Applying parallel computation algorithms in the design of serial algorithms

Megiddo, Nimrod

doi:10.1109/sfcs.1981.11

Cited by 135 publications

(264 citation statements)

References 16 publications

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“…• Our solution to the GNN problem uses range searching data structures, but does not rely on Megiddo's parametric search [18] that usually causes a logarithmic overhead. Parametric search is a powerful but complex technique to reduce optimization problems to decision problems which we substitute by an application of ε-nets (see Section 2).…”

Section: Our Contributionmentioning

confidence: 99%

A Simple Framework for the Generalized Nearest Neighbor Problem

Hrúz

Schöngens

2012

Algorithm Theory – SWAT 2012

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The problem of finding a nearest neighbor from a set of points in R d to a complex query object has attracted considerable attention due to various applications in computational geometry, bio-informatics, information retrieval, etc. We propose a generic method that solves the problem for various classes of query objects and distance functions in a unified way. Moreover, for linear space requirements the method simplifies the known approach based on ray-shooting in the lower envelope of an arrangement.

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Section: Our Contributionmentioning

confidence: 99%

A Simple Framework for the Generalized Nearest Neighbor Problem

Hrúz

Schöngens

2012

Algorithm Theory – SWAT 2012

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“…Although when g(R) = |R| our algorithm in Section 2.2 can be used to solve that problem, simply applying the parametric search technique may just lead to a weakly polynomial time algorithm (i.e., the running time is polynomial with respect to both the size of the input voxel grid Γ and the values of the on-surface and in-region costs of the voxels in Γ). Megiddo's approach 18 may result in an improved strongly polynomial time algorithm (i.e., the running time is polynomial in the size of the input voxel grid and is independent on the values of the voxel costs). However, it requires the design of an efficient parallel algorithm for the maximum flow problem, which is hard to achieve.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

2006

Algorithms and Computation

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In this paper, we study several interesting optimal-ratio region detection (ORD) problems in d-D (d ≥ 3) discrete geometric spaces, which arise in high dimensional medical image segmentation. Given a d-D voxel grid of n cells, two classes of geometric regions that are enclosed by a single or two coupled smooth heighfield surfaces defined on the entire grid domain are considered. The objective functions are normalized by a function of the desired regions, which avoids a bias to produce an overly large or small region resulting from data noise. The normalization functions that we employ are used in real medical image segmentation. To our best knowledge, no previous results on these problems in high dimensions are known. We develop a unified algorithmic framework based on a careful characterization of the intrinsic geometric structures and a nontrivial graph transformation scheme, yielding efficient polynomial time algorithms for solving these ORD problems. Our main ideas include the following. We observe that the optimal solution to the ORD problems can be obtained via the construction of a convex hull for a set of O(n) unknown 2-D points using the hand probing technique. The probing oracles are implemented by computing a minimum s-t cut in a weighted directed graph. The ORD problems are then solved by O(n) calls to the minimum s-t cut algorithm. For the class of regions bounded by a single heighfield surface, our further investigation shows that the O(n) calls to the minimum s-t cut algorithm are on a monotone parametric flow network, which enables to detect the optimal-ratio region in the complexity of computing a single maximum flow.

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“…However, this takes time proportional to the depth of the tree, which may be too large. In an improvement to this method, alternately called centroid decomposition or tree contraction [64,47,52], one alternates this removal of leaves with the removal of vertices having only one child (sometimes with the further restriction that the parent of the vertex also only has one child). Cole and Vishkin [15] and Gazit et al [25] give centroid decomposition algorithms that take logarithmic time with a linear number of EREW operations, matching the bounds for Euler tours.…”

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confidence: 99%

Parallel algorithmic techniques for combinatorial computation

Eppstein¹,

Galil²

1989

Automata, Languages and Programming

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INTRODUCTIONParallel computation offers the promise of great improvements in the solution of problems that, if we were restricted to sequential computation, would take so much time that solution would be impractical.A drawback to the use of parallel computers is that they are harder to program. For this reason, parallel computation is often restricted to simple problems such as matrix multiplication. Certainly this is useful, and in fact we shall see later some non-obvious uses of matrix manipulation, but many important problems are more complex. In particular, problems may be structured as graphs or trees, rather than in the regular order of a matrix.We describe a number of techniques that have been developed for solving such combinatorial problems. Our intent is to show how these tools can be used as building blocks for higher level algorithms, and to provide pointers to the literature for the details of these algorithms. We make no claim to completeness; a number of techniques have been omitted for brevity or because their chief application is not combinatorial. In particular we give very little attention to sorting, although it is used as a subroutine in a number of the algorithms we describe.We use a shared memory model of parallelism; nevertheless we hope that the techniques described will be useful not only on shared memory machines, but also on other types of parallel computers, either through simulations of shared memory [48,70,35,57], or through analogous techniques for different models of parallel computation (e.g. see [30]). THE MODEL OF PARALLELISMThe model of parallel computation we use is the shared memory parallel random access machine (PRAM). This model consists of a collection of identical processors and a separate collection of memory cells; any processor can access any memory cell in unit time. Processors are allowed to perform any other operation that a typical sequential processor could do. Processors can know the size of their input, and the number of processors is assumed to be a function of that size, but the program controlling the processors is the same for all input sizes.Practically, one would like a PRAM algorithm to be as efficient as possible; that is, we want to minimize the number of operations, which may be computed as the product of the algorithm time and the number of processors. Clearly there must be at least as many operations as the best known sequential time; a parallel algorithm that meets this bound is called optimal. A less restrictive condition is that the operations are within a polylogarithmic factor of optimality; we call such algorithms almost optimal.Theoretically, we would like as small a running time as possible. In particular, an important class of algorithms, NC, is defined to be those taking polylogarithmic time with polynomially many processors.Many PRAM algorithms are both in NC and almost optimal. For a number of other problems, the best known NC algorithm takes a number of processors that is a small polynomial of the input size, and so for small problem ...

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Applying parallel computation algorithms in the design of serial algorithms

Cited by 135 publications

References 16 publications

A Simple Framework for the Generalized Nearest Neighbor Problem

A Simple Framework for the Generalized Nearest Neighbor Problem

Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

Parallel algorithmic techniques for combinatorial computation

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