Quasi-planar graphs have a linear number of edges

A graph is calledquasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasi-planar graph withn vertices isO(n).Work on this paper by Pankaj K. Agarwal, Boris Aronov and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work on this paper by Pankaj K. Agarwal has also been supported by NSF Grant CCR-93-01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by an NYI award, and by matching funds from Xerox Corporation. Work on this paper by Boris Aronov has also been supported by NSF Grant CCR-92-11541 and by a Sloan Research Fellowship. Work on this paper by János Pach, Richard Pollack, and Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-94-24398. Work by János Pach was also supported by Grant OTKA-4269 and by a CUNY Research Award. Work by Richard Pollack was also supported by NSF Grants CCR-94-02640 and DMS-94-00293. Work by Micha Sharir was also supported by NSF Grant CCR-93-11127, by a Max-Planck Research Award, and by grants from the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Part of the work on this paper was done during the participation of the first four authors in the Special Semester on Computational and Combinatorial Geometry organized by the Mathematical Research Institute of Tel Aviv University, Spring 1995.     

Vertical decomposition of arrangements of hyperplanes in four dimensions

We show that, for any collection ℋ ofn hyperplanes in ℜ⁴, the combinatorial complexity of thevertical decomposition of the arrangementA(ℋ) of ℋ isO(n ⁴ logn). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them. Work on this paper by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Leonidas Guibas was also supported by National Science Foundation Grant CCR-9215219. Work by Micha Sharir was also supported by National Science Foundation Grant CCR-91-22103, and by grants from the G.I.F.—the German Isreali Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. Work by Jiří Matouŝek was done while he was visiting Tel Aviv University, and its was partially supported by a Humboldt Research Fellowship. Work on this paper by Dan Halperin was carried out while he was at Tel Aviv University.     

A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment

We consider the problem of planning the motion of an arbitraryk-sided polygonal robotB, free to translate and rotate in a polygonal environmentV bounded byn edges. We present an algorithm that constructs a single component of the free configuration space ofB in timeO((kn) ^2+ɛ), for any ɛ>0. This algorithm, combined with some standard techniques in motion planning, yields a solution to the underlying motion-planning problem, within the same running time. Work on this paper by Dan Halperin has been supported by a Rothschild Postdoctoral Fellowship, by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, and by NSF Grant CCR-9215219. Work on this paper by Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary and extended version of the paper has appeared as: D. Halperin and M. Sharir, Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment,Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993, pp. 382–391. Part of the work on the paper was carried out while Dan Halperin was at Tel Aviv University.     

On Overlays and Minimization Diagrams

The overlay of 2≤m≤d minimization diagrams of n surfaces in ℝ ^d is isomorphic to a substructure of a suitably constructed minimization diagram of mn surfaces in ℝ ^d+m−1. This elementary observation leads to a new bound on the complexity of the overlay of minimization diagrams of collections of d-variate semi-algebraic surfaces, a tight bound on the complexity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algorithmic implications are discussed. Work by V. Koltun was supported by NSF CAREER award CCF-0641402. Work by M. Sharir was supported by NSF Grants CCR-00-98246 and CCF-05-14079, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

The detour and spanning ratio of a graph G embedded in measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(nlog n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain O(nlog ² n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in , and show that computing the detour in is at least as hard as Hopcroft’s problem. This research was partly funded by CRM, FCAR, MITACS, and NSERC. P.A. was supported by NSF under grants CCR-00-86013 EIA-99-72879, EIA-01-31905, and CCR-02-04118, by ARO grants W911NF-04-1-0278 and DAAD19-03-1-0352, and by a grant from the U.S.-Israeli Binational Science Foundation. R.K. was supported by DFG grant Kl 655/14-1. M.S. was supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the U.S.-Israeli Binational Science Foundation (jointly with P.A.), by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Some of these results have appeared in a preliminary form in [2, 20].     

Improved bounds on weak ε-nets for convex sets

LetS be a set ofn points in ℝ ^d . A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size , whereβ ₂=0,β ₃=1, andβ _d ≈0.149·2 ^d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log^1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas. Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Efficient Algorithms for Maximum Regression Depth

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n ^d ) time and O(n ^d−1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog ² n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results. A preliminary version of this paper appeared in the proceedings of the 15th Annual ACM Symposium on Computational Geometry (1999) M. van Kreveld partially funded by the Netherlands Organization for Scientific Research (NWO) under FOCUS/BRICKS grant number 642.065.503. J.S.B. Mitchell’s research largely conducted while the author was a Fulbright Research Scholar at Tel Aviv University. The author is partially supported by NSF (CCR-9504192, CCR-9732220), Boeing, Bridgeport Machines, Sandia, Seagull Technology, and Sun Microsystems. M. Sharir supported by NSF Grants CCR-97-32101 and CCR-94-24398, 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 ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University. J. Snoeyink supported in part by grants from NSERC, the Killam Foundation, and CIES while at the University of British Columbia.     

Diameter,width, closest line pair,and parametric searching

We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ε>0, anO(n ^1+ε) algorithm for computing the diameter of a point set in 3-space, anO(^8/5+ε) algorithm for computing the width of such a set, and onO(n ^8/5+ε) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic. Work by Bernard Chazelle was supported by NSF Grant CCR-90-02352. Work by Herbert Edelsbrunner was supported by NSF Grant CCR-89-21421. Work by Leonidas Guibas and Micha Sharir was supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Micha Sharir was also supported by ONR Grant N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from 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.     

