Gary L. Miller

Carnegie Mellon University | CMU · Computer Science Department

Spectral clustering is one of the most popular clustering algorithms that has stood the test of time. It is simple to describe, can be implemented using standard linear algebra, and often finds better clusters than traditional clustering algorithms like $k$-means and $k$-centers. The foundational algorithm for two-way spectral clustering, by Shi an...

We examine the connection between the second eigenvalue of a weighted Laplacian and the isoperimetric constant of a Lipschitz function whose domain is Euclidean space. We prove both Cheeger and Buser-type inequalities for appropriate definitions of the "weighted Laplacian" and "isoperimetric constant." We show that given a positive-valued Lipschitz...

We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality.

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in $O(\text{polylog}(n))$ time per elimination and query. We then study the problem of using this dat...

We study faster algorithms for producing the minimum degree ordering used to speed up Gaussian elimination. This ordering is based on viewing the non-zero elements of a symmetric positive definite matrix as edges of an undirected graph, and aims at reducing the additional non-zeros (fill) in the matrix by repeatedly removing the vertex of minimum d...

Over the last several years there have been major breakthroughs in the design of approximation algorithms for such classic problems as finding the maximum flow in a graph. Maximum flow for undirected graphs can now be approximately solved in almost linear time. This result by researchers at Berkeley and MIT, I claim, is only the beginning of a new...

We use exponential start time clustering to design faster parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of...

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay...

Methods and apparatuses for constructing a multi-level solver, comprising decomposing a graph into a plurality of pieces, wherein each of the pieces has a plurality of edges and a plurality of interface nodes, and wherein the interface nodes in the graph are fewer in number than the edges in the graph; producing a local preconditioner for each of t...

We give a generalized definition of stretch that simplifies the efficient construction of low-stretch embeddings suitable for a variety of efficient graph algorithms. The generalization is based on discounting highly stretched edges by some exponent $p < 1$, and stems from observations about the performances of existing algorithms. We show that an...

We present an efficient algorithm for solving a linear system arising from the 1-Laplacian corresponding to a collapsible simplicial complex with a known collapsing sequence. When combined with a result of Chillingworth, our algorithm is applicable to convex simplicial complexes embedded in R3. The running time of our algorithm is nearly-linear in...

A method of partitioning a weighted combinatorial graph representative of a dataset consists of the steps of generating a generalized Laplacian matrix corresponding to the combinatorial graph, computing the eigenstructure of the generalized Laplacian matrix, determining if an end criterion is satisfied using the eigenstructure, and if the end crite...

Efficiently scaling shortest path algorithms to multiple processors is one of the major open problems in parallel computing. Despite three decades of progresses in theory and practice, obtaining speedups proportional to the number of processors over sequential algorithms remains a challenging problem. We give algorithms that compute $(1 + \epsilon)...

Methods and apparatuses for solving a system on a symmetric diagonally dominant matrix. The method includes constructing an equivalent symmetric diagonally dominant linear system Ax=b from the system on a symmetric diagonally dominant matrix, wherein the matrix A of the equivalent linear system Ax=b has negative off-diagonal entries and zero row su...

We show an improved parallel algorithm for decomposing an undirected unweighted graph into small diameter pieces with a small fraction of the edges in between. These decompositions form critical subroutines in a number of graph algorithms. Our algorithm builds upon the shifted shortest path approach introduced in [Blelloch, Gupta, Koutis, Miller, P...

We describe a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is O(f log n log Δ) where f is the output complexity of the Voronoi diagram and Δ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the...

Spectral Graph Theory is the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solver for Symmetric Diagonally Dominate systems (SDD). This seemingly restrictive class of systems has received much interest in the last 15 years. Both algorithm design theory and practical implem...

We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2O(d)(n log n + m)), where n is the input size, m is the output point set size, and...

We describe a new algorithm for computing the Voronoi diagram of a set of $n$ points in constant-dimensional Euclidean space. The running time of our algorithm is $O(f \log n \log \Delta)$ where $f$ is the output complexity of the Voronoi diagram and $\Delta$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite t...

