Probabilistic analysis of an algorithm for solving thek-dimensional all-nearest-neighbors problem by projection

A well-known simple heuristic algorithm for solving the all-nearest-neighbors problem in thek-dimensional Euclidean spaceE ^k ,k>1, projects the given point setS onto thex-axis. For each pointq S a nearest neighbor inS under anyL _p -metric (1 p ) is found by sweeping fromq into two opposite directions along thex-axis. If _q denotes the distance betweenq and its nearest neighbor inS the sweep process stops after all points in a vertical 2 _q -slice centered aroundq have been examined. We show that this algorithm solves the all-nearest-neighbors problem forn independent and uniformly distributed points in the unit cube [0,1] ^k in (n ^2–1/k ) expected time, while its worst-case performance is (n ²).     

A sparse graph almost as good as the complete graph on points inK dimensions

A setV ofn points ink-dimensional space induces a complete weighted undirected graph as follows. The points are the vertices of this graph and the weight of an edge between any two points is the distance between the points under someL _p metric. Let ε≤1 be an error parameter and letk be fixed. We show how to extract inO(n logn+ε ^−k log(1/ε)n) time a sparse subgraphG=(V, E) of the complete graph onV such that: (a) for any two pointsx, y inV, the length of the shortest path inG betweenx andy is at most (1+∈) times the distance betweenx andy, and (b)|E|=O(ε^−k n).     

Maintaining the minimal distance of a point set in polylogarithmic time

A dynamic data structure is given that maintains the minimal distance in a set ofn points ink-dimensional space inO((logn) ^k log logn) amortized time per update. The size of the data structure is bounded byO(n(logn) ^k ). Distances are measured in the MinkowskiL _t -metric, where 1 t . This is the first dynamic data structure that maintains the minimal distance in polylogarithmic time for fully on-line updates.This work was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Halfspace range search: An algorithmic application ofk-sets

Given a fixed setS ofn points inE ³ and a query plane, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side of. We prove that withO(n(logn)⁸ (loglogn)⁴) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE ³ isO(nk ⁵); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.Supported in part by NSF grants MCS 83-03925 and the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146 and ARPA Order No. 4786.Supported in part by Joint Services Electronics Program under Contract N00014-79-C-0424.     

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.     

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.     

Quasi-optimal range searching in spaces of finite VC-dimension

The range-searching problems that allow efficient partition trees are characterized as those defined by range spaces of finite Vapnik-Chervonenkis dimension. More generally, these problems are shown to be the only ones that admit linear-size solutions with sublinear query time in the arithmetic model. The proof rests on a characterization of spanning trees with a low stabbing number. We use probabilistic arguments to treat the general case, but we are able to use geometric techniques to handle the most common range-searching problems, such as simplex and spherical range search. We prove that any set ofn points inE ^d admits a spanning tree which cannot be cut by any hyperplane (or hypersphere) through more than roughlyn ^1–1/d edges. This result yields quasi-optimal solutions to simplex range searching in the arithmetic model of computation. We also look at polygon, disk, and tetrahedron range searching on a random access machine. Givenn points inE ², we derive a data structure of sizeO(n logn) for counting how many points fall inside a query convexk-gon (for arbitrary values ofk). The query time isO(kn logn). Ifk is fixed once and for all (as in triangular range searching), then the storage requirement drops toO(n). We also describe anO(n logn)-size data structure for counting how many points fall inside a query circle inO(n log² n) query time. Finally, we present anO(n logn)-size data structure for counting how many points fall inside a query tetrahedron in 3-space inO(n ^2/3 log² n) query time. All the algorithms are optimal within polylogarithmic factors. In all cases, the preprocessing can be done in polynomial time. Furthermore, the algorithms can also handle reporting within the same complexity (adding the size of the output as a linear term to the query time).Portions of this work have appeared in preliminary form in Partition trees for triangle counting and other range searching problems (E. Welzl),Proc. 4th Ann. ACM Symp. Comput. Geom. (1988), 23–33, and Tight Bounds on the Stabbing Number of Spanning Trees in Euclidean Space (B. Chazelle), Comput. Sci. Techn. Rep. No. CS-TR-155-88, Princeton University, 1988. Bernard Chazelle acknowledges the National Science Foundation for supporting this research in part under Grant CCR-8700917. Emo Welzl acknowledges the Deutsche Forschungsgemeinschaft for supporting this research in part under Grant We 1265/1-1.     

