A Trace Bound for the Hereditary Discrepancy

Let A be the incidence matrix of a set system with m sets and n points, m ≤ n , and let t= \mathop \rm tr M , where M= A ^T A . Finally, let σ = \mathop \rm tr M ² be the sum of squares of the elements of M . We prove that the hereditary discrepancy of the set system is at least . This general trace bound allows us to resolve discrepancy-type questions for which spectral methods had previously failed. Also, by using this result in conjunction with the spectral lemma for linear circuits, we derive new complexity bounds for range searching. We show that the (red—blue) discrepancy of the set system formed by n points and n lines in the plane is Ω(n ^1/6 ) in the worst case and always ¹ \tilde O(n ^1/6 ) . We give a simple explicit construction of n points and n halfplanes with hereditary discrepancy \tildeΩ(n ^1/4 ) . We show that in any dimension d= Ω( log n/\kern -1ptlog log n ) , there is a set system of n points and n axis-parallel boxes in \bf R ^d with discrepancy n ^{Ω(1/\kern -1pt log log n)} . Applying these discrepancy results together with a new variation of the spectral lemma, we derive a lower bound of Ω(nlog n) on the arithmetic complexity of off-line range searching for points and lines (for nonmonotone circuits). We also prove a lower bound of Ω(nlog n/\kern -1ptlog log n) on the complexity of orthogonal range searching in any dimension Ω(log n/\kern -1ptlog log n) . ¹ We use the notation \tilde O(m) and \tilde Ω(m) as shorthand for O(mlog ^c m) and Ω(m/\kern -1ptlog ^c m) , respectively, for some constant c>0 . Received April 5, 2000, and in revised form October 17, 2000. Online publication June 22, 2001.     

The Discrepancy of Boxes in Higher Dimension

We prove that the red—blue discrepancy of the set system formed by n points and n axis-parallel boxes in <bo>R</bo> ^d can be as high as n ^Ω(1) in any dimension d= Ω(log n) . This contrasts with the fixed-dimensional case d=O(1) , where the discrepancy is always polylogarithmic. More generally we show that in any dimension 1<d= O(log n) the maximum discrepancy is 2 ^Ω(d) . Our result also leads to a new lower bound on the complexity of off-line orthogonal range searching. This is the problem of summing up weights in boxes, given n weighted points and n boxes in <bo>R</bo> ^d . We prove that the number of arithmetic operations is at least Ω(nd+ nlog log n) in any dimension d=O(log n) . Received June 30, 2000, and in revised form November 9, 2000. Online publication April 6, 2001.     

Kinetic Connectivity for Unit Disks

We describe a kinetic data structure (KDS) that maintains the connected components of the union of a set of unit-radius disks moving in the plane. We assume that the motion of each disk can be specified by a low-degree algebraic trajectory; this trajectory, however, can be modified in an on-line fashion. While the disks move continuously, their connectivity changes at discrete times. Our main result is an O(n) space data structure that takes O(log n\slash \kern -1pt log log n) time per connectivity query of the form ``are disks A and B in the same connected component?' A straightforward approach based on dynamically maintaining the overlap graph requires Ω (n ² ) space. Our data structure requires only linear space and must deal with O(n ^{2 + ε} ) updates in the worst case, each requiring O(log ² n) amortized time, for any ε>0 . This number of updates is close to optimal, since a set of n moving unit disks can undergo Ω (n ² ) connectivity changes. Received September 20, 2000, and in revised form January 19, 2001. Online publication April 6, 2001.     

A note on the two-sided two-dimensional moment problem

It is shown that there is a positive semidefinite two-sided two-dimensional sequence f, which is not a moment sequence, such that log f(2m,2n) =O(m ² + n ²) as (m,n) →∞. This revised version was published online in June 2006 with corrections to the Cover Date.     

Local connectivity of a random graph

A graph is locally connected if for each vertex ν of degree ≧2, the subgraph induced by the vertices adjacent to ν is connected. In this paper we establish a sharp threshold function for local connectivity. Specifically, if the probability of an edge of a labeled graph of order n is p = ((3/2 +?_n) log n/n)^1/2 where ?_n = (log log n + log(3/8) + 2x)/(2 log n), then the limiting probability that a random graph is locally connected is exp(-exp(-x)).     

Approximation Algorithms for Minimum-Width Annuli and Shells

Let S be a set of n points in \re ^d . The ``roundness' of S can be measured by computing the width ω ^* =ω ^* (S) of the thinnest spherical shell (or annulus in \re ² ) that contains S . This paper contains two main results related to computing an approximation of ω ^* : (i) For d=2 , we can compute in O(n log n) time an annulus containing S whose width is at most 2ω ^* (S) . We extend this algorithm, so that, for any given parameter ε >0 , an annulus containing S whose width is at most (1+ε )ω ^* is computed in time O(n log n + n/ε ² ) . (ii) For d \geq 3 , given a parameter ε > 0 , we can compute a shell containing S of width at most (1+ε)ω ^* either in time O ( n / ε ^d ) log ( \Delata / ω ^* ε ) or in time O ( n / ε ^d-2 ) log n + 1 / εlog \Delata / ω ^* ε , where Δ is the diameter of S . Received July 6, 1999, and in revised form April 17, 2000. Online publication August\/ 11, 2000.     