We show faster algorithms for solving regression problems based on estimating statistical leverage scores. A growing number of applications involve large, sparse, n*d matrices A where n>>d. For many of these applications the more expensive operations involve the d*d matrix A^TA. When n is much larger than d, the running time bottleneck is in the co...

There has been significant interest and progress recently in algorithms that solve regression problems involving tall and thin matrices in input sparsity time. These algorithms find shorter equivalent of a n*d matrix where n >> d, which allows one to solve a poly(d) sized problem instead. In practice, the best performances are often obtained by inv...

We present faster algorithms for approximate maximum flow in undirected graphs with good separator structures, such as bounded genus, minor free, and geometric graphs. Given such a graph with $n$ vertices, $m$ edges along with a recursive $\sqrt{n}$-vertex separator structure, our algorithm finds an $1-\epsilon$ approximate maximum flow in time $\t...

The solution of linear systems is a problem of fundamental theoretical importance but also one with a myriad of applications in numerical mathematics, engineering, and science. Linear systems that are generated by real-world applications frequently fall into special classes. Recent research led to a fast algorithm for solving symmetric diagonally d...

This document describes four projects that, taken together, serve as a proof by example of the efficacy of a perception-motivated strategy for making graphics tools. More specifically, this strategy involves first selecting local features motivated by human perception and domain understanding, and then building algorithms that allow users to intera...

We present a new algorithm to mesh an arbitrary piece-wise linear complex in three dimensions. The algorithm achieves an O(n log ∆ + m) runtime where n, m, and ∆ are the input size, the output size, and spread respec-tively. This represents the first non-trivial runtime guar-antee for this class of input. The new algorithm extends prior work on run...

The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a $1-\epsilon$ approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem...

We study the geometric properties of point sets that arise in the generation of bounded aspect-ratio meshes and present a constructive formulation to define distributions that allow arbitrary refinements. This formulation can be used to define distributions with one or more singularities, which do not occur in the uniform case but do occur in mesh...

We present the design and analysis of a near linear-work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input of a SDD $n$-by-$n$ matrix $A$ with $m$ non-zero entries and a vector $b$, our algorithm computes a vector $\tilde{x}$ such that $\norm[A]{\tilde{x} - A^+b} \leq \vareps \cdot \norm[A]{A^+b}$ in $O(m\l...

We study theoretical runtime guarantees for a class of optimization problems that occur in a wide variety of inference problems. These problems are motivated by the LASSO framework and have applications in machine learning and computer vision. Our work shows a close connection between these problems and core questions in algorithmic graph theory. W...

We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparison-based algorithm runs in O(n log n + m) time, where n is the input size and m is the output size, and with constants depending only on the dimension and the desired element q...

The development of cancer is largely driven by the gain or loss of subsets of the genome, promoting uncontrolled growth or disabling defenses against it. Denoising array-based Comparative Genome Hybridization (aCGH) data is an important computational problem central to understanding cancer evolution. In this article, we propose a new formulation of...

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n×n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax̅ = b for some (unknown) vector x̅, our algorithm computes a vector x such that ∥x-x̅∥A≤ϵ∥x̅∥A1 in time Õ (m log n log (1/ϵ))2. The solver utilize...

We present an algorithm that on input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries constructs in time ${\tilde O}(m\log n)$ in the RAM model a solver which on input of a vector $b$ computes a vector ${x}$ satisfying $||{x}-A^{+}b||_A

In this work, we introduce the notion of triangle sparsifiers, i.e., sparse graphs which are approximately the same to the original graph with re-spect to the triangle count. This results in a practical triangle counting method with strong theoretical guarantees. For instance, for unweighted graphs we show a randomized algorithm for approximately c...

Memory bandwidth is a major limiting factor in the scalability of parallel iterative algorithms that rely on sparse matrix-vector multiplication (SpMV). This paper introduces Hierarchical Diagonal Blocking (HDB), an approach which we believe captures many of the existing optimization techniques for SpMV in a common representation. Using this repres...

The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link...

We present the IteratedTverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set S⊂Rd with running time sub-exponential in d. The algorithm is a derandomization of the IteratedRadon algorithm of Clarkson et al. (International Journal of Computational Geometry and Applications 6 (3) (1996) 357–377) and i...

