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.     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

Order preserving nonexpansive operators inL 1

We study the limit behaviour ofT ^k f and of Cesaro averagesA _n f of this sequence, whenT is order preserving and nonexpansive inL ₁ ⁺ . IfT contracts also theL _∞-norm, the sequenceT ⁿ f converges in distribution, andA _n f converges weakly inL _p (1<p<∞), and also inL ₁ if the measure is finite. “Speed limit” operators are introduced to show that strong convergence ofA _n f need not hold. The concept of convergence in distribution is extended to infinite measure spaces. Much of this work was done during a visit of the first author at Ben Gurion University of the Negev in Beer Sheva, supported by the Deutsche Forschungsgemeinschaft.     

Small-dimensional linear programming and convex hulls made easy

We present two randomized algorithms. One solves linear programs involvingm constraints ind variables in expected timeO(m). The other constructs convex hulls ofn points in ℝ ^d ,d>3, in expected timeO(n ^[d/2]). In both boundsd is considered to be a constant. In the linear programming algorithm the dependence of the time bound ond is of the formd!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses. Large portions of the research reported here were conducted while the author visited DIMACS at Princeton University. The author was supported by NSF Presidential Young Investigator Award CCR-9058440.     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

A sublogarithmic convex hull algorithm

We present a parallel algorithm for finding the convex hull of a sorted set of points in the plane. Our algorithm runs inO(logn/log logn) time usingO(n log logn/logn) processors in theCommon crcw pram computational model, which is shown to be time and cost optimal. The algorithm is based onn ^1/3 divide-and-conquer and uses a simple pointer-based data structure.Part of this work was done when the last three authors were at the Department of Computer and Information Science, Linköping University. The research of the second author was supported by the Academy of Finland.     

Accounting for boundary effects in nearest-neighbor searching

Givenn data points ind-dimensional space, nearest-neighbor searching involves determining the nearest of these data points to a given query point. Most averagecase analyses of nearest-neighbor searching algorithms are made under the simplifying assumption thatd is fixed and thatn is so large relative tod thatboundary effects can be ignored. This means that for any query point the statistical distribution of the data points surrounding it is independent of the location of the query point. However, in many applications of nearest-neighbor searching (such as data compression by vector quantization) this assumption is not met, since the number of data pointsn grows roughly as 2 ^d .Largely for this reason, the actual performances of many nearest-neighbor algorithms tend to be much better than their theoretical analyses would suggest. We present evidence of why this is the case. We provide an accurate analysis of the number of cells visited in nearest-neighbor searching by the bucketing andk-d tree algorithms. We assumem ^dpoints uniformly distributed in dimensiond, wherem is a fixed integer ≥2. Further, we assume that distances are measured in theL _∞ metric. Our analysis is tight in the limit asd approaches infinity. Empirical evidence is presented showing that the analysis applies even in low dimensions. A preliminary version of this paper appeared in theProceedings of the 11th Annual ACM Symposium on Computational Geometry, 1995, pp. 336–344. Part of this research was conducted while the first author was visiting the Max-Planck-Institut für Informatik, Saarbrücken, Germany. The first author was supported by the ESPRIT Basic Research Actions Program, under Contract No. 7141 (project ALCOM II). The support of the National Science Foundation under Grant CCR-9310705 is gratefully acknowledged by the second author. The third author was supported in part by AT&T Bell Laboratories and the Society of Fellows at Harvard University.     

Illuminating high-dimensional convex sets

A setL of points in thed-spaceE ^d is said toilluminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d if for every setS _i inF and every pointx in the boundary ofS _i there is a pointv inL such thatv illuminatesx, i.e. the line segment joiningv tox intersects the union of the elements ofF in exactly {x}.The problem we treat is the size of a setS needed to illuminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d . We also treat the problem of putting these convex sets in mutually disjoint convex polytopes, each one having at most a certain number of facets.     

Improved heuristics for the minimum weight triangulation problem

Given a planar point setS, a triangulation ofS is a maximal set of non-intersecting line segments connecting the points. The minimum weight triangulation problem is to find a triangulation ofS such that the sum of the lengths of the line segments in it is the smallest. No polynomial time algorithm is known to produce the optimal or even a constant approximation of the optimal solution, and it is also unknown whether the problem is NP-hard. In this paper, we propose two improved heuristics, which triangulate a set ofn points in a plane inO(n ³) time and never do worse than the minimum spanning tree triangulation algorithm given by Lingas and the greedy spanning tree triangulation algorithm given by Heath and Pemmaraju. These two algorithms both produce an optimal triangulation if the points are the vertices of a convex polygon, and also do the same in some special cases.     

