## Papers and Talks by Kenneth L. Clarkson

November 2016.
We investigate algorithms that learn to run faster: given a series of instances of a computational problem, these algorithms use characteristics of past instances to sharpen their performance for new instances. We have found such *self-improving* algorithms for sorting a list of numbers, and for some problems on planar point sets: computing Delaunay triangulations, coordinate-wise maxima, and convex hulls. Under the assumption that input instances are random variables, each algorithm begins with a training phase during which it adjusts itself to the distribution of the input instances; this is followed by a stationary regime in which the algorithm runs in its optimized version. In this setting, an algorithm must take expected time proportional to the input size $n$, and to the entropy $E$ of the output: the input must be touched, and the output must be communicated. Our algorithms achieve $O(n+E)$ expected running times in the stationary regime for all the problems mentioned except convex hulls, where $O(n\log\log n + E)$ expected time is needed.
Notes: This work was done 2008-2012 by Nir Ailon, Bernard Chazelle, Ding Liu, Wolfgang Mulzer, C. Seshadhri, and the speaker.

From the excruciatingly difficult to the achingly elegant, Bernard Chazelle's work on algorithms, especially geometric or natural ones, has been profoundly influential. I'll sketch a few examples that have been inspiring to me, including 1-dimensional range queries, low-stabbing spanning trees, high-order Voronoi diagram construction, deterministic constructions, and the s-energy of a system.
Notes: Talk for Bernard's birthday; I didn't talk about everything in the abstract.

with David P. Woodruff.
In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 310--329. IEEE, 2015.
In the subspace approximation problem, we seek a $k$-dimensional subspace $F$ of $\R^d$ that minimizes the sum of $p$-th powers of Euclidean distances to a given set of $n$ points $a_1, \ldots, a_n \in \R^d$, for $p \geq 1$. More generally than minimizing $\sum_i \dist(a_i,F)^p$, we may wish to minimize $\sum_i M(\dist(a_i,F))$ for some loss function $M()$, for example, $M$-Estimators, which include the Huber and Tukey loss functions. Such subspaces provide alternatives to the singular value decomposition (SVD), which is the $p=2$ case, finding such an $F$ that minimizes the sum of squares of distances. For $p \in [1,2)$, and for typical $M$-Estimators, the minimizing $F$ gives a solution that is more robust to outliers than that provided by the SVD. We give several algorithmic results for these robust subspace approximation problems. We state our results as follows, thinking of the $n$ points as forming an $n \times d$ matrix $A$, and letting $\nnz{A}$ denote the number of non-zero entries of $A$. Our results hold for $p\in [1,2)$. We use $\poly(n)$ to denote $n^{O(1)}$ as $n\rightarrow\infty$.
• For minimizing $\sum_i \dist(a_i,F)^p$, we give an algorithm running in $O(\nnz{A} + (n+d)\poly(k/\eps) + \exp(\poly(k/\eps)))$ time which outputs a $k$-dimensional subspace $F$ whose cost is at most a $(1+\eps)$-factor larger than the optimum.
• We show that the problem of minimizing $\sum_i \dist(a_i, F)^p$ is NP-hard, even to output a $(1+1/\poly(d))$-approximation. This extends work of Deshpande et al. (SODA, 2011) which could only show NP-hardness or UGC-hardness for $p > 2$; their proofs critically rely on $p > 2$. Our work resolves an open question of [Kannan Vempala, NOW, 2009]. Thus, there cannot be an algorithm running in time polynomial in $k$ and $1/\eps$ unless P = NP. Together with prior work, this implies that the problem is NP-hard for all $p \neq 2$.
• For loss functions for a wide class of $M$-Estimators, we give a problem-size reduction: for a parameter $K=(\log n)^{O(\log k)}$, our reduction takes $O(\nnz{A}\log n + (n+d)\poly(K/\eps))$ time to reduce the problem to a constrained version involving matrices whose dimensions are $\poly(K\eps^{-1}\log n)$. We also give bicriteria solutions.
• Our techniques lead to the first $O(\nnz{A} + \poly(d/\eps))$ time algorithms for $(1+\eps)$-approximate regression for a wide class of convex $M$-Estimators. This improves prior results, which were $(1+\eps)$-approximation for Huber regression only, and $O(1)$-approximation for a general class of $M$-Estimators.

Tutorial for Gene Golub Summer School, July 2015.

with David P. Woodruff.
In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 921--939. SIAM, 2015.
We give algorithms for the $M$-estimators $\min_x \norm{Ax-b}_G$, where $A\in\R^{n\times d}$ and $b\in\R^n$, and $\norm{y}_G$ for $y\in\R^n$ is specified by a cost function $G:\R\mapsto \R^{\geq 0}$, with $\norm{y}_G \equiv \sum_i G(y_i)$. The $M$-estimators generalize $\ell_p$ regression, for which $G(x)=|x|^p$. We first show that the Huber measure can be computed up to relative error $\epsilon$ in $O(\nnz{A}\log n+ \poly(d (\log n)/\eps))$ time, where $\nnz{A}$ denotes the number of non-zero entries of the matrix $A$. Huber is arguably the most widely used $M$-estimator, enjoying the robustness properties of $\ell_1$ as well as the smoothness properties of $\ell_2$. We next develop algorithms for general $M$-estimators. We analyze the $M$-sketch, which is a variation of a sketch introduced by Verbin and Zhang in the context of estimating the earthmover distance. We show that the $M$-sketch can be used much more generally for sketching any $M$- estimator provided $G$ has growth that is at least linear and at most quadratic. Using the $M$-sketch we solve the $M$-estimation problem in $O(\nnz{A} + \poly(d \log n))$ time for any such $G$ that is convex, making a single pass over the matrix and finding a solution whose residual error is within a constant factor of optimal, with high probability.
Notes: Results for regression generalized in following paper

with David P. Woodruff.
In Annual ACM Symposium on Theory of Computing, pages 81--90, 2013.
Co-winner, Best Paper Award.
We design a new distribution over $\poly(r \eps^{-1}) \times n$ matrices S so that for any fixed $n\times d$ matrix A of rank r, with probability at least 9/10, $\norm{SAx}_2 = (1 \pm \eps)\norm{Ax}_2$ simultaneously for all $x\in\R^d$. Such a matrix $S$ is called a subspace embedding Furthermore, SA can be computed in $O(\nnz{A})+ \poly(d \eps^{-1})$ time, where $\nnz{A}$ is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at least $\Omega(nd\log d)$ time to achieve this property. We call our matrices $S$ sparse embedding matrices. Using our sparse embedding matrices, we obtain the fastest known algorithms for $(1+\eps)$-approximation for overconstrained least-squares regression, low-rank approximation, approximating all leverage scores, and $\ell_p$-regression. The leading order term in the time complexity of our algorithms is $O(\nnz{A})$ or $O(\nnz{A}\log n)$. We optimize the low-order $\poly(d/\eps)$ terms in our running times (or for rank-$k$ approximation, the $n*\poly(k/\eps)$ term), and show various tradeoffs. For instance, we also use our methods to design new preconditioners that improve the dependence on $\eps$ in least squares regression to $\log 1/\eps$. Finally, we provide preliminary experimental results which suggest that our algorithms are competitive in practice.
Notes: See also Jelani Nelson and Huy L. Nguyen's OSNAP work, and a related paper by Xiangrui Meng and Michael Mahoney.