Rectilinear shortest paths in the presence of rectangular barriers

In this paper we address the following shortest-path problem. Given a point in the plane andn disjoint isothetic rectangles (barriers), we want to construct a shortestL ₁ path (not crossing any of the barriers) from the source point to any given query point. A restricted version of this problem (where the source and destination points are knowna priori) had been solved earlier inO(n ²) time. Our approach consists of preprocessing the source point and the barriers to obtain a planar subdivision where a query point can be located and a shortest path connecting it to the source point quickly transvered. By showing that any such path is monotone in at least one ofx ory directions, we are able to apply a plane sweep technique to divide the plane intoO(n) rectangular regions. This leads to an algorithm whose complexity isO(n logn) preprocessing time,O(n) space, andO(logn+k) query time, wherek is the number of turns on the reported path. If only the length of the path is sought,O(logn) query time suffices. Furthermore, we show an (n logn) time lower bound for the case where the source and destination points are known in advance, which implies the optimality of our algorithm in this case.A preliminary version of this paper appeared in theProceedings of the First Symposium on Computational Geometry (1985).Supported in part by CNPq-Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brazil).Supported in part by the National Science Foundation under Grants MCS 8420814 and ECS 8340031.     

Applications of random sampling in computational geometry,II

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 requiresO(A+n logn) expected time, whereA is the number of intersecting pairs reported. The algorithm requiresO(n) space in the worst case. Another algorithm computes the convex hull ofn points inE ^d inO(n logn) expected time ford=3, andO(n ^[d/2]) expected time ford>3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set ofn points inE ³ inO(n logn) expected time, and on the way computes the intersection ofn unit balls inE ³. We show thatO(n logA) expected time suffices to compute the convex hull ofn points inE ³, whereA 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 (k)-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high-order Voronoi diagrams.     

On minimum and maximum spanning trees of linearly moving points

In this paper we investigate the upper bounds on the numbers of transitions of minimum and maximum spanning trees (MinST and MaxST for short) for linearly moving points. Here, a transition means a change on the combinatorial structure of the spanning trees. Suppose that we are given a set ofn points ind-dimensional space,S={p ₁,p ₂, ...p _n }, and that all points move along different straight lines at different but fixed speeds, i.e., the position ofp _i is a linear function of a real parametert. We investigate the numbers of transitions of MinST and MaxST whent increases from-∞ to +∞. We assume that the dimensiond is a fixed constant. Since there areO(n ²) distances amongn points, there are naivelyO(n ⁴) transitions of MinST and MaxST. We improve these trivial upper bounds forL ₁ andL _∞ distance metrics. Letk _p (n) (resp. ) be the number of maximum possible transitions of MinST (resp. MaxST) inL _p metric forn linearly moving points. We give the following results in this paper: κ₁(n)=O(n ^5/2 α(n)),κ _∞(n)=O(n ^5/2 α(n)), , and where α(n) is the inverse Ackermann's function. We also investigate two restricted cases, i.e., thec-oriented case in which there are onlyc distinct velocity vectors for movingn points, and the case in which onlyk points move.     

A distance constrainedp-facility location problem on the real line

LetV = {v ₁,, v _n } be a set ofn points on the real line (existing facilities). The problem considered is to locatep new point facilities,F ₁,, F _p , inV while satisfying distance constraints between pairs of existing and new facilities and between pairs of new facilities. Fori = 1, , p, j = 1, , n, the cost of locatingF _i at pointv _j isc _ij . The objective is to minimize the total cost of setting up the new facilities. We present anO(p ³ n ² logn) algorithm to solve the model.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