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveC ^rd =G, wherer=rankG. Research supported in part by NSERC Canada Grant A7251. Research supported in part by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.     

On Regular Vertices of the Union of Planar Convex Objects

Let be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in intersect in at most s points for some constant s≥4. We show that the maximum number of regular vertices (intersection points of two boundaries that intersect twice) on the boundary of the union U of is O ^*(n ^4/3), which improves earlier bounds due to Aronov et al. (Discrete Comput. Geom. 25, 203–220, 2001). The bound is nearly tight in the worst case. In this paper, a bound of the form O ^*(f(n)) means that the actual bound is C _ε f(n)⋅n ^ε for any ε>0, where C _ε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0). Work by János Pach and Micha Sharir was supported by NSF Grant CCF-05-14079, and by a grant from the U.S.–Israeli Binational Science Foundation. Work by Esther Ezra and Micha Sharir was supported by grant 155/05 from the Israel Science Fund and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the first author has also been supported by an IBM Doctoral Fellowship. A preliminary version of this paper has been presented in Proc. 23nd Annu. ACM Sympos. Comput. Geom., 2007, pp. 220–226. E. Ezra’s current address: Department of Computer Science, Duke University, Durham, NC 27708-0129, USA. E-mail: esther@cs.duke.edu     

On the number of views of polyhedral terrains

We show that the number of topologically different orthographic views of a polyhedral terrain withn edges isO(n ^5+ɛ ), and that the number of topologically different perspective views of such a terrain isO(n ^8+ɛ ), for any ɛ>0. Both bounds are almost tight in the worst case. The proofs are simple consequences of the recent almost-tight bounds of [11] on the complexity of lower envelopes in higher dimensions. Pankaj Agarwal has been supported by National Science Foundation Grant CCR-91-06514. Micha Sharir has been supported by National Science Foundation 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.     

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.     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

On the concentration of eigenvalues of random symmetric matrices

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e ⁻ t ²32 ^s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces. Research supported in part by a USA — Israel BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a USA — Israel BSF grant and by a Bergmann Memorial Grant.     

Additive latin transversals

We prove that for every odd primep, everyk≤p and every two subsets A={a ₁, …,a _k } andB={b ₁, …,b _k } of cardinalityk each ofZ _p , there is a permutationπ ∈S _k such that the sumsa _i +b _π(i) (inZ _p ) are pairwise distinct. This partially settles a question of Snevily. The proof is algebraic, and implies several related results as well. Research supported in part by a State of New Jersey grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

Constructive lower bounds for off-diagonal Ramsey numbers

We describe an explicit construction whicy, for some fixed absolute positive constant ε, produces, for every integers>1 and all sufficiently largem, a graph on at least vertices containing neither a clique of sizes nor an independent set of sizem. Part of this work was done at the Institute for Advanced Study, Princeton, NJ 08540, USA. Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

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.     

Small Hop-diameter Sparse Spanners for Doubling Metrics

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1+ε)-spanners with small hop diameter for metrics of low doubling dimension. In this paper, we show that given any metric with constant doubling dimension k and any 0<ε<1, one can find (1+ε)-spanner for the metric with nearly linear number of edges (i.e., only O(nlog ^* n+n ε ^−O(k)) edges) and constant hop diameter; we can also obtain a (1+ε)-spanner with linear number of edges (i.e., only n ε ^−O(k) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann’s function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal. The conference version of the paper appeared in ACM-SIAM SODA 2006. This research of T.-H.H. Chan was done while the author was at Carnegie Mellon University and was partly supported by the NSF grant CCR-0122581 (the ALADDIN project), the NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship. This research of A. Gupta was partly supported by the NSF grant CCR-0122581 (the ALADDIN project), the NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship.     

The Number Of Orientations Having No Fixed Tournament

Let T be a fixed tournament on k vertices. Let D(n,T ) denote the maximum number of orientations of an n-vertex graph that have no copy of T. We prove that for all sufficiently (very) large n, where t_k−1(n) is the maximum possible number of edges of a graphon n vertices with no K_k, (determined by Turán’s Theorem). The proof is based on a directed version of Szemerédi’s regularity lemma together with some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying this lemma. For the two possible tournaments with three vertices we obtain separate proofs that avoid the use of the regularity lemma and therefore show that in these cases already holds for (relatively) small values of n. * Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

Castles in the air revisited

We show that the total number of faces bounding any one cell in an arrangement ofn (d−1)-simplices in ℝ ^d isO(n ^d−1 logn), thus almost settling a conjecture of Pach and Sharir. We present several applications of this result, mainly to translational motion planning in polyhedral environments. We than extend our analysis to derive other results on complexity in arrangements of simplices. For example, we show that in such an arrangement the total number of vertices incident to the same cell on more than one “side” isO(n ^d−1 logn). We, also show that the number of repetitions of a “k-flap,” formed by intersectingd−k given simplices, along the boundary of the same cell, summed over all cells and allk-flaps, isO(n ^d−1 log² n). We use this quantity, which we call theexcess of the arrangement, to derive bounds on the complexity ofm distinct cells of such an arrangement. Work on this paper by the first author has been partially supported by National Science Foundation Grant CCR-92-11541. Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-90-J-1284, by National Science Foundation Grants CCR-89-01484 and CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F.—the German-Israeli Foundation for Scientific Reseach and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.