with Petros Drineas, Malik Magdon-Ismail, Michael W. Mahoney, Xiangrui Meng, and David P. Woodruff.
In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 466--477. SIAM, 2013.
We provide fast algorithms for overconstrained $\ell_p$ regression and related problems: for an $n\times d$ input matrix $A$ and vector $b\in\R^n$, in $O(nd\log n)$ time we reduce the problem $\min_{x\in\R^d} \norm{Ax-b}_p$ to the same problem with input matrix $\tilde A$ of dimension $s \times d$ and corresponding $\tilde b$ of dimension $s\times 1$. Here, $\tilde A$ and $\tilde b$ are a coreset for the problem, consisting of sampled and rescaled rows of $A$ and $b$; and $s$ is independent of $n$ and polynomial in $d$. Our results improve on the best previous algorithms when $n\gg d$, for all $p\in [1,\infty)$ except $p=2$; in particular, they improve the $O(nd^{1.376+})$ running time of Sohler and Woodruff (STOC, 2011) for $p=1$, that uses asymptotically fast matrix multiplication, and the $O(nd^5\log n)$ time of Dasgupta et al. (SICOMP, 2009) for general $p$, that uses ellipsoidal rounding. We also provide a suite of improved results for finding well-conditioned bases via ellipsoidal rounding, illustrating tradeoffs between running time and conditioning quality, including a one-pass conditioning algorithm for general $\ell_p$ problems. To complement this theory, we provide a detdailed empirical evaluation of implementations of our algorithms for $p=1$, comparing them with several related algorithms. Among other things, our empirical results clearly show that, in the asymptotic regime, the theory is a very good guide to the practical performance of these algorithms. Our algorithms use our faster constructions of well-conditioned bases for $\ell_p$ spaces and, for $p=1$, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a matrix $\Pi: \R^n\mapsto \R^{O(d\log d)}$, found obliviously to $A$, that approximately preserves the $\ell_1$ norms: that is, $\norm{Ax}_1 \approx \norm{\Pi Ax}_1$, for all $x$, with distortion $O(d^{2+\eta} \log d)$, for an arbitrarily small constant $\eta>0$; and, moreover, $\Pi A$ can be computed in $O(nd\log d)$ time. The techniques underlying our Fast Cauchy Transform include fast Johnson-Lindenstrauss transforms, low-coherence matrices, and rescaling by Cauchy random variables.

with Wolfgang Mulzer and C. Seshadhri.
Preliminary version: Self-improving algorithms for coordinate-wise maxima, SoCG '12: Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry, 2012.
Finding the coordinate-wise maxima and the convex hull of a planar point set are probably the most classic problems in computational geometry. We consider these problems in the self-improving setting. Here, we have $n$ distributions $\cD_1, \ldots, \cD_n$ of planar points. An input point set $(p_1, \ldots, p_n)$ is generated by taking an independent sample $p_i$ from each $\cD_i$, so the input is distributed according to the product $\cD = \prod_i \cD_i$. A self-improving algorithm repeatedly gets inputs from the distribution $\cD$ (which is a priori unknown), and it tries to optimize its running time for $\cD$. The algorithm uses the first few inputs to learn salient features of the distribution $\cD$, before it becomes fine-tuned to $\cD$. Let $\OPTMAX_\cD$ (resp. $\OPTCH_\cD$) be the expected depth of an optimal linear comparison tree computing the maxima (resp. convex hull) for $\cD$. Our maxima algorithm eventually achieves expected running time $O(\OPTMAX_\cD + n)$. Furthermore, we give a self-improving algorithm for convex hulls with expected running time $O(\OPTCH_\cD + n\log\log n)$. Our results require new tools for understanding linear comparison trees. In particular, we convert a general linear comparison tree to a restricted version that can then be related to the running time of our algorithms. Another interesting feature is an interleaved search procedure to determine the likeliest point to be extremal with minimal computation. This allows our algorithms to be competitive with the optimal algorithm for $\cD$.
Notes: Simpler and more verifiable algorithm for convex hulls than the previous one of ours, whose proof is buggy.

Tutorial for kickoff workshop of CG Learning Project, June 2011.
The outline:
• Some geometric problems, algorithms, analysis
• Approximation algorithms
• First order methods (no second derivatives)
• Analysis using on-line convex optimization formalism
• But the problems and algorithms are not on-line
• Fast randomized approximate evaluation of dot products
• Alternatives to JL
• Matrix Bernstein and applications
• A primal-dual algorithm
• (Significant) algorithmic results are joint with Elad Hazan and David Woodruff

Tutorial for MADALGO & CTIC Summer School on High-Dimensional Geometric Computing, Aarhus University, Denmark, August 2011.
The idea of a metric space is among the most basic of geometric concepts, and so appears in a great variety of applications and algorithms, sometimes in disguise. This is a light survey of concepts and constructions associated with metric spaces, including:
• Metric transformations
• $\epsilon$-nets, the greedy algorithm, and applications
• A non-greedy algorithm and a few non-$\epsilon$-nets
• Box dimension and coping with finiteness
• Definitions of dimension that make sense for finite sets
• Estimation of dimension using finite samples
• Filtrations
• Of random subsets, k-medians, $\epsilon$-nets
• Neighbors, generalizing "Delaunay neighbors"
• For witness complexes and NN searching
• Interpolation
• Interpolation of scattered data, Laplacians
• Interpolation in metric spaces
• A curious approach via "witness stealing"
A general theme here is "coping with finiteness", in trying to apply concepts developed for infinite sets to a finite setting, or finding frameworks that apply in both settings. Another theme is the extent to which some constructions allow the resolution, or scale of measurement, to be determined by the data.
Notes: This is yet another version of a long continuing interest, with pointers to work on meshes, and nearest neighbor searching. The animations' javascript code is not for the squeamish.

with Elad Hazan and David P. Woodruff.
Journal of the ACM, 59(5), 2012.
Preliminary version in FOCS 2010: Proceedings of the Fifty-First Symposium on Foundations of Computer Science, 2010.
We give sublinear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions of these problems, such as SVDD, hard margin SVM, and $L_2$-SVM, for which sublinear-time algorithms were not known before. These new algorithms use a combination of a novel sampling techniques and a new multiplicative update algorithm. We give lower bounds which show the running times of many of our algorithms to be nearly best possible in the unit-cost RAM model. We also give implementations of our algorithms in the semi-streaming setting, obtaining the first low pass polylogarithmic space and sublinear time algorithms achieving arbitrary approximation factor.

with Wolfgang Mulzer and C. Seshadhri.
In SODA 2010: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 2010.
We describe an algorithm for computing planar convex hulls in the self-improving model: given a sequence $I_1, I_2, \ldots$ of planar $n$-point sets, the upper convex hull $\conv(I)$ of each set $I$ is desired. We assume that there exists a probability distribution $D$ on $n$-point sets, such that the inputs $I_j$ are drawn independently according to $D$. Furthermore, $D$ is such that the individual points are distributed independently of each other. In other words, the $i$'th point is distributed according to $D_i$. The $D_i$'s can be arbitrary but are independent of each other. The distribution $D$ is not known to the algorithm in advance. After a learning phase of $n^\epsilon$ rounds, the expected time to compute $\conv(I)$ is $O(n + H(\conv(I)))$. Here, $H(\conv(I))$ is the entropy of the output, which is a lower bound for the expected running time of any algebraic computation tree that computes the convex hull. (More precisely, $H(\conv(I))$ is the minimum entropy of any random variable that maps $I$ to a description of $\conv(I)$ and to a labeling scheme that proves nonextremality for every point in $I$ not on the hull.) Our algorithm is thus asymptotically optimal for $D$.
Notes: Bug found! Please see our later paper.

with Kun Liu and Evimaria Terzi.
In Link Mining: Models Algorithms and Applications. 2010.

with Barna Saha, Ioana Stanoi.
In ICDE, pages 285--296, 2010.

with Chandra Chekuri and Sariel Har-Peled.
ACM Transactions on Algorithms, 9(1), 2012.
Preliminary version in SoCG '09: Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, 2009.
We consider the set multi-cover problem in geometric settings. Given a set of points $P$ and a collection of geometric shapes (or sets) $\cal F$, we wish to find a minimum cardinality subset of $\cal F$ such that each point $p \in P$ is covered by (contained in) at least $\textrm{d}(p)$ sets. Here $\textrm{d}(p)$ is an integer demand (requirement) for $p$. When the demands $\textrm{d}(p)=1$ for all $p$, this is the standard set cover problem. The set cover problem in geometric settings admits an approximation ratio that is better than that for the general version. In this paper, we show that similar improvements can be obtained for the multi-cover problem as well. In particular, we obtain an $O(\log\ OPT)$ approximation for set systems of bounded VC-dimension, and an $O(1)$ approximation for covering points by half-spaces in three dimensions and for some other classes of shapes.

