A new duality result concerning voronoi diagrams

A new duality between order-k Voronoi diagrams inE ^d and convex hulls inE ^d+1 is established. It implies a reasonably simple algorithm for computing the order-k diagram forn points in the plane inO(k ² n logn) time and optimalO(k(n–k)) space.Research was supported by the Austrian Fond zur Foerderung der wissenschaftlichen Forschung.     

A Fast and Simple Algorithm for Identifying 2-Monotonic Positive Boolean Functions

Consider the problem of identifying min T(f) and max F(f) of a positive (i.e., monotone) Boolean functionf, by using membership queries only, where min T(f) (max F(f)) denotes the set of minimal true vectors (maximum false vectors) off. Moreover, as the existence of a polynomial total time algorithm (i.e., polynomial time in the length of input and output) for this problem is still open, we consider here a restricted problem: given an unknown positive functionfofnvariables, decide whetherfis 2-monotonic or not, and iffis 2-monotonic, output both min T(f) and max F(f). For this problem, we propose a simple algorithm, which is based on the concept of maximum latency, and we show that it usesO(n²m) time andO(n²m) queries, wherem = |min T(f)| + |max F(f)|. This answers affirmatively the conjecture raised in Boroset al.[Lecture Notes in Comput. Sci.557(1991), 104–115], Boroset al.[SIAM J. Comput.26(1997), 93–109], and is an improvement over the two algorithms discussed therein: one usesO(n³m) time andO(n³m) queries, and the other usesO(nm² + n²m) time andO(nm) queries.     

Some simple in-place merging algorithms

The theoretical presentation and analysis is given for two families of simple in-place merging algorithms and their limiting cases. The first family merges stably inO(k·n) time andO(n ^1/k ) additional space with a limiting case running inO(n logn) time and constant space. The second family merges unstably inO (k ·n) time andO(log ^k n) space with a limiting case running inO(nG(n)) time and constant space. HereG(n) is the leastk such thatF(k) n whereF(0)=1 andF(i)=2 ^F(i–1) fori1. Each algorithm gives rise to a corresponding merge sort.     

Triangulating point sets in space

A setP ofn points inR ^d is called simplicial if it has dimensiond and contains exactlyd + 1 extreme points. We show that whenP containsn interior points, there is always one point, called a splitter, that partitionsP intod + 1 simplices, none of which contain more thandn/(d + 1) points. A splitter can be found inO(d ⁴ +nd ²) time. Using this result, we give anO(nd ⁴ log_1+1/d n) algorithm for triangulating simplicial point sets that are in general position. InR ³ we give anO(n logn +k) algorithm for triangulating arbitrary point sets, wherek is the number of simplices produced. We exhibit sets of 2n + 1 points inR ³ for which the number of simplices produced may vary between (n – 1)² + 1 and 2n – 2. We also exhibit point sets for which every triangulation contains a quadratic number of simplices.Research supported by the Natural Science and Engineering Research Council grant A3013 and the F.C.A.R. grant EQ1678.     

An Efficient Algorithm for a Sparse k-Edge-Connectivity Certificate

We present an efficient algorithm for finding a sparse k-edge-connectivity certificate of a multigraph G. Our algorithm runs in O((log kn)(log k)²(log n)²) time using O(k(n + m′)) processors on an ARBITRARY CRCW PRAM, where n and m′ stand for the numbers of vertices in G and edges in the simplified graph of G, respectively.     

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.     

Augmenting Edge-Connectivity over the Entire Range inÕ(nm) Time

For a given undirected graphG = (V, E, c_G) with edges weighted by nonnegative realsc_G:E → R ⁺ , let Λ_G(k) stand for the minimum amount of weights which needs to be added to makeG k-edge-connected, and letG*(k) be the resulting graph obtained fromG. This paper first shows that function Λ_Gover the entire rangek [0, +∞] can be computed inO(nm + n² log n) time, and then shows that allG*(k) in the entire range can be obtained fromO(n log n) weighted cycles, and such cycles can be computed inO(nm + n² log n) time, wherenandmare the numbers of vertices and edges, respectively.     

Efficient Computation of Rectilinear Geodesic Voronoi Neighbor in the Presence of Obstacles

In this paper we present an algorithm to compute the rectilinear geodesic voronoi neighbor of an arbitrary query pointqamong a setSofmpoints in the presence of a set ofnvertical line segment obstacles inside a rectangular floor. The distance between a pair of points α and β is the shortest rectilinear distance avoiding the obstacles in and is denoted by δ(α, β). The rectilinear geodesic voronoi neighbor of an arbitrary query pointq,RGVN(q) is the pointp_i ∈ Ssuch that δ(q, p_i) is minimum. The algorithm suggests a preprocessing of the elements of the setsSand inO((m + n)log(m + n)) time such that for an arbitrary query pointq, theRGVNquery can be answered inO(log(m + n)) time. The space required for storing the preprocessed information isO(n + m log m). If the points inSare placed on the boundary of the rectangular floor, a different technique is adopted to decrease the space complexity toO(m + n). This technique works even if the obstacles are rectangles instead of line segments. Finally, the parallelization of the preprocessing steps for the latter algorithm is suggested, which takesO(log³(m + n)) time, usingO((m + n)^1.5/log²(m + n)) processors andO(log(m + n)) query time.     

