..."
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Inspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph. Specically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph. Since our focus is on sublinear algorithms, these algorithms access the input graph via queries to an adequate oracle. We consider two types of queries. The rst type is standard neighborhood queries (i.e., what is the ith neighbor of vertex v?), whereas the second type are queries regarding the quantities that we need to nd the average of (i.e., what is the degree of vertex v? and what is the distance between u and v?, respectively). Loosely speaking, our results indicate a dierence between the two problems: For approximating the average degree, the standard neighbor queries suce and in fact are preferable to degree queries. In contrast, for approximating average distances, the standard neighbor queries are of little help whereas distance queries are crucial. Note: Part of this work (i.e., the material in Section 3.1) was posted on ECCC (as TR04013).