with David P. Woodruff.
In STOC '09: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009.
We give near-optimal space bounds in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank. In the streaming model, sketches of input matrices are maintained under updates of matrix entries; we prove results for turnstile updates, given in an arbitrary order. We give the first lower bounds known for the space needed by the sketches, for a given estimation error $\epsilon$. We sharpen prior upper bounds, with respect to combinations of space, failure probability, and number of passes. The sketch we use for matrix $A$ is simply $S^TA$, where $S$ is a sign matrix. Our results include the following upper and lower bounds on the bits of space needed for $1$-pass algorithms. Here $A$ is an $n\times d$ matrix, $B$ is an $n\times d'$ matrix, and $c := d+d'$. These results are given for fixed failure probability; for failure probability $\delta>0$, the upper bounds require a factor of $\log(1/\delta)$ more space. We assume the inputs have integer entries specified by $O(\log(nc))$ bits, or $O(\log(nd))$ bits. The Frobenius matrix norm is used.
(Matrix Product)
Output matrix $C$ with $$|| A^TB-C || \leq \epsilon ||A|| ||B||.$$ We show that $\Theta(c\epsilon^{-2}\log(nc))$ space is needed.
(Linear Regression)
For $d'=1$, so that $B$ is a vector $b$, find $x$ so that $$||Ax-b|| \leq (1+\epsilon)\min_{x' \in \reals^d} || Ax'-b ||.$$ We show that $\Theta(d^2\epsilon^{-1}\log(nd))$ space is needed.
(Rank-$k$ Approximation)
Find matrix $\hat A_k$ of rank no more than $k$, so that $$|| A - \hat A_k|| \leq (1+\epsilon)||A-A_k||,$$ where $A_k$ is the best rank-$k$ approximation to $A$. Our lower bound is $\Omega(k\epsilon^{-1}(n+d)\log(nd))$ space, and we give a one-pass algorithm matching this when $A$ is given row-wise or column-wise. For general updates, we give a one-pass algorithm needing $$O(k\epsilon^{-2}(n + d/\epsilon^2)\log(nd))$$ space. We also give upper and lower bounds for algorithms using multiple passes, and a bicriteria low-rank approximation.

Survey talk given at SoCG; please see later slides mostly incorporating this material, 2008.

with Nir Ailon, Bernard Chazelle, Ding Liu, Wolfgang Mulzer, and C. Seshadhri.
SIAM Journal on Computing, 40(2), 2011.
Preliminary version: Self-improving algorithms for Delaunay triangulations, SoCG '08: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, 2008. The prior paper on Delaunay triangulations with C. Seshadhri was combined with the paper by the other authors that introduced the model.
We investigate ways in which an algorithm can improve its expected performance by fine-tuning itself automatically with respect to an unknown input distribution $D$. We assume here that $D$ is of product type. More precisely, suppose that we need to process a sequence $I_1, I_2,\ldots$ of inputs $I = (x_1, x_2, \ldots, x_n)$ of some fixed length $n$, where each $x_i$ is drawn independently from some arbitrary, unknown distribution $D_i$. The goal is to design an algorithm for these inputs so that eventually the expected running time will be optimal for the input distribution $D = D_1 * D_2 * ... * D_n$. We give such self-improving algorithms for two problems: (i) sorting a sequence of numbers and (ii) computing the Delaunay triangulation of a planar point set. Both algorithms achieve optimal expected limiting complexity. The algorithms begin with a training phase during which they collect information about the input distribution, followed by a stationary regime in which the algorithms settle to their optimized incarnations.

In SoCG '08: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, 2008.
We study the problem of two-dimensional Delaunay triangulation in the self-improving algorithms model. We assume that the $n$points of the input each come from an independent, unknown, and arbitrary distribution. The first phase of our algorithm builds data structures that store relevant information about the input distribution. The second phase uses these data structures to efficiently compute the Delaunay triangulation of the input. The running time of our algorithm matches the information-theoretic lower bound for the given input distribution, implying that if the input distribution has low entropy, then our algorithm beats the standard $\Omega(n \log n)$ bound for computing Delaunay triangulations. Our algorithm and analysis use a variety of techniques: $\epsilon$-nets for disks, entropy-optimal point-location data structures, linear-time splitting of Delaunay triangulations, and information-theoretic arguments.

In SoCG '08: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, 2008.
The Johnson-Lindenstrauss random projection lemma gives a simple way to reduce the dimensionality of a set of points while approximately preserving their pairwise distances. The most direct application of the lemma applies to a finite set of points, but recent work has extended the technique to affine subspaces, curves, and general smooth manifolds. Here the case of random projection of smooth manifolds is considered, and a previous analysis is sharpened, reducing the dependence on such properties as the manifold's maximum curvature.
Versions:
• July 22, 2008: Various corrections (actually done March 2008)
• December 2007: First version

ACM Transactions on Algorithms, 6(4), 2010.
Preliminary version in SODA '08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2008.
The problem of maximizing a concave function $f(x)$ in a simplex $S$ can be solved approximately by a simple greedy algorithm, that for given $k$ can find a point $x_{(k)}$ on a $k$-dimensional face such that $f(x_{(k)}) \ge f(x_*) - O(1/k)$, where $f(x_*)$ is the maximum value of $f$ in $S$. This algorithm and analysis were known before, and related to problems of statistics and machine learning, such as boosting, regression, and density mixture estimation. In other work, coming from computational geometry, the existence of $\epsilon$-coresets was shown for the minimum enclosing ball problem, by means of a simple greedy algorithm. Similar greedy algorithms, that are special cases of the Frank-Wolfe algorithm, were described for other enclosure problems. Here these results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.
Versions:
• July 22, 2008: Various corrections (actually done October 2007)
• June 19, 2007: Corrected probabilistic existence proof (per K. Varadarajan), typos
• April 29, 2007: Clearer statement of results, cleaner proof for algorithms with away steps, typos
• March 26, 2007: First version

with John D. Hobby.
In VTC-2007-Fall: IEEE Vehicular Technology Conference, 2007.
Winner, conference Best Paper award.
We give a model for the performance impact on wireless systems of the limitations of certain resources, namely, the base-station power amplifier and the available OVSF codes. These limitations are readily modeled in the loss model formulation as a stochastic knapsack. A simple and well-known recurrence of Kaufman and Roberts allows the predictions of the model to be efficiently calculated. We discuss the assumptions and approximations we have made that allow the use of the model. We have included the model in Ocelot, a Lucent tool for modeling and optimizing cellular phone systems. The model is fast to compute, differentiable with respect to the relevant parameters, and able to model broad ranges of capacity and resource use. These conditions are critical to our application of optimization.

with K. Georg Hampel and John D. Hobby.
In VTC-2007-Fall: IEEE Vehicular Technology Conference, 2007.
to appear, IEEE Transactions on Wireless Communications.
We investigate models for uplink interference in wireless systems. Our models account for the effects of outage probabilities. Such an accounting requires a nonlinear, even nonconvex model, since increasing interference at the receiving base station increases both mobile transmit power and outage probability, and this results in a complex interaction. Our system model always has at least one solution, a fixed point, and it is provably unique under certain reasonable conditions. Our main purpose is to model real wireless systems as accurately as possible, and so we test our models on realistic scenarios using data from a sophisticated simulator. Our algorithm for finding a fixed point works very well on such scenarios, and is guaranteed to find the fixed point when we can prove it is unique. A slightly simplified model reduces the main data structure for a $K$-sector market to $16K^2$ bytes of memory.

In STOC '06: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, 2006.
This work addresses the problem of approximating a manifold by a simplicial mesh, and the related problem of building triangulations for the purpose of piecewise-linear approximation of functions. It has long been understood that the vertices of such meshes or triangulations should be "well-distributed," or satisfy certain "sampling conditions." This work clarifies and extends some algorithms for finding such well-distributed vertices, by showing that they can be regarded as finding $\epsilon$-nets or Delone sets in appropriate metric spaces. In some cases where such Delone properties were already understood, such as for meshes to approximate smooth manifolds that bound convex bodies, the upper and lower bound results are extended to more general manifolds; in particular, under some natural conditions, the minimum Hausdorff distance for a mesh with $n$ simplices to a $d$-manifold $M$ is $$\Theta((\int_M\sqrt{|\kappa(x)|}/n)^{2/d})$$ as $n\rightarrow\infty$, where $\kappa(x)$ is the Gaussian curvature at point $x\in M$. We also relate these constructions to Dudley's approximation scheme for convex bodies, which can be interpreted as involving an $\epsilon$-net in a metric space whose distance function depends on surface normals. Finally, a novel scheme is given, based on the Steinhaus transform, for scaling a metric space by a Lipschitz function to obtain a new metric. This scheme is applied to show that some algorithms for building finite element meshes and for surface reconstruction can be also be interpreted in the framework of metric space $\epsilon$-nets.
Versions:
• Revised from 20/2/06: patches to upper bound proof, lower bound proof, many typos etc.
• Revised from version of 11/19/05: better upper bound proof, typos in Dudley example, cites peyre/cohen.

