Intersection queries in sets of disks

In this paper we develop some new data structures for storing a set of disks that can answer different types of intersection queries efficiency. If the disks are non-intersecting we obtain a linear size data structure that can report allk disks intersecting a query line segment in timeO(n ⁺ +k), wheren is the number of disks,=log₂(1+5)–1 0.695, and is an arbitrarily small positive constant. If the segment is a full line, the query time becomesO(n +k). For intersecting disks we obtain anO(n logn) size data structure that can answer an intersection query in timeO(n ^2/3 log² n+k). We also present a linear size data structure for ray shooting queries, whose query time isO(n ).The research of the first two authors was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). The work of the third author was supported byDimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center — NSF-STC88-09648.     

Partitioning arrangements of lines II: Applications

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m ^2/3 n ^2/3 · log^2/3 n · log^/3 (m/n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where is a constant <3.33; (ii) anO(m ^2/3 n ^2/3 · log^5/3 n · log^/3 (m/n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n ^4/3 log^(+2)/3 n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n ^4/3 log^{( + 2)/3} n) algorithm to count red-blue intersections between two sets of segments, and (v) anO(n ^3/2 log^/3 n) algorithm to compute spanning trees with low stabbing number for a set ofn points. We also present an algorithm that, given set ofn points in the plane, preprocesses it, in timeO(nm log^+1/2 n), into a data structure of sizeO(m) forn lognmn ², so that the number of points ofS lying inside a query triangle can be computed inO((n/m) log^3/2 n) time.Work on this paper has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation. A preliminary version of this paper appears in theProceedings of the 5th ACM Symposium on Computational Geometry, 1989, pp. 11–22.     

New applications of random sampling in computational geometry

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 requiresO(s ^d+ ) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind 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 pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed >0, with the constant factors dependent ond and .) 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 expectedO(n ^[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s ¹⁺ k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk ² logs) for Lee's algorithm [21], andO(s ² logs+k(s–k) log² s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE ³ hasO(sk ² log⁸ s/(log logs)⁶) distinctj-sets withjk. (ForS E ^d , a setS S with |S| =j is aj-set ofS if there is a half-spaceh ⁺ withS =S h ⁺.) This sharpens with respect tok the previous boundO(sk ⁵) [5]. The proof of the bound given here is an instance of a probabilistic method [15].A preliminary version of this paper appeared in theProceedings of the 18th Annual ACM Symposium on Theory of Computing, Berkeley, CA, 1986.     

Characterizing matchings as the intersection of matroids

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function (G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, (n), for graphs with n vertices. We describe an integer programming formulation for deciding whether (G)k. Using combinatorial arguments, we prove that (n)(log logn). On the other hand, we establish that (n)O(logn/ log logn). Finally, we prove that (n)=4 for n=5,,12, and sketch a proof of (n)=5 for n=13,14,15.An earlier version appears as an extended abstract in the Proceedings of COMB01 [5]. Supported by the Gerhard-Hess-Forschungs-Förderpreis (WE 1462) of the German Science Foundation (DFG) awarded to R. Weismantel.     

Dynamic partition trees

In this paper we study dynamic variants of conjugation trees and related structures that have recently been introduced for performing various types of queries on sets of points and line segments, like half-planar range searching, shooting, intersection queries, etc. For most of these types of queries dynamic structures are obtained with an amortized update time ofO(log² n) (or less) with only minor increases in query times. As an application of the method we obtain an output-sensitive method for hidden surface removal in a set ofn triangles that runs in timeO(nlogn+n ^· k ) where=log₂((1+5)/2) 0.695 andk is the size of the visibility map obtained.Research of the second author was partially supported by the ESPRIT II Basic Research Actions Program of the EC, under contract No. 3075 (project ALCOM).     

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).     

Number of no nisomorphic subgraphs in an n-point graph

LetG(n) be the set of all nonoriented graphs with n enumerated points without loops or multiple lines, and let v_k(G) be the number of mutually nonisomorphic k-point subgraphs of G G(n). It is proved that at least |G(n)| (1–1/n) graphs G G(n) possess the following properties: a) for any k [6log₂n], where c=–c log₂c–(1–c)×log₂(1–c) and c>1/2, we havev _k(G) > C _n ^k (1–1/n²); b) for any k [cn + 5 log₂n] we havev _k(G) = C _n ^k . Hence almost all graphs G G(n) containv(G) 2ⁿ pairwise nonisomorphic subgraphs.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 263–273, March, 1971.     

The local nature of Δ-coloring and its algorithmic applications

Given a connected graphG=(V, E) with |V|=n and maximum degree such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with colors. We generalize this as follows: letG-v be -colored; then,v can be colored by considering the vertices in anO(log n) radius aroundv and by recoloring anO(log n) length augmenting path inside it. Using this, we show that -coloringG is reducible inO(log³ n/log) time to (+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for -coloring.A preliminary version of this paper appeared as part of the paper Improved Distributed Algorithms for Coloring and Network Decomposition Problems, in theProceedings of the ACM Symposium on Theory of Computing pages 581–592, 1992. This research was done when the authors were at the Computer Science Department of Cornell University. The research was supported in part by NSF PYI award CCR-89-96272 with matching funds from UPS and Sun Microsystems.     

A superfast algorithm for multi-dimensional Padé systems

For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n ₀,...,n _k ) inO(n · log²n) block-matrix operations, where n=n ₀+...+n _k . Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(n²). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.Supported in part by NSERC grant No. A8035.Partially supported by NSERC operating grant No. 6194.     