Edge-Coloring Bipartite Graphs

Given a bipartite graph G with n nodes, m edges, and maximum degree Δ, we find an edge-coloring for G using Δ colors in time T + O(m log Δ), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k ≤ Δ. Together with best known bounds for T this implies on edge-coloring algorithm which improves on the algorithm of Hopcroft and Cole. Our algorithm can also be used to find a (Δ + 2)-edge-coloring for G in time O(m log Δ). The previous best approximation algorithm with the same time bound needed Δ + log Δ colors.     

A note on optimal multiway split trees

Split trees are suitable data structures for storing records with different access frequencies. Under assumption that the access frequencies are all distinct, Huang has proposed anO(n ⁴ logm) time algorithm to construct an (m+1)-way split tree for a set ofn keys. In this paper, we generalize Huang's algorithm to deal with the case of non-distinct access frequencies. The technique used in the generalized algorithm is a generalization of Hesteret al.'s, where the binary case was considered. The generalized algorithm runs inO(n ⁵ logm) time.     

Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

A Randomized Parallel Algorithm for Planar Graph Isomorphism

We present a parallel randomized algorithm running on aCRCW PRAM, to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms inO(log² n) time withn^{1 + ε}processors, for any ε > 0). Ifnis the number of vertices, our algorithm takesO(log(n)) time with processors and with a probability of failure of 1/nat most. The algorithm needs 2 · log(m) − log(n) + O(log(n)) random bits. The number of random bits can be decreased toO(log(n)) by increasing the number of processors ton^{3/2 + ε}, for any ε > 0. Our parallel algorithm has significantly improved processor efficiency, compared to the previous logarithmic time parallel algorithm of Miller and Reif (Siam J. Comput.20(1991), 1128–1147), which requiresn⁴randomized processors orn⁵deterministic processors.     

Minimum asymptotic error of algorithms for solving ODE

We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) × ^s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n^−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n^−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f.     

Finding thek smallest spanning trees

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m log(m, n)=k ²); for planar graphs this bound can be improved toO(n+k ²). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k ² n+n logn,k ²+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k ²).     

Disjoint bases in a polymatroid

Let f : 2^N → ⁺ be a polymatroid (an integer‐valued non‐decreasing submodular set function with f(??) = 0). We call S ? N a base if f(S) = f(N). We consider the problem of finding a maximum number of disjoint bases; we denote by m* be this base packing number. A simple upper bound on m* is given by k* = max{k : Σ_iεNf_A(i) ≥ kf_A(N),?A ? N} where f_A(S) = f(A ∪ S) ‐ f(A). This upper bound is a natural generalization of the bound for matroids where it is known that m* = k*. For polymatroids, we prove that m* ≥ (1 ? o(1))k*/lnf(N) and give a randomized polynomial time algorithm to find (1 ? o(1))k*/lnf(N) disjoint bases, assuming an oracle for f. We also derandomize the algorithm using minwise independent permutations and give a deterministic algorithm that finds (1 ? ε)k*/lnf(N) disjoint bases. The bound we obtain is almost tight because it is known there are polymatroids for which m* < (1 + o(1))k*/lnf(N). Moreover it is known that unless NP ? DTIME(n^{log log n}), for any ε > 0, there is no polynomial time algorithm to obtain a (1 + ε)/lnf(N)‐approximation to m*. Our result generalizes and unifies two results in the literature. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Triangulating a nonconvex polytope

This paper is concerned with the problem of partitioning a three-dimensional nonconvex polytope into a small number of elementary convex parts. The need for such decompositions arises in tool design, computer-aided manufacturing, finite-element methods, and robotics. Our main result is an algorithm for decomposing a nonconvex polytope of zero genus withn vertices andr reflex edges intoO(n +r ²) tetrahedra. This bound is asymptotically tight in the worst case. The algorithm requiresO(n +r ²) space and runs inO((n +r ²) logr) time.This research was supported in part by the National Science Foundation under Grant CCR-8700917.     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).     

Ray Shooting Amidst Convex Polygons in 2D

We consider the problem of ray shooting in a two-dimensional scene consisting ofmconvex polygons with a total ofnedges. We present a data structure that requiresO(mn log m) space and preprocessing time and that answers a ray shooting query inO(log² m log² n) time. If the polygons are pairwise disjoint, the space and preprocessing time can be improved toO((m²+n)log m) andO((m²+n log n)log m), respectively. Our algorithm also works for a collection of disjoint simple polygons. We also show that if we allow onlyO(n) space, a ray shooting query among a collection of disjoint simple polygons can be answered in timeO(m/[formula]¹⁺ log² n) time, for any >0.     

Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^ε) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and ε > 0 is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^ε).     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.