(Survey).
In G. Shakhnarovich, T. Darrell, and P. Indyk, editors, Nearest-Neighbor Methods for Learning and Vision: Theory and Practice, pages 15--59. MIT Press, 2006.
Given a set $S$ of points in a metric space with distance function $D$, the nearest-neighbor searching problem is to build a data structure for $S$ so that for an input query point $q$, the point $s\in S$ that minimizes $D(s,q)$ can be found quickly. We survey approaches to this problem, and its relation to concepts of metric space dimension. Several measures of dimension can be estimated using nearest-neighbor searching, while others can be used to estimate the cost of that searching. In recent years, several data structures have been proposed that are provably good for low-dimensional spaces, for some particular measures of dimension. These and other data structures for nearest-neighbor searching are surveyed.
Notes: Some dimensions discussed: box, packing, Hausdorff, Assouad, pointwise, information, correlation, quantization, energy, Renyi.

Discrete and Computational Geometry, 37(1):43--58, January 2007.
Preliminary version in SoCG '05: Proceedings of the Twenty-First Annual Symposium on Computational Geometry, 2005.
Given a collection $S$ of subsets of some set $U$, and $M \subset U$, the set cover problem is to find the smallest subcollection $C\subset S$ such that $M$ is a subset of the union of the sets in $C$. While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually $U = \reals^d$. Combining previously known techniques [BG,CF], we show that polynomial time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset $R\subset S$ and function $f()$, there is a decomposition of the complement $U\setminus\cup_{Y\in R} Y$ into an expected $f(|R|)$ regions, each region of a particular simple form. Under this condition, a cover of size $O(f(|C|))$ can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions $f(c)$ that are nearly linear, we obtain $o(\log c)$ approximation algorithms for covering by fat triangles, by pseudodisks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in $\reals^3$, and for guarding an $x$-monotone polygonal chain.
Notes: While no great claim of novelty was made for Theorem 2.2, it should be pointed out that the proof is very close to that by de Berg and Cheong (n Schwarzkopf). (The related proof appears only in the journal version of their paper. Thanks to Sariel Har-Peled for pointing this out.) Moreover, recent work (circa 2008) tightens some of the nonlinear cases; see also here.

In SODA '05: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2005.
Given an $n\times d$ matrix $A$ and an $n$-vector $b$, the $L_1$ regression problem is to find the vector $x$ minimizing the objective function $||Ax-b||_1$, where $||y||_1 \equiv \sum_i |y_i|$ for vector $y$. This paper gives an algorithm needing $O(n\log n)d^{O(1)}$ time in the worst case to obtain an approximate solution, with objective function value within a fixed ratio of optimum. Given $\epsilon>0$, a solution whose value is within $1+\epsilon$ of optimum can be obtained either by a deterministic algorithm using an additional $O(n)(d/\epsilon)^{O(1)}$ time, or by a Monte Carlo algorithm using an additional $O((d/\epsilon)^{O(1)})$ time. The analysis of the randomized algorithm shows that weighted coresets exist for $L_1$ regression. The algorithms use the ellipsoid method, gradient descent, and random sampling.

Let $S$ be a set of $n$ points in $d$ dimensions. A $k$-set of $S$ is a subset of size $k$ that is the intersection of $S$ with some open halfspace. This note shows that if the points of $S$ are random, with a coordinate-wise independent distribution, then the expected number of $k$-sets of $S$ is $O((k\log(en/k))^{d-1})2^d/(d-1)!$, as $k\log n->\infty$, with a constant independent of the dimension.

with John D. Hobby.
In VTC-2004-Spring: IEEE Vehicular Technology Conference, volume 3, pages 1534--1538, 2004.
We introduce a simple model of the effect of temporal variation in signal strength on active-set membership, for cellular phone systems that use the soft-handoff algorithm of IS-95a. This model is based on a steady-state calculation, and its applicability is confirmed by Monte Carlo studies.

Preliminary version presented at ALENEX99, 2003.
Given a set $S$ of $n$ sites (points), and a distance measure $d$, the nearest neighbor searching problem is to build a data structure so that given a query point $q$, the site nearest to $q$ can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distance function as a "black box". The structure is able to speed up nearest neighbor searching in a variety of settings, for example: points in low-dimensional or structured Euclidean space, strings under Hamming and edit distance, and bit vector data from an OCR application. The data structures are observed to need linear space, with a modest constant factor. The preprocessing time needed per site is observed to match the query time. The data structure can be viewed as an application of a "kd-tree" approach in the metric space setting, using Voronoi regions of a subset in place of axis-aligned boxes.

This paper gives an algorithm for solving linear systems, using a randomized version of incomplete $LU$ factorization together with iterative improvement. The factorization uses Gaussian elimination with partial pivoting, and preserves sparsity during factorization by randomized rounding of the entries. The resulting approximate factorization is then applied to estimate the solution. This simple technique, combined with iterative improvement, is demonstrated to be effective for a range of linear systems. When applied to medium-sized sample matrices for PDEs, the algorithm is qualitatively like multigrid: the work per iteration is typically linear in the order of the matrix, and the number of iterations to achieve a small residual is typically on the order of fifteen to twenty. The technique is also tested for a sample of asymmetric matrices from the Matrix Market, and is found to have similar behavior for many of them.

with Simon C. Borst, John Graybeal, Harish Viswanathan, and Phillip Whiting.
Bell Labs Technical Journal, 8(2):33--47, 2003.
3G wireless systems such as 3G-1X, 1xEV-DO and 1xEV-DV provide support for a variety of high-speed data applications. The success of these services critically relies on the capability to ensure an adequate QoS experience to users at an affordable price. With wireless bandwidth at a premium, traffic engineering and network planning play a vital role in addressing these challenges. We present models and techniques that we have developed for quantifying the QoS perception of 1xEV-DO users generating FTP or Web browsing sessions. We show how user-level QoS measures may be evaluated by means of a Processor-Sharing model which explicitly accounts for the throughput gains from multi-user scheduling. The model provides simple analytical formulas for key performance metrics such as response times, blocking probabilities and throughput. Analytical models are especially useful for network deployment and in-service tuning purposes due to the intrinsic difficulties associated with simulation-based optimization approaches. We discuss the application of our results in the context of Ocelot, which is a Lucent tool for wireless network planning and optimization.

with K. Georg Hampel, John D. Hobby, and Paul A. Polakos.
In VTC-2003-Fall: IEEE Vehicular Technology Conference, pages 927--932, 2003.
For 3G cellular networks, capacity is an important objective, along with coverage, when characterizing the performance of high-data-rate services. In live networks, the effective network capacity heavily depends on the degree that the traffic load is balanced over all cells, so changing traffic patterns demand dynamic network reconfiguration to maintain good performance. Using a four-cell sample network, and antenna tilt, cell power level and pilot fraction as adjustment variables, we study the competitive character of network coverage and capacity in such a network optimization process, and how it compares to the CDMA-intrinsic coverage-capacity tradeoff driven by interference. We find that each set of variables provides its distinct coverage-capacity tradeoff behavior with widely varying and application-dependent performance gains. The study shows that the impact of dynamic load balancing highly depends on the choice of the tuning variable as well as the particular tradeoff range of operation.

with John D. Hobby.
2003.
We give a simple analytic model of coverage probability for CDMA cellular phone systems under lognormally distributed shadow fading. Prior analyses have generally considered the coverage probability of a single antenna; here we consider the probability of coverage by an ensemble of antennas, using some independence assumptions, but also modeling a limited form of dependency among the antenna fades. We use the Fenton-Wilkinson approach of approximating the external interference $I_0$ as lognormally distributed. We show that our model gives a coverage probability that is generally within a few percent of Monte Carlo estimates, over a wide regime of antenna strengths and other relevant parameters.