On the distribution of comparisons in sorting algorithms

We considered the following natural conjecture: For every sorting algorithm every key will be involved in(logn) comparisons for some input. We show that this is true for most of the keys and prove matching upper and lower bounds. Every sorting algorithm for some input will involven–n /2+1 keys in at leastlog₂ n comparisons,>0. Further, there exists a sorting algorithm that will for every input involve at mostn–n ^/c keys in greater thanlog₂ n comparisons, wherec is a constant and>0. The conjecture is shown to hold for natural algorithms from the literature.     

Lower bounds on moving a ladder in two and three dimensions

It is shown that (n ²) distinct moves may be necessary to move a line segment (a ladder) in the plane from an initial to a final position in the presence of polygonal obstacles of a total ofn vertices, and that (n ⁴) moves may be necessary for the same problem in three dimensions. These two results establish lower bounds on algorithms that solve the motion-planning problems by listing the moves of the ladder. The best upper bounds known areO(n ² logn) in two dimensions, andO(n ⁵ logn) in three dimensions.This work was partially supported by NSF Grants DCR-83-51468 and grants from Martin Marietta, IBM, and General Motors.     

Random Lazy Random Walks on Arbitrary Finite Groups

This paper considers lazy random walks supported on a random subset of k elements of a finite group G with order n. If k=a log₂ n where a>1 is constant, then most such walks take no more than a multiple of log₂ n steps to get close to uniformly distributed on G. If k=log₂ n+f(n) where f(n) and f(n)/log₂ n0 as n, then most such walks take no more than a multiple of (log₂ n) ln(log₂ n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdös and Rényi and of Pak.     

Representations of integers as sums of powers with increasing exponents

Let(n) be the least integer such thatn may be represented in the formn=x ₁ ² +x ₂ ³ +...+x _(n) ⁽ⁿ⁾⁺¹ wherex ₁,x ₂, ...,x _(n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

Many triangulated spheres

Lets(d, n) be the number of triangulations withn labeled vertices ofS ^d–1, the (d–1)-dimensional sphere. We extend a construction of Billera and Lee to obtain a large family of triangulated spheres. Our construction shows that logs(d, n)C ₁(d)n ^[(d–1)/2], while the known upper bound is logs(d, n)C ₂(d)n ^[d/2] logn.Letc(d, n) be the number of combinatorial types of simpliciald-polytopes withn labeled vertices. (Clearly,c(d, n)s(d, n).) Goodman and Pollack have recently proved the upper bound: logc(d, n)d(d+1)n logn. Combining this upper bound forc(d, n) with our lower bounds fors(d, n), we obtain, for everyd5, that lim _n(c(d, n)/s(d, n))=0. The cased=4 is left open. (Steinitz's fundamental theorem asserts thats(3,n)=c(3,n), for everyn.) We also prove that, for everyb4, lim _d(c(d, d+b)/s(d, d+b))=0. (Mani proved thats(d, d+3)=c(d, d+3), for everyd.)Lets(n) be the number of triangulated spheres withn labeled vertices. We prove that logs(n)=2^{0.69424n(1+o(1))}. The same asymptotic formula describes the number of triangulated manifolds withn labeled vertices.Research done, in part, while the author visited the mathematics research center at AT&T Bell Laboratories.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Matrix functions with arbitrarily prescribed left and right partial indices

We prove that ifn2 and , are two given vectors inZ ⁿ, then there exists a matrix function inL ^n×n (T) which has a right Wiener-Hopf factorization inL ² with the partial indices and a left Wiener-Hopf factorization inL ² with the partial indices .     

Storing line segments in partition trees

We design two variants of partition trees, calledsegment partition trees andinterval partition trees, that can be used for storing arbitrarily oriented line segments in the plane in an efficient way. The raw structures useO(n logn) andO(n) storage, respectively, and their construction time isO(n logn). In our applications we augment these structures by certain (simple) auxiliary structures, which may increase the storage and preprocessing time by a polylogarithmic factor. It is shown how to use these structures for solving line segment intersection queries, triangle stabbing queries and ray shooting queries in reasonably efficient ways. If we use the conjugation tree as the underlying partition tree, the query time for all problems isO(n ), where=log₂(1+5)–10.695. The techniques are fairly simple and easy to understand.Research of the first author was partially supported by the ESPRIT II Basic Research Action of the EC under contract No. 3075 (Project ALCOM).Work by the third author has been supported in part by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grants DCR-83-20085 and CCR-89-01484, and by grants from the Digital Equipment Corporation, the IBM Corporation, the U.S.-Israeli Binational Science Foundation, the NCRD — the Israeli National Council for Research and Development, and the Fund for Basic Research in Electronics, Computers and Communication, administered by the Israeli Academy of Sciences.     

The distribution of sublattices of ℤ m

Lattices , are similar if one can be transformed into the other by an angle-preserving linear map. Similarity classes of lattices of rankn may be parametrized by a fundamental domain of the action ofGL _n () on the generalized upper half-plane _n . Given 1<nm and, letN(D,T) be the number of sublattices of _n which have rankn, similarity class inD, and determinant T. Our most basic result will be thatN(D,T)c ₁(m, n)(D)T ^m asT for suitable setsD, where is the invariant measure on _n . The casen=2 had been dealt with by Roelcke and by Maass using the theory of modular forms.Herrn Professor Hlawka zum achtzigsten Geburtstag gewidmetSupported in part by NSF-DMS-9401426     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .