On the expected number of k-sets

Given a setS ofn points inR ^d , a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE _d (k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onR ^d . WhenP is spherically symmetric we prove thatE _d (k, n)≤cn ^d−1 WhenP is uniform on a convex bodyK⊂R ² we prove thatE ₂ (k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR ² for whichE ₂((n−2)/2,n) iscn logn. The authors express gratitude to the NSF DIMACS Center at Rutgers and Princeton. The research of I. Bárány was supported in part by Hungarian National Science Foundation Grants 1907 and 1909, and W. Steiger's research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR ^d , there arek hyperplanes so that each orthant contains a fraction 1/2 ^k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 ^k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2 ^k−1. We are able to prove that Δ(j, k)=⌈j(^2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 ⁿ withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5). This research was supported by the National Science Foundation under Grant CCR-9118874.     

Thed-step conjecture for polyhedra of dimensiond<6

Two functions Δ and Δ _b , of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ(d, n) is the maximum diameter of convex polyhedra of dimensiond withn faces of dimensiond−1; similarly, Δ _b (d,n) is the maximum diameter of bounded polyhedra of dimensiond withn faces of dimensiond−1. The diameter of a polyhedronP is the smallest integerl such that any two vertices ofP can be joined by a path ofl or fewer edges ofP. It is shown that the boundedd-step conjecture, i.e. Δ _b (d,2d)=d, is true ford≤5. It is also shown that the generald-step conjecture, i.e. Δ(d, 2d)≤d, of significance in linear programming, is false ford≥4. A number of other specific values and bounds for Δ and Δ _b are presented. This revised version was published online in November 2006 with corrections to the Cover Date.     

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.     

A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

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).     

Points and triangles in the plane and halving planes in space

We prove that for any setS ofn points in the plane andn ^3−α triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn ^3−3α/(c log⁵ n) of the triangles. This implies that any set ofn points in three-dimensional space defines at most halving planes. Work on this paper by Boris Aronov and Rephael Wenger has been supported by DIMACS under NSF Grant STC-88-09648. Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-87-14565. Micha Sharir has been supported by ONR Grant N00014-87-K-0129, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Israeli National Council for Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in E ^d (where the dimensiond is fixed). An ε-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all E ^d and such that the interior of any simplex is intersected by at mostεn hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(r ^d ) simplices (which is asymptotically optimal). Forr≤n ^1−σ, where δ>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn) ^O(1) r ^d−1). In the plane we achieve a time boundO(nr) forr≤n ^1−δ, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆E ^d and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ϒ R; R is a simplex}), In timeO(n(logn) ^O(1) r ^d−1 +r ^O(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ε-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly. A preliminary version of this paper appeared inProceedings of the Sixth ACM Symposium on Computational Geometry, Berkeley, 1990, pp. 1–9. Work on this paper was supported by DIMACS Center.     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

An elementary approach to lower bounds in geometric discrepancy

For eachd≥2, it is possible to placen points ind-space so that, given any two-coloring of the points, a half-space exists within which one color outnumbers the other by as much ascn ^1/2−1/2d , for some constantc>0 depending ond. This result was proven in a slightly weaker form by Beck and the bound was later tightened by Alexander. It was recently shown to be asymptotically optimal by Matoušek. We present a proof of the lower bound, which is based on Alexander's technique but is technically simpler and more accessible. We present three variants of the proof, for three diffrent cases, to provide more intuitive insight into the “large-discrepancy” phenomenon. We also give geometric and probabilistic interpretations of the technique. Work by Bernard Chazelle has been supported in part by NSF Grant CCR-90-02352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. Work by Jiří Matoušek has been supported by Charles University Grant No. 351, by Czech Republic Grant GAČR 201/93/2167 and in part by DIMACS. Work by Micha Sharir has been supported by NSF Grant CCR-91-22103, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Lower bounds for off-line range searching

This paper proves three lower bounds for variants of the following rangesearching problem: Given n weighted points inR ^d andn axis-parallel boxes, compute the sum of the weights within each box: (1) if both additions and subtractions are allowed, we prove that Ω(n log logn) is a lower bound on the number of arithmetic operations; (2) if subtractions are disallowed the lower bound becomes Ω(n(logn/loglogn)^d-1), which is nearly optimal; (3) finally, for the case where boxes are replaced by simplices, we establish a quasi-optimal lower bound of Ω(n^2-2/(d+1))/polylog(n). A preliminary version of this paper appeared inProc. 27th Ann. ACM Symp. on Theory of Computing, May 1995, pp. 733–740. This work was supported in part by NSF Grant CCR-93-01254 and the Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc.     