An Optimal Deterministic Algorithm for Computing the Diameter of a Three-Dimensional Point Set

We describe a deterministic algorithm for computing the diameter of a finite set of points in R ³ , that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(nlog n) for a set of n points. The first optimal, but randomized, algorithm for this problem was proposed more than 10 years ago by Clarkson and Shor [11] in their ground-breaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matoušek [25] for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos [31] and Bespamyatnikh [7], both with running time O(nlog ² n) . The diameter algorithm appears to be the last one in Clarkson and Shor's paper that up to now had no deterministic counterpart with a matching running time. Received May 10, 2000, and in revised form November 3, 2000. Online publication June 22, 2001.     

The complexity and depth of Boolean circuits for multiplication and inversion in some fields <Emphasis Type="Italic">GF</Emphasis>(2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Let n = (p − 1) · p ^k , where p is a prime number such that 2 is a primitive root modulo p, and 2 ^p−1 − 1 is not a multiple of p ². For a standard basis of the field GF(2 ⁿ ), a multiplier of complexity O(log log p)n log n log log _p n and an inverter of complexity O(log p log log p)n log n log log _p n are constructed. In particular, in the case p = 3 the upper bound

On a Problem of Dobrowolski and Williams and the Pólya-Vinogradov Inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum _n=a+1 ^a+N f(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.     

An efficient algorithm for the three-dimensional diameter problem

We explore a new approach for computing the diameter of n points in \Bbb R ³ that is based on the restriction of the furthest-point Voronoi diagram to the convex hull. We show that the restricted Voronoi diagram has linear complexity. We present a deterministic algorithm with O(nlog ² n) running time. The algorithm is quite simple and is a good candidate to be implemented in practice. Using our approach the chromatic diameter and all-furthest neighbors in \Bbb R ³ can be found in the same running time. Received February 18, 2000, and in revised form June 27, 2000. Online publication January 17, 2001.     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Proximity‐preserving labeling schemes

This article considers informative labeling schemes for graphs. Specifically, the question introduced is whether it is possible to label the vertices of a graph with short labels in such a way that the distance between any two vertices can be inferred from inspecting their labels. A labeling scheme enjoying this property is termed a proximity‐preserving labeling scheme. It is shown that, for the class of n‐vertex weighted trees with M‐bit edge weights, there exists such a proximity‐preserving labeling scheme using O(M log n + log²n) bit labels. For the family of all n‐vertex unweighted graphs, a labeling scheme is proposed that using O(log² n · κ · n^1/κ) bit labels can provide approximate estimates to the distance, which are accurate up to a factor of In particular, using O(log³n) bit labels, the scheme can provide estimates accurate up to a factor of $\sqrt{2 \log n}$. (For weighted graphs, one of the log n factors in the label size is replaced by a factor logarithmic in the network's diameter.) © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 167–176, 2000     

The divisor problem for d 4 (n) in short intervals

In this paper we establish an asymptotic formula for the sum

Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)^33/31+o(1) for infinitely many n, and

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

A fast algorithm for computing minimum 3-way and 4-way cuts

For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k=3,4. The algorithm runs in O(n ^k-1 F(n,m))=O(mn ^k log(n ² /m)) time and O(n ²) space for k=3,4, where F(n,m) denotes the time bound required to solve the maximum flow problem in G. The bound for k=3 matches the current best deterministic bound ?(mn ³) for weighted graphs, but improves the bound ?(mn ³) to O(n ² F(n,m))=O(min{mn ^8/3,m ^3/2 n ²}) for unweighted graphs. The bound ?(mn ⁴) for k=4 improves the previous best randomized bound ?(n ⁶) (for m=o(n ²)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system. Received: April 1999 / Accepted: February 2000?Published online August 18, 2000     

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

Estimating a density and its derivatives via the minimum distance method

Summary This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the L _p distance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m-th derivative. For the case p=2, it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the L ₁ error. Estimation of derivatives of the density is considered as well.In a class parametrized by entire functions, it is proved that the rate of convergence of the MD density estimator (and its derivatives) to the unknown density (its derivatives) is of order in expected L ₁ and L ₂ distances. In the same class of distributions, MD estimators of unknown density and its derivatives are proved to achieve an extraordinary rate (log log n/n)^1/2 of strong consistency.     

On Approximate Geometric k -Clustering

For a partition of an n -point set into k subsets (clusters) S ₁ ,S ₂ ,. . .,S _k , we consider the cost function , where c(S _i ) denotes the center of gravity of S _i . For k=2 and for any fixed d and ε >0 , we present a deterministic algorithm that finds a 2-clustering with cost no worse than (1+ε) -times the minimum cost in time O(n log n); the constant of proportionality depends polynomially on ε . For an arbitrary fixed k , we get an O(n log ^k n) algorithm for a fixed ε , again with a polynomial dependence on ε . Received October 19, 1999, and in revised form January 19, 2000.     

Rates in the empirical bayes estimation problem with non-identical components

This paper considers empirical Bayes estimation of the mean θ of the univariate normal densityf ₀ with known variance where the sample sizesm(n) may vary with the component problems but remain bounded by <∞. Let {(θ _n ,X _n =(X _n,1,...,X _{n, m(n)} ))} be a sequence of independent random vectors where theθ _n are unobservable and iidG and, givenθ _n =θ has densityf _θ ^m(n) . The first part of the paper exhibits estimators for the density of and its derivative whose mean-squared errors go to zero with rates and respectively. LetR ^m(n+1)(G) denote the Bayes risk in the squared-error loss estimation ofθ _n+1 usingX _n+1. For given 0<a<1, we exhibitt _n (X₁,...,X _n ;X _n+1) such that . forn>1 under the assumption that the support ofG is in [0, 1]. Under the weaker condition that E[|θ|^2+γ]<∞ for some γ>0, we exhibitt _n ^* (X ₁,...,X _n ;X _n+1) such that forn>1.