We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental sparsifier (G) over cap with n - 1 + m/k edges, such that the condition number of G with (G) over cap is bounded above by (O) over tilde (k log(2) n), with probability 1 - p. The algorithm runs in time (O) over tilde ((m log n + n log...

We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud $P$ in Euclidean space $\R^d$. Classical approaches rely on the \v Cech, Rips, $\alpha$-complex, or witness complex filtrations of $P$, whose complexities scale up very badly with $d$....

Let $P=(P_1, P_2, \ldots, P_n)$, $P_i \in \field{R}$ for all $i$, be a signal and let $C$ be a constant. In this work our goal is to find a function $F:[n]\rightarrow \field{R}$ which optimizes the following objective function: $$ \min_{F} \sum_{i=1}^n (P_i-F_i)^2 + C\times |\{i:F_i \neq F_{i+1} \} | $$ The above optimization problem reduces to sol...

We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\em incremental sparsifier} $\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\hat{G}$ is bounded above by $\tilde{O}(k\log^2 n)$, with probability $1-p$. The algorithm runs in time $$\tilde{O}((m \log{n} + n\...

We apply ideas from mesh generation to improve the time and space complexity of computing the persistent homology of a point set in R d . The traditional approach to persistence starts with the α-complex of the point set and thus incurs the O(n ⌊d/2⌋) size of the Delaunay triangulation. The common alternative is to use a Rips complex and then to tr...

Counting the number of triangles in a graph is a beautiful algo- rithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the tran- sitivity ratio involve the execution of a triangle counting algorithm. Furthermore, sev...

We present the Iterated-Tverberg algorithm, the first de- terministic algorithm for computing an approximate center- point of a set S ∈ Rd with running time sub-exponential in d. The algorithm is a derandomization of the Iterated- Radon algorithm of Clarkson et al and is guaranteed to terminate with an O(1/d2)-center. Moreover, it returns a polynom...

Triangle counting is an important problem in graph mining. Clustering coefficients of vertices and the transitivity ratio of the graph are two metrics often used in complex network analysis. Furthermore, triangles have been used successfully in several real-world applications. However, exact triangle counting is an expensive computation. In this pa...

Typicalvolumemeshesinthreedimensionsaredesignedtoconformtoanunderlyingtwo-dimensional surface mesh, with volume mesh element size growing larger away from the surface. The surface mesh may be uniformly spaced or highly graded, and may have fine resolution due to extrinsic mesh size concerns. When we desire that such a volume mesh have good aspect r...

There are many depth measures on point sets that yield centerpoint theorems. These theorems guarantee the existence of points of a specified depth, a kind of geo- metric median. However, the deep point guaranteed to exist is not guaranteed to be among the input, and often, it is not. The �-wedge depth of a point with respect to a point set is a nat...

We consider the problem of decomposing a weighted graph with n vertices into a collection P of vertex disjoint clus- ters such that, for all clusters C 2 P, the graph induced by the vertices in C and the edges leaving C, has conduc- tance bounded below by . We show that for planar graphs we can compute a decomposition P such that |P| < n/ , where i...

Most modern meshing algorithms produce asymptoti- cally optimal size output. However, the size of the opti- mal mesh may not be bounded by any function of n. In this paper, we introduce well-paced point sets and prove that these will produce linear size outputs when meshed with any "size-optimal" meshing algorithm. This work generalizes all previou...

Provably correct algorithms for meshing difficult domains in three dimensions have been recently developed in the literature. These algorithms handle the problem of sharp angles (< / 2) between segments and between facets by constructing protective collars around these regions. The collars are approximately sized according to the local fea- ture si...

We present a new meshing algorithm for the plane, Overlay Stitch Meshing (OSM), that accepts as input an arbitrary Planar Straight Line Graph and produces a triangulation with all angles smaller than 170-. The output triangulation has size that is competitive with any optimal size mesh having bounded largest angle. The competitive ratio is O(log(L=...

The authors recently introduced the technique of sparse mesh re- finement to produce the first near-optimal sequential time bo unds of O(n lg L/s+m) for inputs in any fixed dimension with piecewise- linear constraining (PLC) features. This paper extends that work to the parallel case, refining the same inputs in time O(lg(L/s) lg m) on an EREW PRAM...

