Gary L. Miller

Carnegie Mellon University | CMU · Computer Science Department

Abstract We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR, dloglogn) time. Our algorithm has been used in mesh,partitioning methods,and,can be used in the construction of high breakdown,estimators for multivariate datasets in statistics. It has the potential to improve res...

We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method's novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain classes of "well-shaped" finite element meshes ha...

We show that testing reachability in a planar DAG can be performed in parallel in O(log n log time (O(log n) time using randomization) using O(n) processors. In general we give a paradigm for reducing a planar DAG to a constant size and then expanding it back. This paradigm is developed from a property of planar directed graphs we refer to as the P...

Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there is not much theoretical analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods "work well in practice." We present an initia...

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d . A point c 2 IR d is a fi-center point of P if every closed halfspace containing c contains at least fin points of P . Every point set has a 1=(d + 1)-center point; our algorithm finds an OmegaGamma/ =d 2 )-center point w...

We show that testing reachability in a planar DAG can be performed in parallel in O(log n log n) time (O(log n) time using randomization) using O(n) processors. In general we give a paradigm for contracting a planar DAG to a point and then expanding it back. This paradigm is developed from a property of planar directed graphs we refer to as the Poi...

We give a deterministic linear time algorithm for finding a "good" sphere separator of a k-ply neighborhood system Phi in any fixed dimension, where a k-ply neighborhood system in IR d is a collection of n balls such that no points in the space is covered by more than k balls. The separating sphere intersects at most O Gamma k 1=d n 1Gamma1=d Delta...

We describe a practical and provably good algorithm forapproximating center points in any number of dimensions. Herec is a center point of a point setP in Rd if every closed halfspace containingc contains at least P/d+1 points of P. Ouralgorithm has a small constant factor and is the first approximatecenter point algorithm whose complexity is subex...

We give a deterministic linear time algorithm for finding a small cost sphere separator of a k-ply neighborhood system &Fgr; in any fixed dimension, where a k-ply neighborhood system in Rd is a collection of n balls such that no points in the space is covered by more than k balls. The sphere separator intersects at most Ok1dnd-1d balls of &Fgr; an...

This paper describes an efficient approach to partitioning unstructured meshes that occur naturally in the finite element and finite difference methods. The approach makes use of the underlying geometric structure of a given mesh and finds a provably good partition in random O(n) time. It applies to meshes in both two and three dimensions. The new...

Abstract An O(log n) time, n processor randomized algorithm for computing the k - nearest neighbor graph of n points in d dimensions, for fixed d and k is presented The method is based on the use of sphere separators Probability bounds are proved using the moment generating function technique

This paper applies the parallel tree contraction techniques developed in Miller and Reif's paper (Randomness and Computation, Vol. 5, S. Micali, ed., JAI Press, 1989, pp. 47-72) to a number of fundamental graph problems. The paper presents an O(1ogn) time and nllogn processor, a 0-sided randomized algorithm for testing the isomorphism of trees, and...

A class of graphs called k -overlap graphs is proposed. Special cases of k-overlap graphs include planar graphs, k -nearest neighbor graphs, and earlier classes of graphs associated with finite element methods. A separator bound is proved for k -overlap graphs embedded in d dimensions. The result unifies several earlier separator results. All the a...

We propose a class of graphs that would occur naturally in three-dimensional finite-element Problems, and we prove an O(If2/3) bound on separators for this class of graphs. We also propose a simple randomized algorithm to find this separator in O(N) time. Such an algorithm could beused aa a preprocessing step for the domain decomposition method of...

We show that every graph that is the 1-skeleton of a simplicial complex K in 3-dimensions has a separator of size O(c 2/3 + v̄), where c is the number of 3-simplexes in K and v̄ is the number of 0-simplexes on the boundary of K, if every 3-simplex has bounded aspect-ratio. This is natural generalization of the separator results for planar graphs, s...

In this paper we show that every 2-connected embedded planar graph with faces of sizes d 1.....d f has a simple cycle separator of size 1.58 \(\sqrt {d_1^2 + \cdots + d_f^2 }\)and we give an almost linear time algorithm for finding these separators, O(no(n,n)). We show that the new upper bound expressed as a function of ‖G‖=\(\sqrt {d_1^2 + \cdots...

This paper presents a parallel algorithm that computes the breadth-first search (BFS) numbering of a directed graph in O(log²n) time using M(n) processors on the exclusive-read exclusive-write (EREW) parallel random access machine (PRAM) model, where M(n) denotes the number of processors needed to multiply two n x n integer matrices over the ring (...

This paper presents a parallel algorithm that computes the breadth-first search (BFS) numbering of a directed graph in O(log2n) time using M(n) processors on the exclusive-read exclusive-write (EREW) parallel random access machine (PRAM) model, where M(n) denotes the number of processors needed to multiply two n x n integer matrices over the ring (...

We present a randomized parallel algorithm for finding a simple cycle separator in a planar graph. The size of the separator is O(√n) and it separates the graph so that the largest part contains at most 2/8 ¿ n vertices. Our algorithm takes T = O(log2(n)) time and P = O(n + f1+ε) processors, where n is the number of vertices, f is the number of fa...

The main goal of this paper is to prove a new additivity theorem for the genus of a graph. The theorem is true only by making a natural modification to the definition of the genus of an embedding. The (generalized) genus of a graph G will equal the connectivity of the minimum connected surface for which G is embeddable. The connectivity of a surfac...

This paper presents a sublinear time parallel algorithm for computing the greatest common divisor of two integers. Its running time on two n bit integers is O(n log log n/log n) using the weak concurrent read concurrent write model. Topics considered include algorithms, parallel processing, computer calculations, time dependence, and readout system...

We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called ‘open ear decomposition’. A parallel implementation of the algorithm on a CRCW PRAM runs inO(log2n) parallel time usingO(n+m) processors, wheren is the number of vertices andm is the number of edge...

The dynamic parallel complexity of general computational circuits (defined in introduction) is discussed. We exhibit some relationships between parallel circuit evaluation and some uniform closure properties of a certain class of unary functions and present a systematic method for the design of processor efficient parallel algorithms for circuit ev...

A deterministic parallel algorithm for parallel tree contraction is presented in this paper. The algorithm takes T time and uses (P processors, where n the number of vertices in a tree using an Exclusive Read and Exclusive Write (EREW) Parallel Random Access Machine (PRAM). This algorithm improves the results of Miller and who use the CRCW randomiz...

A new parallel algorithm is given to evaluate a straight-line program. The algorithm evaluates a program over a commutative semi-ring R of degree d and size n in time O((Iog n)(log nd)) using M(n) processors, where M(n) is the number of processors required for multiplying n x n matrices over the semi-ring R in O(log n)time. AMS(MOS) subject classif...

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W.T. Tutte showed that if G is an arc transitive connected cubic graph then the automorphism group of G is in fact regular on s-arcs for some s ≤ 5. We analyze these arc transitive cubic graphs using the unifying concepts of the infinite cubic tree, ᴦ3 and coverings. We are able to answer a large number of questions, open and otherwise. As an examp...

We present an algorithm which determines isomorphism of graphs in vO(g)steps where v is the number of vertices and g is the genus of the graphs. In [FMR 79] an algorithm was presented for embedding graph on surfaces of genus g in vO(g) steps. Here we show how to extend this algorithm to isomorphism testing for graphs of small genus. This result is...

In this paper we present an algorithm which on input a graph G and a positive integer g finds an embedding of G on a surface on genius g, if such an embedding exists. This algorithm runs in (v) O(g) steps where v is the number of vertices of G. We believe that removing the nondiscrete topological definitions (i.e., the notation or differentiability...

This article describes Archimedes, an automated system for solving partial differential equations on geometrically complex domains using distributed memory supercomputers. The tasks of such a system are manifold. First, Archimedes discretizes the object or domain being modeled by generating an unstructured mesh that fills the region. Then, the doma...

Linear systems and eigen-calculations on symmetric diagonally dominant matrices (SDDs) occur ubiquitously in computer vision, computer graphics, and machine learning. In the past decade a multitude of specialized solvers have been developed to tackle restricted instances of SDD systems for a diverse collection of problems including segmentation, gr...

1 Why Fat Voronoi Diagrams? Voronoi refinement is a powerful tool for efficiently gen-erating meshes for finite element simulation. The clas-sic definition of quality in a mesh can be achieved by bounding the aspect ratio of the Voronoi cells measured as the ratio of the circumscribing and inscribing radii as measured from the site. There are tight...

Memory bandwidth is a major limiting factor in the scalability of parallel algorithms. In this paper, we introduce hierarchical diagonal blocking, a sparse matrix representation which we believe captures most of optimization techniques in a common representation. It can take advantage of symmetry while still being easy to parallelize. It takes adva...

In this paper we show how to find a support-tree preconditioner for any Laplacian matrix A, i.e., any matrix that can be viewed as the weighted adjacency matrix of an undirected graph G with non-negative edge weights (where the diagonal entries of A are set so that its row and column sums are zero). The preconditioner is used to accelerate the conv...

Purpose: Eye motion during scan aquisition of SD-OCT volumes produces motion artifacts, and missing or poorly sampled regions, that ultimately hinder diagnosis and interfere with registration of multiple scans in longitudinal studies. We propose a method of combining multiple scans to produce a single rectified volume. The method can also be used t...

We generalize the Tukey depth to use cones instead of halfspaces. We prove a generalization of the center point theorem that for SRd, there is a point s 2 S, with depth at least n d+1 for cones of half-angle 45 ◦ . This gives a notion of data depth for which an approximate median can always be found among the original set.

Spectral algorithms exploit information on the graph spectrum to gain computational speedups. Advances in spectral algorithms, such as spectral sparsifiers, fast symmetric diagonally dominant (SDD) system solvers of [KMP11] have created very powerful tools for the algorithmist. Most of the current fast technology are only available for weighted und...