TY - JOUR
AB - This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry. Among these applications we find problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points. All results are either combinatorial or computational in nature. They are based on the combinatorial analysis in two companion papers [PS] and [E2] and a divide-and-conquer algorithm for computing envelopes described in this paper.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
AU - Sharir, Micha
ID - 4081
IS - 1
JF - Discrete & Computational Geometry
TI - The upper envelope of piecewise linear functions: Algorithms and applications
VL - 4
ER -
TY - JOUR
AB - Sweeping a collection of figures in the Euclidean plane with a straight line is one of the novel algorithmic paradigms that have emerged in the field of computational geometry. In this paper we demonstrate the advantages of sweeping with a topological line that is not necessarily straight. We show how an arrangement of n lines in the plane can be swept over in O(n2) time and O(n) space by a such a line. In the process each element, i.e., vertex, edge, or region, is visited once in a consistent ordering. Our technique makes use of novel data structures which exhibit interesting amortized complexity behavior; the result is an algorithm that improves upon all its predecessors either in the space or the time bounds, as well as being eminently practical. Numerous applications of the technique to problems in computational geometry are given—many through the use of duality transforms. Examples include solving visibility problems, detecting degeneracies in configurations, computing the extremal shadows of convex polytopes, and others. Even though our basic technique solves a planar problem, its applications include several problems in higher dimensions.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
ID - 4082
IS - 1
JF - Journal of Computer and System Sciences
TI - Topologically sweeping an arrangement
VL - 38
ER -
TY - JOUR
AB - It is shown that, given a set S of n points in $R^3 $, one can always find three planes that form an eight-partition of S, that is, a partition where at most ${n / 8}$ points of S lie in each of the eight open regions. This theorem is used to define a data structure, called an octant tree, for representing any point set in $R^3 $. An octant tree for n points occupies $O(n)$ space and can be constructed in polynomial time. With this data structure and its refinements, efficient solutions to various range query problems in two and three dimensions can be obtained, including (1) half-space queries: find all points of S that lie to one side of any given plane; (2) polyhedron queries: find all points that lie inside (outside) any given polyhedron; and (3) circle queries in $R^2 $: for a planar set S, find all points that lie inside (outside) any given circle. The retrieval time for all these queries is $T(n) = O(n^\alpha + m)$, where $\alpha = 0.8988$ (or 0.8471 in case (3)), and m is the size of the output. This performance is the best currently known for linear-space data structures that can be deterministically constructed in polynomial time.
AU - Yao, F. Frances
AU - Dobkin, David P
AU - Herbert Edelsbrunner
AU - Paterson,Michael S
ID - 4083
IS - 2
JF - SIAM Journal on Computing
TI - Partitioning space for range queries
VL - 18
ER -
TY - JOUR
AB - A tour of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in . It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour.
In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O(n2 log n) time. If a tour is given, then we can test in O(n2) time whether or not this tour is a necklace-tour. Both algorithms can be generalized to ƒ-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.
AU - Herbert Edelsbrunner
AU - Rote,Günter
AU - Welzl, Emo
ID - 4084
IS - 2
JF - Theoretical Computer Science
TI - Testing the necklace condition for shortest tours and optimal factors in the plane
VL - 66
ER -
TY - CONF
AB - Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces of C with respect to any fixed viewpoint x is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.
AU - Herbert Edelsbrunner
ID - 4085
TI - An acyclicity theorem for cell complexes in d dimension
ER -
TY - JOUR
AB - This note proves that the maximum number of faces (of any dimension) of the upper envelope of a set ofn possibly intersectingd-simplices ind+1 dimensions is (n d (n)). This is an extension of a result of Pach and Sharir [PS] who prove the same bound for the number ofd-dimensional faces of the upper envelope.
AU - Herbert Edelsbrunner
ID - 4086
IS - 4
JF - Discrete & Computational Geometry
TI - The upper envelope of piecewise linear functions: Tight bounds on the number of faces
VL - 4
ER -
TY - CONF
AB - This paper offers combinatorial results on extremum problems concerning the number of tetrahedra in a tetrahedrization of n points in general position in three dimensions, i.e. such that no four points are coplanar. It also presents an algorithm that in O(nlog n) time constructs a tetrahedrization of a set of n points consisting of at most 3n–11 tetrahedra.