In SODA '03: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2003.
Given a set of points $P\subset R^d$ and value $\epsilon>0$, an $\epsilon$-core-set $S \subset P$ has the property that the smallest ball containing $S$ is within $\epsilon$ of the smallest ball containing $P$. This paper shows that any point set has an $\epsilon$-core-set of size $\lceil 1/\epsilon\rceil$, and this bound is tight in the worst case. A faster algorithm given here finds an $\epsilon$-core-set of size at most $2/\epsilon$. These results imply the existence of small core-sets for solving approximate $k$-center clustering and related problems. The sizes of these core-sets are considerably smaller than the previously known bounds, and imply faster algorithms; one such algorithm needs $O(d n/\epsilon+(1/\epsilon)^{5})$ time to compute an $\epsilon$-approximate minimum enclosing ball (1-center) of $n$ points in $d$ dimensions. A simple gradient-descent algorithm is also given, for computing the minimum enclosing ball in $O(d n / \epsilon^{2})$ time. This algorithm also implies slightly faster algorithms for computing approximately the smallest radius $k$-flat fitting a set of points.
Notes: The ideas and algorithm have seen application in machine learning, including support vector regression, and computational biology (mentioned in the "supplementary notes" of the latter).

Manuscript, 2002.
Given a set of points $P\subset \reals^d$ and value $\epsilon>0$, an $\epsilon$-core-set $S \subset P$ has the property that the smallest ball containing $S$ is within $\epsilon$ of the smallest ball containing $P$. This paper shows that any point set has an $\epsilon$-core-set of size $\lceil 1/\epsilon\rceil$, and this bound is tight in the worst case. Some experimental results are also given, comparing this algorithm with a previous one, and with a more powerful, but slower one.
Versions:
• Revised May 2006: Removed dependence on size of smallest ball, consider variant, etc.

with Wim Sweldens and Alice Zheng.
IEEE Trans. Commun., 49(2):253--261, 2001.
We present an algorithm based on lattice reduction for the fast decoding of diagonal differential modulation across multiple antenna. While the complexity of the maximum likelihood algorithm is exponential both in the number of antenna and the rate, the complexity of our approximate lattice algorithm is polynomial in the number of antennas and the rate. We show that the error performance of our lattice algorithm is very close to the maximum likelihood algorithm.

Discrete and Computational Geometry, 22:63--93, 1999.
Preliminary version in STOC '97: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 1997.
Given a set $S$ of $n$ sites (points), and a distance measure $d$, the nearest neighbor searching problem is to build a data structure so that given a query point $q$, the site nearest to $q$ can be found quickly. This paper gives data structures for this problem when the sites and queries are in a metric space. One data structure, $D(S)$, uses a divide-and-conquer recursion. The other data structure, $M(S,Q)$, is somewhat like a skiplist. Both are simple and implementable. The data structures are analyzed when the metric space obeys a certain sphere-packing bound, and when the sites and query points are random and have distributions with an exchangeability property. This property implies, for example, that query point $q$ is a random element of $S\cup\{q\}$. Under these conditions, the preprocessing and space bounds for the algorithms are close to linear in $n$. They depend also on the sphere-packing bound, and on the logarithm of the distance ratio $\Upsilon(S)$ of $S$, the ratio of the distance between the farthest pair of points in $S$ to the distance between the closest pair. The data structure $M(S,Q)$ requires as input data an additional set $Q$, taken to be representative of the query points. The resource bounds of $M(S,Q)$ have a dependence on the distance ratio of $S\cup Q$. While $M(S,Q)$ can return wrong answers, its failure probability can be bounded, and is decreasing in a parameter $K$. Here $K\leq |Q|/n$ is chosen when building $M(S,Q)$. The expected query time for $M(S,Q)$ is $O(K\log n)\log\Upsilon(S\cup Q)$, and the resource bounds increase linearly in $K$. The data structure $D(S)$ has expected $O(\log n)^{O(1)}$ query time, for fixed distance ratio. The preprocessing algorithm for $M(S,Q)$ can be used to solve the all-nearest-neighbor problem for $S$ in $O(n(\log n)^2(\log\Upsilon(S))^2)$ expected time.
Notes: The assumption of a sphere-packing bound is equivalent to a bounded doubling constant, as discussed by Krauthgamer and Lee, which is more general than the growth-restricted or doubling measure assumption of Karger and Ruhl. See also Har-Peled and Mendel, and a survey.

Lecture notes, 1997.

The technique of randomized incremental construction allows a variety of geometric structures to be built quickly. This note shows that once such a structure is built, it is possible to store the geometric input data for it so that the structure can be built again by a randomized algorithm even more quickly. Except for the randomization, this generalizes the technique of Snoeyink and van Kreveld that applies to planar problems.

Given points moving with constant, but possibly different, velocities, the minimum moving diameter problem is to find the minimum, over all time, of the maximum distance between a pair of points at each moment. This note gives a randomized algorithm requiring $O(n \log n )$ expected time for this problem, in two and three dimensions. Also briefly noted is a randomized $O(n\log n\log\log n)$ expected-time algorithm for the discrete 1-center problem in three dimensions; in this problem, a member $p$ of a set $S$ of points is desired, whose maximum distance to $S$ is minimum over all points of $S$.
Notes: Slight foreshadowing of sublinear geometric algorithms.

In FOCS '94: Proceedings of the Thirty-Fifth Symposium on Foundations of Computer Science, pages 695--702, 1994.
A simple idea for speeding up the computation of extrema of a partially ordered set turns out to have a number of interesting applications in geometric algorithms; the resulting algorithms generally replace an appearance of the input size $n$ in the running time by an output size $A\leq n$. In particular, the $A$ coordinate-wise minima of a set of $n$ points in $R^d$ can be found by an algorithm needing $O(nA)$ time. Given $n$ points uniformly distributed in the unit square, the algorithm needs $n+O(n^{5/8})$ point comparisons on average. Given a set of $n$ points in $R^d$, another algorithm can find its $A$ extreme points in $O(nA)$ time. Thinning for nearest-neighbor classification can be done in time $O(n\log n)\sum_i A_i n_i$, finding the $A_i$ irredundant points among $n_i$ points for each class $i$, where $n=\sum_i n_i$ is the total number of input points. This sharpens a more obvious $O(n^3)$ algorithm, which is also given here. Another algorithm is given that needs $O(n)$ space to compute the convex hull of $n$ points in $O(nA)$ time. Finally, a new randomized algorithm finds the convex hull of $n$ points in $O(n\log A)$ expected time, under the condition that a random subset of the points of size $r$ has expected hull complexity $O(r)$. All but the last of these algorithms has polynomial dependence on the dimension $d$, except possibly for linear programming.
Notes: There is some overlap with the work of Chan and Ottman et al, in particular, for finding extreme points.

Merges two papers below, 1994.
This paper gives an algorithm for polytope covering: let $L$ and $U$ be sets of points in $R^d$, comprising $n$ points altogether. A cover for $L$ from $U$ is a set $C\subset U$ with $L$ a subset of the convex hull of $C$. Suppose $c$ is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than $c(5d\ln c)$, for $c$ large enough. The algorithm requires $O(c^2n^{1+\delta})$ expected time. (In this paper, $\delta$ will denote any fixed value greater than zero.) More exactly, the time bound is $$O(cn^{1+\delta}+c(nc)^{ 1/(1+\gamma/(1+\delta))}),$$ where $\gamma\equiv 1/\lfloor d/2\rfloor$. The previous best bounds were $c O(\log n)$ cover size in $O(n^d)$ time [MiS]. A variant algorithm is applied to the problem of approximating the boundary of a polytope with the boundary of a simpler polytope. For an appropriate measure, an approximation with error $\epsilon$ requires $c=O(1/\epsilon)^{(d-1)/2}$ vertices, and the algorithm gives an approximation with $c({5d^3\ln(1/\epsilon)})$ vertices. The algorithms apply ideas previously used for small-dimensional linear programming. The final result here applies polytope approximation to the the post office problem: given $n$ points (called sites) in $d$ dimensions, build a data structure so that given a query point $q$, the closest site to $q$ can be found quickly. The algorithm given here is given also a relative error bound $\epsilon$, and depends on a ratio $\rho$, which is no more than the ratio of the distance between the farthest pair of sites to the distance between the closest pair of sites. The algorithm builds a data structure of size $O(n(\log\rho)/\epsilon^{d/2}$ in time $O(n^2(\log\rho))/\epsilon^d$. With this data structure, closest-point queries can be answered in $O(\log n)/\epsilon^{d/2}$ time.
Notes: Brönnimann and Goodrich show that the same iterative randomized algorithm applies in the general setting of range spaces of bounded VC-dimension, and that a linear-sized $\epsilon$-net implies a constant-factor approximation algorithm. Their results are extended by this later paper.