The recent Sparse Voronoi Refinement (SVR) Algorithm for mesh generation has the fastest theoretical bounds for runtime and memory usage. We present a robust practical software implementation of the SVR for meshing a piecewise linear complex in 3 dimensions. Our software is competitive in runtime with state of the art freely available pac kages on...

We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector x such that ||x--x||A ≤ ε, in O(n1/6+clog(1/ε)) parallel time, doing O(nlog(1/ε)) work, where c is any positive c...

A new topological representation of surfaces in higher dimensions, "cell-chains" is devel- oped. The representation is a generalization of Brisson's cell-tuple data structure. Cell-chains are iden- tical to cell-tuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for m...

We introduce a family of spectral partitioning methods. Edge separators of a graph are produced by iteratively reweighting the edges until the graph disconnects into the prescribed number of components. At each iteration a small number of eigenvectors with small eigenvalue are computed and used to determine the reweighting. In this way spectral rou...

We present a new algorithm, Sparse Voronoi Refinement, that produces a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in output-sensitive time O(nlog(L/s) + m), with con- stants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including...

In 1995, Gremban, Miller, and Zagha introduced support-tree preconditioners and a parallel algorithm called support-tree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n x n Laplacian matrix. A Laplacian is a symmetric matrix in which the off-diagonal entries are non-positive, and the row and column sums are...

An "adaptive" variant of Ruppert's Algorithm for producing quality triangular planar meshes is introduced. The algorithm terminates for arbitrary Planar Straight Line Graph (PSLG) input. The algorithm outputs a Delaunay mesh where no triangle has minimum angle smaller than about 26.45° except "across" from small angles of the input. No angle of the...

Given a general graph G, a fundamental problem is to find a spanning tree H that best approximates G by some measure. Often this measure is some combination of the congestion and dilation of an embedding of G into H. One example is the routing time p(G, H) ≤ O (congestion + dilation), the number of steps necessary to route pairwise demands G on net...

In this paper we present a Delaunay refinement algorithm for generating good aspect ratio and optimal size triangulations. This is the first algorithm known to have sub-quadratic running time. The algorithm is based on the extremely popular Delaunay refinement algorithm of Ruppert. We know of no prior refinement algorithm with an analyzed subquadra...

Point location in dynamic Delaunay triangulations is a prob- lem that as yet has no elegant solution. Current approaches either only give guarantees against a weakened adversary, or require superlinear space. In this paper we propose that we should seek intuition from balanced binary search trees, where rotations are used to maintain a shallow wors...

In this paper we present an application of our B ezier-based approach to moving meshes (1) to Navier-Stokes simulations with several immersed elastic membranes. By a moving mesh we mean one that moves with the material and is adapted to maintain good aspect ratio triangles of minimal size. The adaptations we employ include point insertion and remov...

An "adaptive" variant of Ruppert's Algorithm for producing quality triangular planar meshes is introduced. The algorithm terminates for arbitrary Planar Straight Line Graph (PSLG) input. The algorithm outputs a Delaunay mesh where no triangle has minimum angle smaller than 26.45 deg. except "across" from small angles of the input. No angle of the o...

We present a new framework for maintaining the quality of two dimensional triangular moving meshes. The use of curved elements is the key idea that allows us to avoid excessive refinement and still obtain good quality meshes consisting of a low number of well shaped elements. We use B-splines curves to model object boundaries, and objects are meshe...

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 nonnegative 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 conve...

The classical meshing problem is to construct a triangulation of a region that conforms to the boundary, is as coarse as possible, and is constructed from simplices having bounded aspect ratio. In this paper we present a fully incremental Delaunay re nement algorithm. The algorithm is an extension of one introduced by Ruppert. The algorithm is full...

Many important phenomena in science and engineering, including our motivating problem of microstructural blood flow, can be modeled as flows with dynamic interfaces. The major challenge faced in simulating such flows is resolving the interfacial motion. Lagrangian methods are ideally suited for such problems, since interfaces are naturally represen...

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(log n(log nd)) using M(n) processors, where M(n) is the number of processors required for multiplying n Theta n matrices over the semi-ring R in O(log n) time.