Off-Line Maintenance of Planar Configurations

We achieve anO(log n) amortized time bound per operation for the off-line version of the dynamic convex hull problem in the plane. In this problem, a sequence ofninsert,delete, andqueryinstructions are to be processed, where each insert instruction adds a new point to the set, each delete instruction removes an existing point, and each query requests a standard convex hull search. We process the entire sequence in totalO(n log n) time andO(n) space. Alternatively, we can preprocess a sequence ofninsertions and deletions inO(n log n) time and space, then answer queries in history inO(log n) time apiece (a query in history means a query comes with a time parametert, and it must be answered with respect to the convex hull present at timet). The same bounds also hold for the off-line maintenance of several related structures, such as the maximal vectors, the intersection of half-planes, and the kernel of a polygon. Achieving anO(log n) per-operation time bound for theon-lineversions of these problems is a longstanding open problem in computational geometry.     

A new technique to compute polygonal schema for 2-manifolds with application to null-homotopy detection

We provide a new technique for deriving optimal-sized polygonal schema for triangulated compact 2-manifolds without boundary inO(n) time, wheren is the size of the given triangulationT. We first derive a polygonal schemaP embedded inT using Seifert-Van Kampen's theorem. A reduced polygonal schemaQ of optimal size is computed fromP, where a surjective map from the vertices ofP is retained to the vertices ofQ. This helps detecting null-homotopic (contractible to a point) cycles. Given a cycle of lengthk, we determine if it is null-homotopic inO(n+k logg) time and in θ(n+k) space, whereg is the genus of the given 2-manifold. The actual contraction for a null-homotopic cycle can be computed in θ(nk) time and space. This is a considerable improvement over the previous best-known algorithm for this problem.     

Hyperbolic 0–1 programming and query optimization in information retrieval

Unconstrained hyperbolic 0–1 programming can be solved in linear time when the numerator and the denominator are linear and the latter is always positive. It is NP-hard, and finding an approximate solution with a value equal to a positive multiple of the optimal one is also NP-hard, if this last hypothesis does not hold. Determining the optimal logical form of a query in information retrieval, given the attributes to be used, can be expressed as a parametric hyperbolic 0–1 program and solved in O(n logn) time, wheren is the number of elementary logical conjunctions of the attributes. This allows to characterize the optimal queries for the Van Rijsbergen synthetic criterion.This research was done in part during a visit of the first author to the Pontifical Catholic University of Rio de Janeiro in July and August 1987, sponsored by CNPq. It was also supported in part by grants 0271 and 0066 of the AFOSR to Rutgers University. The second author was with Centro de Análise de Sistemas Navais, Rio de Janeiro.     

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.     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

Inspheres and inner products

Among normed linear spacesX of dimension ≧3, finite-dimensional Hilbert spaces are characterized by the condition that for each convex bodyC inX and each ballB of maximum radius contained inC,B’s center is a convex combination of points ofB ∩ (boundary ofC). Among reflexive Banach spaces of dimension ≧3, general Hilbert spaces are characterized by a related but weaker condition on inscribed balls. Research of the first author was partially supported by the U.S. National Science Foundation. Research of the second and third authors was supported by the Consiglio Nazionale delle Ricerche and the Ministero della Pubblica Istruzione of Italy, while they were visiting the University of Washington, Seattle, USA.     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

A note on the circle containment problem

We extend a result due to Bárányet al. and prove the following theorem: given any setS ofn points in the plane, there are pointsx andy inS, such that every circle that containsx andy contains at least [5/84(n – 2)] other points ofS.     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

Optimization of a time-variable,constrained control matrix

The problem posed is to choose, in a optimal manner, a time-variable, bounded, linear transformation defining the velocity of a state point inn-dimensional space in terms of the state. The two-point boundary-value problem which arises from an application of the Pontryagin maximum principle is explicitly solvable; hence, a formula is derived showing that the optimal trajectories in state space are equiangular spirals in two-dimensional subspaces ofR ⁿ and also describing the boundary of the set of attainability. This formula is used to solve the problem of minimal-time transfer between any two given points, and the optimal control is specified both as an open-loop and a closed-loop controller. The solutions to the problem of maximizing a linear payoff function of the final state and of maximizing the angle of rotation of the state vector about the origin are also given.