In SoCG '94: Proceedings of the Tenth Annual Symposium on Computational Geometry, pages 160--164, 1994.

In WADS '93: Proceedings of the Third Workshop on Algorithms and Data Structures, pages 246--252, 1993.

with David Eppstein, Gary L. Miller, Carl Sturtivant, and Shang-Hua Teng.
International Journal of Computational Geometry and Applications, 6(3):357--377, 1996.
Preliminary version in SoCG '93: Proceedings of the Ninth Annual Symposium on Computational Geometry, 1993.
We give a practical and provably good Monte Carlo algorithm for approximating center points. Let $P$ be a set of $n$ points in $\reals^d$. A point $c \in \reals^d$ is a $\beta$-center point of $P$ if every closed halfspace containing $c$ contains at least $\beta n$ points of $P$. Every point set has a $1/(d+1)$-center point; our algorithm finds an $\Omega(1/d^2)$-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in $d$. Moreover, it can be optimally parallelized to require $O(\log^2d\log\log n)$ time. Our algorithm has been used in mesh partitioning methods and can be used in constructing high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak $\epsilon$-nets. We derive a variant of our algorithm whose time bound is fully polynomial in $d$ and linear in $n$, and show how to combine our approach with previous techniques to compute high quality center points more quickly.

Presentation at Fifth MSI-Stony Brook Workshop on Computational Geometry, 1996.
This talk sketches: a convenient method for computing the volumes of Voronoi regions; a proof that "area-stealing" natural neighbor interpolation works; a scheme for smoother natural neighbor interpolation alternative to Sibson's method; the interpolation scheme used in the "finite volume element" method; and the observation that the minimax piecewise-linear interpolant of a convex function is the (lower) convex hull.

In FOCS '92: Proceedings of the Thirty-First Symposium on Foundations of Computer Science, pages 387--395, Pittsburgh, PA, October 1992.
The problem of evaluating the sign of the determinant of a small matrix arises in many geometric algorithms. Given an $n\times n$ matrix $A$ with integer entries, whose columns are all smaller than $M$ in Euclidean norm, the algorithm given here evaluates the sign of the determinant $\det A$ exactly. The algorithm requires an arithmetic precision of less than $1.5n+2\lg M$ bits. The number of arithmetic operations needed is $O(n^3)+O(n^2)\log\OD(A)/\beta$, where $\OD(A)|\det A|$ is the product of the lengths of the columns of $A$, and $\beta$ is the number of "extra" bits of precision, $$\min\{\lg(1/\mathbf{u})-1.1n-2\lg n-2,\lg N - \lg M - 1.5n - 1\},$$ where $\mathbf{u}$ is the roundoff error in approximate arithmetic, and $N$ is the largest representable integer. Since $\OD(A)\leq M^n$, the algorithm requires $O(n^3\lg M)$ time, and $O(n^3)$ time when $\beta=\Omega(\log M)$.

Discrete and Computational Geometry, 10:227--233, 1993.
This paper shows that the $i$-level of an arrangement of hyperplanes in $E^d$ has at most $\binom{i+d-1}{d-1}$ local minima. This bound follows from ideas previously used to prove bounds on $(\le k)$-sets. Using linear programming duality, the Upper Bound Theorem is obtained as a corollary, giving yet another proof of this celebrated bound on the number of vertices of a simple polytope in $E^d$ with $n$ facets.
Notes: This paper by Mulmuley seems closely related, and probably should have been cited.

(Survey).
In F.~K. Hwang and D.~Z. Hu, editors, Computers and Euclidean Geometry. World Scientific Publishing, 1992.
This paper surveys some of the applications of randomization to computational and combinatorial geometry. Randomization provides a general way to divide-and-conquer geometric problems, and gives a simple incremental approach to building geometric structures. The paper discusses closest-point problems, convex hulls, Voronoi diagrams, trapezoidal diagrams of line segments, linear programming in small dimension, range queries, and bounds for point-line incidences and for $(\le k)$-sets. Relations to the Vapnik-Chervonenkis dimension, PAC-learnability of geometric concepts, and the Hough transform are briefly noted.

with Kurt Mehlhorn and Raimund Seidel.
Comp. Geom.: Theory and Applications, pages 185--121, 1993.
Preliminary version in Proc. Symp. Theor. Aspects of Comp. Sci., 1992.
We prove four results on randomized incremental constructions (RICs):
• an analysis of the expected behavior under insertion and deletions,
• a fully dynamic data structure for convex hull maintenance in arbitrary dimensions,
• a tail estimate for the space complexity of RICs,
• a lower bound on the complexity of a game related to RICs.
Notes: Among other things, this paper extends Seidel's "backwards analysis" approach (not far from the "leave one out" technique of learning theory) to a general version of RIC; this involves a kind of "searching in history" and exploitation of the exchangeability of members of a random sample. The talk slides are missing a few figures.

with Richard Cole and Robert E. Tarjan.
Int. J. Comp. Geom. and Applications, pages 117--133, 1992.
Preliminary version in SoCG '91: Proceedings of the Seventh Annual Symposium on Computational Geometry, 1991.
We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal $O(A+n\log n)$ expected work and optimal $O(\log n)$ time, where $A$ is the number of intersecting pairs of segments. If the segments form a simple chain, we give an algorithm requiring optimal $O(n)$ expected work and $O(\log n\log\log n\log^*n)$ expected time, and a simpler algorithm requiring $O(n\log^*n)$ expected work. The serial algorithm corresponding to the latter is among the simplest known algorithms requiring $O(n\log^*n)$ expected operations. For a set of segments forming $K$ chains, we give an algorithm requiring $O(A+n\log^*n+K\log n)$ expected work and $O(\log n\log\log n\log^* n)$ expected time. The parallel time bounds require the assumption that enough processors are available, with processor allocations every $\log n$ steps.
Notes: The serial version of our basic algorithm, as applied to non-intersecting segments, is a bit simpler than the divide-and-conquer scheme of the earlier paper, and not very far from Seidel's algorithm, independently discovered at the same time as this one. (His algorithm uses planar point location for a key task, while we use a sweepline algorithm.)

In SODA '91: Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, January 1991.
This paper gives a partitioning scheme for the geometric, planar traveling salesman problem, under the Euclidean metric: given a set $S$ of $n$ points in the plane, find a shortest closed tour (path) visiting all the points. The scheme employs randomization, and gives a tour that can be expected to be short, if $S$ satisfies the condition that a random subset $R\subset S$ has on average a tour much shorter than an optimal tour of $S$. This condition holds for points independently, identically distributed in the plane, for example, for which a tour within $1+\epsilon$ of shortest can be found in expected time $nk^2 2^k$, where $k=O(\log\log n)^3/\epsilon^2$. One algorithm employed in the scheme is of interest in its own right: when given a simple polygon $P$, it finds a Steiner triangulation of the interior of $P$. If $P$ has $n$ sides and perimeter $L_P$, the edges of the triangulation have total length $L_PO(\log n)$. If this algorithm is applied to a simple polygon induced by a minimum spanning tree of a point set, the result is a Steiner triangulation of the set with total length within a factor of $O(\log n)$ of the minimum possible.
Notes: A better partitioning scheme was given by Eppstein, using quadtrees, and of course Arora's algorithm makes the whole approach moot. There is a certain foreshadowing here of "pseudo-triangulations", however.

with Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and Emo Welzl.
Discrete and Computational Geometry, 5(2):99--160, 1990.
Preliminary version in FOCS '88: Proceedings of the Twenth-Ninth Symposium on Foundations of Computer Science, 1988.

with Jon L. Bentley and David B. Levine.
Algorithmica, pages 168--183, 1993.
Preliminary version in SODA '90: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, 1990.
Notes: Related to this paper.