AU - Herbert Edelsbrunner
AU - Preparata, Franco P
AU - West, Douglas B
ID - 4087
TI - Tetrahedrizing point sets in three dimensions
VL - 358
ER -
TY - JOUR
AB - Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n 2) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space1 and (n 3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n 1/3)+O(K) time, given(n 4/3) space and (n 5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m 2/3 n 2/3+n), which is nearly optimal.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas
AU - Hershberger, John
AU - Seidel, Raimund
AU - Sharir, Micha
AU - Snoeyink, Jack
AU - Welzl, Emo
ID - 4088
IS - 1
JF - Discrete & Computational Geometry
TI - Implicitly representing arrangements of lines or segments
VL - 4
ER -
TY - JOUR
AB - Motivated by a number of motion-planning questions, we investigate in this paper some general topological and combinatorial properties of the boundary of the union ofn regions bounded by Jordan curves in the plane. We show that, under some fairly weak conditions, a simply connected surface can be constructed that exactly covers this union and whose boundary has combinatorial complexity that is nearly linear, even though the covered region can have quadratic complexity. In the case where our regions are delimited by Jordan acrs in the upper halfplane starting and ending on thex-axis such that any pair of arcs intersect in at most three points, we prove that the total number of subarcs that appear on the boundary of the union is only (n(n)), where(n) is the extremely slowly growing functional inverse of Ackermann's function.
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas
AU - Hershberger, John
AU - Pach, János
AU - Pollack, Richard
AU - Seidel, Raimund
AU - Sharir, Micha
AU - Snoeyink, Jack
ID - 4089
IS - 1
JF - Discrete & Computational Geometry
TI - On arrangements of Jordan arcs with three intersections per pair
VL - 4
ER -
TY - CONF
AU - Chazelle, Bernard
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
AU - Sharir, Micha
ID - 4092
TI - A singly exponential stratification scheme for real semi-algebraic varieties and its applications
VL - 372
ER -
TY - JOUR
AB - This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed.
AU - Chazelle, Bernard
AU - Herbert Edelsbrunner
AU - Guibas, Leonidas J
ID - 4093
IS - 1
JF - Discrete & Computational Geometry
TI - The complexity of cutting complexes
VL - 4
ER -
TY - JOUR
AB - Three methods for estimating the average level of gene flow in natural population are discussed and compared. The three methods are FST, rare alleles, and maximum likelihood. All three methods yield estimates of the combination of parameters (the number of migrants [Nm] in a demic model or the neighborhood size [4πDσ2] in a continuum model) that determines the relative importance of gene flow and genetic drift. We review the theory underlying these methods and derive new analytic results for the expectation of FST in stepping-stone and continuum models when small sets of samples are taken. We also compare the effectiveness of the different methods using a variety of simulated data. We found that the FST and rare-alleles methods yield comparable estimates under a wide variety of conditions when the population being sampled is demographically stable. They are roughly equally sensitive to selection and to variation in population structure, and they approach their equilibrium values at approximately the same rate. We found that two different maximum-likelihood methods tend to yield biased estimates when relatively small numbers of locations are sampled but more accurate estimates when larger numbers are sampled. Our conclusion is that, although FST and rare-alleles methods are expected to be equally effective in analyzing ideal data, practical problems in estimating the frequencies of rare alleles in electrophoretic studies suggest that FST is likely to be more useful under realistic conditions.
AU - Slatkin, Montgomery
AU - Nicholas Barton
ID - 4309
IS - 7
JF - Evolution; International Journal of Organic Evolution
TI - A comparison of three methods for estimating average levels of gene flow
VL - 43
ER -
TY - JOUR
AU - Nicholas Barton
AU - Turelli, Michael
ID - 4312
JF - Annual Review of Genetics
TI - Evolutionary quantitative genetics: how little do we know ?
VL - 23
ER -
TY - CHAP
AU - Nicholas Barton
ED - Otte, Daniel
ED - Endler, John A
ID - 4313
T2 - Speciation and its consequences
TI - Founder effect speciation
ER -
TY - JOUR
AB - Polygenic variation can be maintained by a balance between mutation and stabilizing selection. When the alleles responsible for variation are rare, many classes of equilibria may be stable. The rate at which drift causes shifts between equilibria is investigated by integrating the gene frequency distribution W2N II (pq)4N mu-1. This integral can be found exactly, by numerical integration, or can be approximated by assuming that the full distribution of allele frequencies is approximately Gaussian. These methods are checked against simulations. Over a wide range of population sizes, drift will keep the population near an equilibrium which minimizes the genetic variance and the deviation from the selective optimum. Shifts between equilibria in this class occur at an appreciable rate if the product of population size and selection on each locus is small (Ns alpha 2 less than 10). The Gaussian approximation is accurate even when the underlying distribution is strongly skewed. Reproductive isolation evolves as populations shift to new combinations of alleles: however, this process is slow, approaching the neutral rate (approximately mu) in small populations.