Arcs fixed byA 5 andA 6

The complete list of thek-arcsK inPG(n, q) fixed by a projective group isomorphic toA ₅ orA ₆, which acts primitively on the points ofK, is presented. This leads to new classes of 10-arcs inPG(n, q), 3 n 5. Our results also show that the non-classical 10-arc inPG(4, 9), discovered by D.G. Glynn [3], belongs to an infinite class of 10-arcs inPG(4, 3^h),h 2, fixed by a projective group isomorphic toA ₆.The first author wishes to thank the Belgian National Fund for Scientific Research for financial supportSenior research assistant of the Belgian National Fund for Scientific ResearchDedicated to Professor M. Scafati Tallini on the occasion of her sixtyfifth birthday     

A sharp form of Poincaré type inequalities on balls and spheres

Summary Letu be a real valued function on ann-dimensional Riemannian manifoldM ⁿ. We consider an inequality between theL ^q-norm ofu minus its mean value overM ⁿ and theL ^p-norm of the gradient ofu.The best constant in such inequality is exhibited in the following cases: i)M ⁿ is an open ball inIR ⁿ andp=1, 0<qn/(n–1); ii)M ⁿ is a sphere inIR ⁿ +1 and eitherp=1, 0<qn/(n–1) orp>n,q=.

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Sunto Siau una funzione a valori reali dafinita su una varietà riemannianan-dimensionaleM ⁿ. Si considera una disuguaglianza tra la normaL ^q diu meno il suo valor medio suM ⁿ e la normaL ^p del gradiente diu.Si determina la costante ottimale in tale disuguaglianza nei seguenti casi: i)M ⁿ è un disco aperto inIR ⁿ ep=1, 0<qn/(n–1); ii)M ⁿ è una sfera inIR ⁿ +1 ep=1, 0<qn/(n–1) oppurep>n,q=.

     

Partitions of finite vector spaces into subspaces

Let V_n(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V_n(q) is a partition of V_n(q) if every nonzero element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

On the number of queries necessary to identify a permutation

Let p and q be two permutations over {1, 2,…, n}. We denote by m(p, q) the number of integers i, 1 ≤ i ≤ n, such that p(i) = q(i). For each fixed permutation p, a query is a permutation q of the same size and the answer a(q) to this query is m(p, q). We investigate the problem of finding the minimum number of queries required to identify an unknown permutation p. A polynomial-time algorithm that identifies a permutation of size n by O(n · log₂n) queries is presented. The lower bound of this problem is also considered. It is proved that the problem of determining the size of the search space created by a given set of queries and answers is #P-complete. Since this counting problem is essential for the analysis of the lower bound, a complete analysis of the lower bound appears infeasible. We conjecture, based on some preliminary analysis, that the lower bound is Ω(n · log₂n).     

Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inE ^d in timeO(F _d (N,N) log ^d N), whereF _d (n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inE ^d . IfF _d (N,N)=Ω(N ^1+ε), for some fixed ɛ>0, then the running time improves toO(F _d (N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)^2/3+m log² n+n log² m) inE ₃, which yields anO(N ^4/3 log^4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE ³. Ind≥4 dimensions we obtain expected timeO((nm)^{1−1/([d/2]+1)+ε}+m logn+n logm) for the bichromatic closest pair problem andO(N ^{2−2/([d/2]+1)ε}) for the Euclidean minimum spanning tree problem, for any positive ɛ. The first, second, and fourth authors acknowledge support from the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The second author's work was supported by the National Science Foundation under Grant CCR-8714565. The third author's work was supported by the Deutsche Forschungsgemeinschaft under Grant A1 253/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”. The last two authors' work was also partially supported by the ESPRIT II Basic Research Action of the EC under Contract No. 3075 (project ALCOM).     

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.     

AnO(log*n) Approximation Algorithm for the Asymmetricp-Center Problem

The input to the asymmetricp-center problem consists of an integerpand ann × ndistance matrixDdefined on a vertex setVof sizen, whered_ijgives the distance fromitoj. The distances are assumed to obey the triangle inequality. For a subsetS Vthe radius ofSis the minimum distanceRsuch that every point inVis at a distance at mostRfrom some point inS. Thep-center problem consists of picking a setS Vof sizepto minimize the radius. This problem is known to be NP-complete.For the symmetric case, whend_ij = d_ji, approximation algorithms that deliver a solution to within 2 of the optimal are known. David Shmoys, in his article [[11]], mentions that nothing was known about the asymmetric case. We present an algorithm that achieves a ratio ofO(log*n).