Journal of the ACM, 42(2):488--499, 1995.
Preliminary version in FOCS '88: Proceedings of the Twenty-Ninth Symposium on Foundations of Computer Science, 1988.
This paper gives an algorithm for solving linear programming problems. For a problem with $n$ constraints and $d$ variables, the algorithm requires an expected $$O(d^2n)+(\log n)O(d)^{d/2+O(1)} +O(d^4\sqrt{n}\log n)$$ arithmetic operations, as $n\rightarrow\infty$. The constant factors do not depend on $d$. Also, an algorithm is given for integer linear programming. Let $\varphi$ bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let $n$ and $d$ be as before. Then the algorithm requires expected $$O(2^ddn+8^dd\sqrt{n\ln n}\ln n) +d^{O(d)}\varphi\ln n$$ operations on numbers with $d^{O(1)}\varphi$ bits, as $n->oo$, where the constant factors do not depend on $d$ or $\varphi$. The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any given input. The technique can be extended to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of $n$ points in $E^d$ has the same time bound.
Notes: The bound given for integer programming is not quite right, as corrected by F. Eisenbrand, here.

(The accompanying talk is the one given at FOCS in 1988; note that it gives the results of the paper in terms of the Smallest Enclosing Sphere (or Minimum Enclosing Ball) problem.)

The delay between conference and journal publication is not the fault of the journal.

Developments between 1988 and 1995, roughly as discussed at the conclusion of the journal version of the paper:

Several developments have occurred since the conference version of this paper appeared. Adler and Shamir have shown that these ideas can be applied to general convex programming. Chazelle and Matou{\v s}ek have derandomized the recursive algorithm, obtaining a deterministic algorithm requiring $d^{O(d)}n$ time. Alon and Megiddo have applied and extended the ideas of this paper to a parallel setting. Seidel gave a different randomized algorithm, requiring $O(d!n)$ expected time, with a somewhat simpler analysis; Matousek, Sharir and Welzl found a variant of Seidel's algorithm requiring time subexponential in $d$. Their algorithm is a randomized instance of the simplex algorithm. Kalai was the first to find a subexponential simplex algorithm. Problem instances have long been known for which versions of the simplex algorithm require at least $2^d$ operations.[KM] These results cast new light on the complexity of the simplex algorithm, and on the possibility that linear programming problems can be solved in "strongly polynomial" time; such an algorithm would need $(nd)^{O(1)}$ operations, with the number of operations independent of the size of the numbers specifying a problem instance.
Some more recent related results: Vapnik's leave-one-out error estimate for support vector machines is a version of Lemma 3.2, generalized to quadratic programming. (Such deleted error estimates were found in the sixties.)

Chazelle et al. observe that one of these algorithms can be the basis for sublinear geometric algorithms; another paper observes that the approach works well from the standpoint of multi-pass algorithms. Lemma 3.2 (or its generalization to convex programming), was rediscovered recently: Calafiore and Campi, Theorem 1. Their proof rediscovers "backwards analysis".

with Robert E. Tarjan and C. J. Van Wyk.
Discrete and Computational Geometry, 4(1):423--432, 1989.
Preliminary version in SoCG '88: Proceedings of the Fourth Annual Symposium on Computational Geometry, 1988.
We present an algorithm that triangulates a simple polygon on $n$ vertices in $O(n \log ^* n)$ expected time. The algorithm uses random sampling on the input, and its running time does not depend on any assumptions about a probability distribution from which the polygon is drawn.
Notes: The first algorithm to apply randomization to the problem, and obtain essentially linear time. See here also. Chazelle got rid of the "essentially" and the randomization, at the cost of some complexity; Amato et al. took out some of that complexity, at the cost of putting the randomization back in.

with P. W. Shor.
Discrete and Computational Geometry, 4(1):387--421, 1989.
Merges two papers below.
We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires $O(A+n\log n)$ expected time, where $A$ is the number of intersecting pairs reported. The algorithm requires $O(n)$ space in the worst case. Another algorithm computes the convex hull of $n$ points in $E^d$ in $O(n\log n)$ expected time for $d=3$, and $O(n^{\lfloor d/2\rfloor})$ expected time for $d>3$. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of $n$ points in $E^3$ in $O(n\log n)$ expected time, and on the way computes the intersection of $n$ unit balls in $E^3$. We show that $O(n\log A)$ expected time suffices to compute the convex hull of $n$ points in $E^3$, where $A$ is the number of input points on the surface of the hull. Algorithms for halfspace range reporting are also given. In addition, we give asymptotically tight bounds for $(\le k)$-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high order Voronoi diagrams.

with P. W. Shor.
In SoCG '88: Proceedings of the Fourth Annual Symposium on Computational Geometry, Urbana, Illinois, June 1988.

In SoCG '88: Proceedings of the Fourth Annual Symposium on Computational Geometry, Urbana, Illinois, June 1988.

Algorithmica, 4:461--469, 1989.
Included in PhD Thesis.
We describe an algorithm for finding a minimum spanning tree of the weighted complete graph induced by a set of $n$ points in Euclidean $d$-space. The algorithm requires nearly linear expected time for points that are independently uniformly distributed in the unit $d$-cube. The first step of the algorithm is the spiral search procedure described by Bentley, Weide, and Yao [BWY] for finding a supergraph of the MST that has $O(n)$ edges. (The constant factor in the bound depends on $d$.) The next step is that of sorting the edges of the supergraph by weight using a radix distribution, or "bucket," sort. These steps require linear expected time. Finally, Kruskal's algorithm is used with the sorted edges, requiring $O(n\alpha(cn,n))$ time in the worst case, with $c>6$. Since the function $\alpha(cn,n)$ grows very slowly, this step requires linear time for all practical purposes. This result improves the previous best $O(n\log\log^*n)$, and employs a much simpler algorithm. Also, this result demonstrates the robustness of bucket sorting, which requires $O(n)$ expected time in this case despite the probability dependency between the edge weights.

with S. Kapoor and P. Vaidya.
In SoCG '87: Proceedings of the Third Annual Symposium on Computational Geometry, Waterloo, Ontario, June 1987.
The problem of finding a rectilinear shortest path amongst obstacles may be stated as follows: Given a set of obstacles in the plane find a shortest rectilinear ($L_1$) path from a point $s$ to a point $t$ which avoids all obstacles. The path may touch an obstacle but may not cross an obstacle. We study the rectilinear shortest path problem for the case where the obstacles are non-intersecting simple polygons, and present an $O( n \log^2 n )$ algorithm for finding such a path, where $n$ is the number of vertices of the obstacles. This algorithm requires $O(n \log n )$ space. Another algorithm is given that requires $O(n ( \log n ) ^{ 3/2} )$ time and space. We also study the case of rectilinear obstacles in three dimensions, and show that $L_1$ shortest paths can be found in $O( n^2 \log^ 3 n )$ time.

In STOC '87: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, New York, New York, May 1987.
This paper gives approximation algorithms for solving the following motion planning problem: Given a set of polyhedral obstacles and points $s$ and $t$, find a shortest path from $s$ to $t$ that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an $\epsilon$-short path, that is, a path with length within $(1+\epsilon)$ of shortest. Let $n$ be the total number of faces of the polyhedral obstacles, and $\epsilon$ a given value satisfying $0<\epsilon\leq\pi$. The algorithm for the planar case requires $O(n\log n)/\epsilon$ time to build a data structure of size $O(n/\epsilon)$. Given points $s$ and $t$, an $\epsilon$-short path from $s$ to $t$ can be found with the use of the data structure in time $O(n/\epsilon+n\log n)$. The data structure is associated with a new variety of Voronoi diagram. Given obstacles $S\subset E^3$ and points $s,t\in E^3$, an $\epsilon$-short path between $s$ and $t$ can be found in $$O(n^2\lambda(n)\log(n/\epsilon)/\epsilon^4 +n^2\log n\rho\log(n\log \rho))$$ time, where $\rho$ is the ratio of the length of the longest obstacle edge to the distance between $s$ and $t$. The function $\lambda(n)=\alpha(n)^{O(\alpha(n)^{O(1)})}$, where the $\alpha(n)$ is a form of inverse of Ackermann's function. For $\log(1/\epsilon)$ and $\log\rho$ that are $O(\log n)$, this bound is $O(n^2(\log^2n)\lambda(n)/\epsilon^4)$.
Notes: This paper introduces the observation that Yao's fan of cones can be used to build a spanner. This observation was independently made by Keil and Gutwin, and by Ruppert and Seidel. The resulting spanners have seen recent (circa 2005) application in the wireless literature, where they are called Yao graphs.

with L. J. Guibas, Jorge Stolfi.
Theoretical Computer Science, 49:81--84, 1987.