. In this paper we present a sphere-packing technique for Delaunay-based mesh generation, refinement and coarsening. We have previously established [10] that a bounded radius of ratio of circumscribed sphere to smallest tetrahedra edge is sufficient to get optimal rates of convergence for approximate solutions of Poisson's equation constructed usin...

We study the geometric properties of point sets that arise in the generation of bounded aspect-ratio meshes and present a constructive formulation to define distributions that allow arbitrary refinements. This formulation can be used to define distributions with one or more singularities, which do not occur in the uniform case but do occur in mesh...

In this paper we show that the control volume algorithm for the solution of Poisson's equation in three dimensions will converge even if the mesh contains a class of very flat tetrahedra (slivers). These tetrahedra are characterized by the fact that they have modest ratios of diameter to shortest edge, but large circumscribing to inscribed sphere r...

In this paper we are concerned with developing a practical parallel algorithm for Delaunay triangulation that works well on general distributions, particularly those that arise in Scientific Computation. Although there have been many theoretical algorithms for the problem, and some implementations based on bucketing that work well for uniform distr...

A hierarchical gradient of an unstructured mesh M 0 is a sequence of meshes M 1 ; . . . ; M k such that jM k j is smaller than a given threshold mesh size b. The gradient is well-conditioned if for each i in the range 1 i k, (1) M i is well-shaped, namely, elements of M i have a bounded aspect ratio; and (2) M i is a coarsened approximation of M iG...

Graphs that arise from the nite element or nite dierence methods often include geometric information such as the coordinates of the nodes of the graph. The geometric separator algorithm of Miller, Teng, Thurston, and Vavasis uses some of the available geometric information to nd small node separators of graphs. The algorithm utilizes a random sampl...

A bounded aspect-ratio coarsening sequence of an unstructured mesh M 0 is a sequence of meshes M 1 ,⋯,M k such that: M i is bounded aspect-ratio mesh, and M i is an approximation of M i-1 that has fewer elements, where a mesh is called a bounded aspect-ratio mesh if all its elements are of bounded aspect-ratio. The sequence is node-nested if the se...

In this paper we demonstrate that tradeoffs can be made between parallelism and fill in nested dissection algorithms for Gaussian elimination, both in theory and in practice. We present a new "less parallel nested dissection" algorithm (LPND), and prove that, unlike the standard nested dissection algorithm, when applied to a chordal graph LPND find...

New estimates are established for the error between a function and its linear interpolant over a triangular domain. Results previously established using compactness arguments are established here using combinatorial arguments which provide explicit estimates of the constants. New results include new path embeddings for “convex” graphs and bounds on...

This paper is a report on ongoing work in developing automated systems for the partitioning, placement, and routing of data that is necessary for the efficient parallel solution of large problems in scientific computing, specifically the numerical solution of partial differential equations. Many of these problems have as an iterated inner loop the...

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an nn Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix #; the best possible bound...

We propose a class of graphs that would occur naturally in finite-element and finite-difference problems and we prove a bound on separators for this class of graphs. Graphs in this class are embedded in $d$-dimensional space in a certain manner. For d-dimensional graphs our separator bound is O(n(d-1)d), which is the best possible bound. We also pr...

Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there has not been much prior 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...

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(log n(log nd)) using M(n) processors, where M(n) is the number of processors required for multiplying n Theta n matrices over the semi-ring R in O(log n) time. Appears in SIAM J. Co...

We present new geometrical and numerical analysis structure theorems for the Delaunay diagram of point sets in IR d for a fixed d where the point sets arise naturally in numerical methods. In particular, we show that if the largest ratio of the circum-radius to the length of smallest edge over all simplexes in the Delaunay diagram of P , DT (P ), i...

tween peak and achievable performance. One of the conclusions that can be drawn from these benchmarks is that machines with high communication bandwidth perform well across the board, whereas peak floating-point performance is relevant only on embarrassingly parallel problems. The benchmarks, however, still do not reveal the full cost of inadequate...

A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system Γ, there is a sphere S that intersects at most O(k1/dn1−1/d) balls of Γ and divides the remainder of Γ into two parts: those in the interior and those in the exterior of the sphere S, respectivel...

An abstract is not available.