@article{4088, abstract = {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.}, author = {Edelsbrunner, Herbert and Guibas, Leonidas and Hershberger, John and Seidel, Raimund and Sharir, Micha and Snoeyink, Jack and Welzl, Emo}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {433 -- 466}, publisher = {Springer}, title = {{Implicitly representing arrangements of lines or segments}}, doi = {10.1007/BF02187742}, volume = {4}, year = {1989}, } @article{4089, abstract = {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.}, author = {Edelsbrunner, Herbert and Guibas, Leonidas and Hershberger, John and Pach, János and Pollack, Richard and Seidel, Raimund and Sharir, Micha and Snoeyink, Jack}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {523 -- 539}, publisher = {Springer}, title = {{On arrangements of Jordan arcs with three intersections per pair}}, doi = {10.1007/BF02187745}, volume = {4}, year = {1989}, } @inproceedings{4092, author = {Chazelle, Bernard and Edelsbrunner, Herbert and Guibas, Leonidas and Sharir, Micha}, booktitle = {16th International Colloquium on Automata, Languages, and Programming}, location = {Stresa, Italy}, pages = {179 -- 193}, publisher = {Springer}, title = {{A singly exponential stratification scheme for real semi-algebraic varieties and its applications}}, doi = {10.1007/BFb0035760}, volume = {372}, year = {1989}, } @article{4093, abstract = {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.}, author = {Chazelle, Bernard and Edelsbrunner, Herbert and Guibas, Leonidas}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {139 -- 181}, publisher = {Springer}, title = {{The complexity of cutting complexes}}, doi = {10.1007/BF02187720}, volume = {4}, year = {1989}, } @article{4309, abstract = {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.}, author = {Slatkin, Montgomery and Barton, Nicholas H}, issn = {1558-5646}, journal = {Evolution; International Journal of Organic Evolution}, number = {7}, pages = {1349 -- 1368}, publisher = {Wiley-Blackwell}, title = {{A comparison of three methods for estimating average levels of gene flow}}, doi = {10.1111/j.1558-5646.1989.tb02587.x }, volume = {43}, year = {1989}, } @article{4312, author = {Barton, Nicholas H and Turelli, Michael}, issn = {1545-2948}, journal = {Annual Review of Genetics}, pages = {337 -- 370}, publisher = {Annual Reviews}, title = {{Evolutionary quantitative genetics: how little do we know?}}, doi = {10.1146/annurev.ge.23.120189.002005}, volume = {23}, year = {1989}, } @inbook{4313, author = {Barton, Nicholas H}, booktitle = {Speciation and its consequences}, editor = {Otte, Daniel and Endler, John}, isbn = {‎ 978-0878936571}, publisher = {Sinauer Press}, title = {{Founder effect speciation}}, year = {1989}, } @article{4314, abstract = {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.}, author = {Barton, Nicholas H}, issn = {1469-5073}, journal = {Genetical Research}, number = {1}, pages = {59 -- 78}, publisher = {Cambridge University Press}, title = {{The divergence of a polygenic system under stabilising selection, mutation and drift}}, doi = {10.1017/S0016672300028378}, volume = {54}, year = {1989}, }