Discrete and Computational Geometry, 2:195--222, 1987.
Preliminary version: Further applications of random sampling to computational geometry, STOC '86: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, 1986.
This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requires $O(s ^ {d + \epsilon})$ expected preprocessing time to build a search structure for an arrangement of $s$ hyperplanes in $d$ dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query point $p$, the cell of the arrangement containing $p$ can be found in $O( \log s)$ worst-case time. (The bound holds for any fixed $\epsilon >0$, with the constant factors dependent on $d$ and $\epsilon$.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expected $O(n ^{\lfloor d/2\rfloor} )$ time, where $n$ is the total number of vertices of the two polytopes. This matches previous results [DoK] for the case $d=3$ and extends them. Another algorithm samples points in the plane to determine their order $k$ Voronoi diagram, and requires expected $O(s^{1+\epsilon}k)$ time for $s$ points. (It is assumed that no four of the points are cocircular.) This sharpens the bound $O(sk ^ 2 \log s)$ for Lee's algorithm [Lee], and $O(s ^ 2 \log s + k(s-k) \log ^ 2 s)$ for Chazelle and Edelsbrunner's algorithm [ChE]. Finally, random sampling is used to show that any set of $s$ points in $E^3$ has $O(sk^2\log ^ 8 s / ( \log \log s) ^ 6 )$ distinct $j$-sets with $j\leq k$. (For $S \subset E^d$, a set $S'\subset S$ with $|S'|=j$ is a $j$-set of $S$ if there is a halfspace $h^+$ with $S'=S \cap h^+$.) This sharpens with respect to $k$ the previous bound $O(s k ^ 5 )$ [ChP]. The proof of the bound given here is an instance of a "probabilistic method" [ErS].
Notes: This paper extends the ideas of an earlier one to a much more general setting; This framework is slightly less general than that of range spaces of finite VC-dimension, which were introduced to geometric algorithms at the same time by Haussler and Welzl (in the same issue of the journal). The scheme here is very close to the sample compression framework in the learning theory literature. It is not clear that there are any natural applications of the VC-dimension for which this framework is not also applicable. The framework is much the same as this later one, which gives bounds that hold on average; here the bounds hold with high probability, but with an extra factor of $\log r$. The arrangement search structure uses a construction that sharpens an idea of Megiddo, and in turn was later sharpened (removing the $\epsilon$) and called cuttings. The polytope separation distance algorithm was made obsolete by Gaertner, who gave an $O(n)$-time algorithm by application and extension of these ideas, and later improvements. The $k$-set bounds are sharpened here, with a much cleaner argument. The construction used for the $k$-set bounds, considered in a dual arrangement of planes, is an approximate level, always above an $n/r$ level and always below a slightly higher level; that is, it is a crude kind of "shallow cutting".

Information Processing Letters, 22:21--24, January 1986.

SIAM Journal on Computing, pages 830--847, 1988.
Preliminary version: A probabilistic algorithm for the post office problem, STOC '85: Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, 1985.
An algorithm for closest-point queries is given. The problem is this: given a set $S$ of $n$ points in $d$-dimensional space, build a data structure so that given an arbitrary query point $p$, a closest point in $S$ to $p$ can be found quickly. The measure of distance is the Euclidean norm. This is sometimes called the post-office problem [Kn]. The new data structure will be termed an RPO tree, from Randomized Post Office. The expected time required to build an RPO tree is $O(n^{\lceil d/2\rceil (1+ \epsilon )} )$, for any fixed $\epsilon > 0$, and a query can be answered in $O(\log n )$ worst-case time. An RPO tree requires $O(n^{\lceil d/2\rceil (1+ \epsilon )} )$ space in the worst case. The constant factors in these bounds depend on $d$ and $\epsilon$. The bounds are average-case due to the randomization employed by the algorithm, and hold for any set of input points. This result approaches the $\Omega(n^{\lceil d/2\rceil } )$ worst-case time required for any algorithm that constructs the Voronoi diagram of the input points, and is a considerable improvement over previous bounds for $d>3$. The main step of the construction algorithm is the determination of the Voronoi diagram of a random sample of the sites, and the triangulation of that diagram.
Notes: A key result proven and used is that with high probability, every Delaunay ball of a random sample of the sites each contains few sites of the full set. This is the first appearance of the kind of analysis done in more generality here; the analysis essentially reduces to the observation that there are $r^{O(1)}$ balls associated with a sample of size $r$, that might be Delaunay, and an exponentially small probability that a given such ball with many points in it would be Delaunay in the sample, and finally, the union bound.

PhD thesis, Stanford University, January 1985.
This dissertation reports a variety of new algorithms for solving closest-point problems.  The input to these algorithms is a set or sets of points in $d$-dimensional space, with an associated $L_p$ metric. The problems considered are:
• The all nearest neighbors problem.  For point set $A$, find the nearest neighbors in $A$ of each point in $A$.
• The nearest foreign neighbor problem.  For point sets $A$ and $B$, find the closest point in $B$ to each point in $A$.
• The geometric minimum spanning tree problem.  For point set $A$, find the minimum spanning tree for the complete weighted undirected graph associated with $A$, where the vertices of the graph correspond to the points of $A$, and the weight of an edge is the distance between the points defining the edge.
These problems arise in routing, statistical classification, data compression, and other areas.  Obvious algorithms for them require a running time quadratic in $n$, the number of points in the input.  In many cases they can be solved with algorithms requiring $O(n \log ^{O(1)} n)$ time.

In this work, approximation algorithms for some cases of these problems have been found.  For example, for the minimum spanning tree problem with the $L_1$ metric, an algorithm has been devised that requires $O(n \log^d (1/\rho))$ time to find a spanning tree with weight within $1+\rho$ of the minimum.  Several other algorithms have been found with time bounds dependent on $\log(1/\rho)$ for attaining error $\rho$.

Algorithms have also been found that require linear expected time, for independent identically distributed random input points with a probability density function satisfying weak conditions.  One such algorithm depends on the fact that under certain conditions, values that are identically distributed, but dependent, can be bucket sorted in linear expected time.

An algorithm has been found for the all nearest neighbors problem that requires $O(n \log n)$ expected time for any input set of points, where the expectation is on the random sampling performed by the algorithm.  This algorithm involves the construction of a new data structure, a compressed form of digital trie.

In STOC '84: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, Washington, DC, April 1984.
Included in PhD Thesis.

In FOCS '83: Proceedings of the Twenty-Fourth Symposium on Foundations of Computer Science, Tucson, AZ, November 1983.
Included in PhD Thesis.
We present new algorithms for the all nearest neighbors problem:
Given a set $S$ of $n$ sites (points) in $d$-dimensional space, find the nearest neighbors in set $S$ of each site in $S$.
Our results:
• An algorithm for solving the all nearest neighbors problem in $O(n\log\sigma)$ time, where $\sigma$ is the ratio of the distance between the farthest pair of sites to the distance between the closest pair of sites. A similar algorithm is described for finding all $k$'th nearest neighbors.
• An algorithm for building a celltree, a compressed form of digital trie, in $O(n\log n)$ probabilistic time. The logarithm, floor, and bitwise exclusive-or functions are assumed available at unit cost.
• An algorithm for solving the all nearest neighbors problem in $O(n)$ worst-case time, given a celltree for the sites.
• An algorithm for building a celltree in linear expected time, assuming the sites are independently identically distributed random variables, with an unknown probability density function obeying some very weak conditions.
Notes: A "celltree" is now more commonly called a "compressed quadtree" or "compressed hyperoctree", and this paper is apparently the first appearance of such a construction. Vaidya refined the algorithm here to avoid randomness and bit-twiddling, at some cost in dependence on $d$; his algorithm, and this one, and that of Gabow, Bentley, and Tarjan, all use the same basic geometrical observation, which implies that the total number of nearest neighbors is $O(n)$, even up to approximate neighbors. Callahan and Kosaraju used similar ideas for "well-separated pairs decomposition". The $\sigma$ ratio is now more commonly called the spread.

Information Processing Letters, 16:23--25, January 1983.

(Survey).
In P. Cohen and E. Feigenbaum, editors, The Handbook of Artificial Intelligence. William Kaufman, Inc., Los Altos, CA, 1982.

In Proceedings of the DARPA Image Understanding Workshop, Maclean, VA: Science Applications, Inc., 1981.

Technical report, Claremont Graduate School, Claremont, CA, 1977.

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