AU - Nicholas Barton
ID - 4314
IS - 1
JF - Genetical Research
TI - The divergence of a polygenic system under stabilising selection, mutation and drift
VL - 54
ER -
TY - CONF
AB - A real-time temporal logic for the specification of reactive systems is introduced. The novel feature of the logic, TPTL, is the adoption of temporal operators as quantifiers over time variables; every modality binds a variable to the time(s) it refers to. TPTL is demonstrated to be both a natural specification language and a suitable formalism for verification and synthesis. A tableau-based decision procedure and model-checking algorithm for TPTL are presented. Several generalizations of TPTL are shown to be highly undecidable.
AU - Alur, Rajeev
AU - Thomas Henzinger
ID - 4596
TI - A really temporal logic
ER -
TY - JOUR
AB - Asymmetrical displacement currents and Na currents of single myelinated nerve fibers of Xenopus laevis were studied in the temperature range from 5 to 24 degrees C. The time constant of the on-response at E = 4 mV, tau on, was strongly temperature dependent, whereas the amount of displaced charge at E = 39 mV, Qon, was only slightly temperature dependent. The mean Q10 for tau on-1 was 2.54, the mean Q10 for Qon was 1.07. The time constant of charge immobilization, tau i, at E = 4 mV varied significantly (alpha = 0.001) with temperature. The mean Q10 for tau i-1 was 2.71 +/- 0.38. The time constants of immobilization of gating charge and of fast inactivation of Na permeability were similar in the temperature range from 6 to 22 degrees C. The Qoff/Qon ratio for E = 4 mV pulses of 0.5 msec duration decreased with increasing temperature. The temperature dependence of the time constant of the off-response could not be described by a single Q10 value, since the Q10 depended on the duration of the test pulse. Increasing temperature shifted Qon (E) curves to more negative potentials by 0.51 mV K-1, but shifted PNa (E) curves and h infinity (E) curves to more positive potentials by 0.43 and 0.57 mV K-1, respectively. h infinity (E = -70 mV) increased monotonously with increasing temperature. The present data indicate that considerable entropy changes may occur when the Na channel molecule passes from closed through open to inactivated states.
AU - Peter Jonas
ID - 3465
IS - 3
JF - Journal of Membrane Biology
TI - Temperature dependence of gating current in myelinated nerve fibers
VL - 112
ER -
TY - JOUR
AB - Amphibian myelinated nerve fibers were treated with collagenase and protease. Axons with retraction of the myelin sheath were patch-clamped in the nodal and paranodal region. One type of Na channel was found. It has a single-channel conductance of 11 pS (15 degrees C) and is blocked by tetrodotoxin. Averaged events show the typical activation and inactivation kinetics of macroscopic Na current. Three potential-dependent K channels were identified (I, F, and S channel). The I channel, being the most frequent type, has a single-channel conductance of 23 pS (inward current, 105 mM K on both sides of the membrane), activates between -60 and -30 mV, deactivates with intermediate kinetics, and is sensitive to dendrotoxin. The F channel has a conductance of 30 pS, activates between -40 and 60 mV, and deactivates with fast kinetics. The former inactivates within tens of seconds; the latter inactivates within seconds. The third type, the S channel, has a conductance of 7 pS and deactivates slowly. All three channels can be blocked by external tetraethylammonium chloride. We suggest that these distinct K channel types form the basis for the different components of macroscopic K current described previously.
AU - Peter Jonas
AU - Bräu, Michael E
AU - Hermsteiner, Markus
AU - Vogel, Werner
ID - 3466
IS - 18
JF - PNAS
TI - Single-channel recording in myelinated nerve fibers reveals one type of Na channel but different K channels
VL - 86
ER -
TY - CONF
AU - Herbert Edelsbrunner
ID - 3549
TI - Spatial triangulations with dihedral angle conditions
ER -
TY - JOUR
AB - Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models of frequency-dependence, single locus clines are stabilized with a maximum slope of between {complex}s/{complex}8 {sigma} and {complex}s/{complex}12 {delta}, where {sigma} is the dispersal distance. These clines are similar to those maintained by heterozygote disadvantage. Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations (``linkage disequilibria''), and heterozygote deficits. The effects of changing the order of reproduction, migration, and selection, are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except that the disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis, as is likely in nature, the error made in estimating selection is usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequences and of gametic correlations from a hybrid zone.
AU - Mallet, James L
AU - Nicholas Barton
ID - 3652
IS - 4
JF - Genetics
TI - Inference from clines stabilized by frequency-dependent selection
VL - 122
ER -