Playing with admissibility spectra

Jensen showed that any countable sequenceA ofA-admissibles is the initial part of the admissibility spectrum of a realR. His construction generalizes straightforwardly to Σ _n -admissibles. This adaptation makes admissibles not inA R-inadmissible. We strengthen Jensen’s theorem by requiring that Σ _n (A)-admissibles not inA be Σ _m (R)-admissible or Σ _m (R)-non-projectible, form <n. The contents of this paper formed part of the author’s doctoral thesis. He would like to thank Professor Sy Friedman for his capable advising, and the NSF and MIT for their financial support.     

Generalized Ham-Sandwich Cuts

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S ₁,…,S _d in R ^d and constants β _i ∈[0,1], there exists a unique hyperplane h with the property that Vol (h ⁺∩S _i )=β _i ⋅Vol (S _i ); h ⁺ is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut.” We give a discrete analogue for a set S of n points in R ^d which are partitioned into a family S=P ₁∪⋅⋅⋅∪P _d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n) ^d−3) algorithm which, given positive integers a _i ≤|P _i |, finds the unique hyperplane h incident with a point in each P _i and having |h ⁺∩P _i |=a _i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse.     

Worst-case minimum rectilinear steiner trees in all dimensions

It is proved that the length of the longest possible minimum rectilinear Steiner tree ofn points in the unitd-cube is asymptotic toβ _dn^(d−1)/d , whereβ _d is a constant that depends on the dimensiond≥2. A method of Chung and Graham (1981) is generalized to dimensiond to show that 1≤β _d≤d4^(1−d)/d . In addition to replicating Chung and Graham's exact determination ofβ ₂=1, this generalization yields new bounds such as 1≤β ₃<1.191 and .     

Almost tight upper bounds for lower envelopes in higher dimensions

We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d≥3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(n ^d−1+∈), for any ∈>0; the constant of proportionality depends on ∈, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(n ^d-2λ _q (n)) for some constantq depending on the shape and degree of the surfaces (where λ _q (n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n ^2+∈), and give several applications of the new bounds. Work on this paper has been supported by NSF Grant CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

A near-linear algorithm for the planar segment-center problem

LetP be a set ofn points in the plane and lete be a segment of fixed length. The segment-center problem is to find a placement ofe (allowing translation and rotation) which minimizes the maximum euclidean distance frome to the points ofP. We present an algorithm that solves the problem in timeO(n ^1+ε), for any ε>0, improving the previous solution of Agarwalet al. [3] by nearly a factor ofO(n). Work on this paper by the second author has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

MAXIMAL CALDERON-ZYGMUND SINGULAR INTEGRAL ON RBMO

Given a positive Radon measure μ on R^d satisfying the linear growth condition μ（B（x,r））≤C0r^n,x∈R^d,r〉0,（1） where n is a fixed number and O〈n≤d. When d-1〈n,it is proved that if Tt,N1=0,then the corresponding maximal Calderon-Zygmund singular integral is bounded from RBMO to itself only except that it is infinite μ-a. e. on R^d.     

The distribution of the root degree of a random permutation

Given a permutation ω of {1, …, n}, let R(ω) be the root degree of ω, i.e. the smallest (prime) integer r such that there is a permutation σ with ω = σ ^r . We show that, for ω chosen uniformly at random, R(ω) = (lnlnn − 3lnlnln n + O ^p (1))⁻¹ lnn, and find the limiting distribution of the remainder term. Research supported in part by NSF grants CCR-0225610, DMS-0505550 and ARO grant W911NF-06-1-0076. Research supported by NSF grant DMS-0406024.