@inbook{3565, abstract = {We investigate the complexity of determining the shape and presentation (i.e. position with orientation) of convex polytopes in multi-dimensional Euclidean space using a variety of probe models.}, author = {Dobkin, David and Edelsbrunner, Herbert and Yap, Chee}, booktitle = {Autonomous Robot Vehicles}, editor = {Cox, Ingemar and Wilfong, Gordon}, isbn = {978-1-4613-8997-2}, pages = {328 -- 341}, publisher = {Springer}, title = {{Probing convex polytopes}}, doi = {10.1007/978-1-4613-8997-2_25}, year = {1990}, } @article{3649, abstract = {Selection on polygenic characters is generally analyzed by statistical methods that assume a Gaussian (normal) distribution of breeding values. We present an alternative analysis based on multilocus population genetics. We use a general representation of selection, recombination, and drift to analyze an idealized polygenic system in which all genetic effects are additive (i.e., both dominance and epistasis are absent), but no assumptions are made about the distribution of breeding values or the numbers of loci or alleles. Our analysis produces three results. First, our equations reproduce the standard recursions for the mean and additive variance if breeding values are Gaussian; but they also reveal how non-Gaussian distributions of breeding values will alter these dynamics. Second, an approximation valid for weak selection shows that even if genetic variance is attributable to an effectively infinite number of loci with only additive effects, selection will generally drive the distribution of breeding values away from a Gaussian distribution by creating multilocus linkage disequilibria. Long-term dynamics of means can depart substantially from the predictions of the standard selection recursions, but the discrepancy may often be negligible for short-term selection. Third, by including mutation, we show that, for realistic parameter values, linkage disequilibrium has little effect on the amount of additive variance maintained at an equilibrium between stabilizing selection and mutation. Each of these analytical results is supported by numerical calculations.}, author = {Turelli, Michael and Barton, Nicholas H}, issn = {0040-5809}, journal = {Theoretical Population Biology}, number = {1}, pages = {1 -- 57}, publisher = {Academic Press}, title = {{Dynamics of polygenic characters under selection}}, doi = {10.1016/0040-5809(90)90002-D}, volume = {38}, year = {1990}, } @article{3650, abstract = {Hybrid zones can yield estimates of natural selection and gene flow. The width of a cline in gene frequency is approximately proportional to gene flow (σ) divided by the square root of per-locus selection ( &s). Gene flow also causes gametic correlations (linkage disequilibria) between genes that differ across hybrid zones. Correlations are stronger when the hybrid zone is narrow, and rise to a maximum roughly equal to s. Thus cline width and gametic correlations combine to give estimates of gene flow and selection. These indirect measures of σ and s are especially useful because they can be made from collections, and require no field experiments. The method was applied to hybrid zones between color pattern races in a pair of Peruvian Heliconius butterfly species. The species are Mullerian mimics of one another, and both show the same changes in warning color pattern across their respective hybrid zones. The expectations of cline width and gametic correlation were generated using simulations of clines stabilized by strong frequency-dependent selection. In the hybrid zone in Heliconius erato, clines at three major color pattern loci were between 8.5 and 10.2 km wide, and the pairwise gametic correlations peaked at R & 0.35. These measures suggest that s & 0.23 per locus, and that σ & 2.6 km. In erato, the shapes of the clines agreed with that expected on the basis of dominance. Heliconius melpomene has a nearly coincident hybrid zone. In this species, cline widths at four major color pattern loci varied between 11.7 and 13.4 km. Pairwise gametic correlations peaked near R & 1.00 for tightly linked genes, and at R & 0.40 for unlinked genes, giving s & 0.25 per locus and σ & 3.7 km. In melpomene, cline shapes did not perfectly fit theoretical shapes based on dominance; this deviation might be explained by long-distance migration and/or strong epistasis. Compared with erato, sample sizes in melpomene are lower and the genetics of its color patterns are less well understood. In spite of these problems, selection and gene flow are clearly of the same order of magnitude in the two species. The relatively high per locus selection coefficients agree with ``major gene'' theories for the evolution of Mullerian mimicry, but the genetic architecture of the color patterns does not. These results show that the genetics and evolution of mimicry are still only sketchily understood.}, author = {Mallet, James and Barton, Nicholas H and Lamas, Gerado and Santisteban, José and Muedas, Manuel and Eeley, Harriet}, issn = {0016-6731}, journal = {Genetics}, number = {4}, pages = {921 -- 936}, publisher = {Genetics Society of America}, title = {{Estimates of selection and gene flow from measures of cline width and linkage disequilibrium in Heliconius hybrid zones}}, doi = {10.1093/genetics/124.4.921}, volume = {124}, year = {1990}, } @article{3651, abstract = {It is widely held that each gene typically affects many characters, and that each character is affected by many genes. Moreover, strong stabilizing selection cannot act on an indefinitely large number of independent traits. This makes it likely that heritable variation in any one trait is maintained as a side effect of polymorphisms which have nothing to do with selection on that trait. This paper examines the idea that variation is maintained as the pleiotropic side effect of either deleterious mutation, or balancing selection. If mutation is responsible, it must produce alleles which are only mildly deleterious (s & 10(-3)), but nevertheless have significant effects on the trait. Balancing selection can readily maintain high heritabilities; however, selection must be spread over many weakly selected polymorphisms if large responses to artificial selection are to be possible. In both classes of pleiotropic model, extreme phenotypes are less fit, giving the appearance of stabilizing selection on the trait. However, it is shown that this effect is weak (of the same order as the selection on each gene): the strong stabilizing selection which is often observed is likely to be caused by correlations with a limited number of directly selected traits. Possible experiments for distinguishing the alternatives are discussed.}, author = {Barton, Nicholas H}, issn = {0016-6731}, journal = {Genetics}, number = {3}, pages = {773 -- 782}, publisher = {Genetics Society of America}, title = {{Pleiotropic models of quantitative variation}}, doi = {10.1093/genetics/124.3.773 }, volume = {124}, year = {1990}, } @article{4060, abstract = {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 co-planar, It also presents an algorithm that in O(n log n) time constructs a tetrahedrization of a set of n points consisting of at most 3n-11 tetrahedra.}, author = {Edelsbrunner, Herbert and Preparata, Franco and West, Douglas}, issn = {1095-855X}, journal = {Journal of Symbolic Computation}, number = {3-4}, pages = {335 -- 347}, publisher = {Elsevier}, title = {{Tetrahedrizing point sets in three dimensions}}, doi = {10.1016/S0747-7171(08)80068-5}, volume = {10}, year = {1990}, } @article{4063, abstract = {This paper describes a general-purpose programming technique, called Simulation of Simplicity, that can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task of providing a consistent treatment for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than those that do not use it. We believe that this technique will become a standard tool in writing geometric software.}, author = {Edelsbrunner, Herbert and Mücke, Ernst}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {1}, pages = {66 -- 104}, publisher = {ACM}, title = {{Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms}}, doi = {10.1145/77635.77639}, volume = {9}, year = {1990}, } @article{4064, abstract = {Given a set of data points pi = (xi, yi ) for 1 ≤ i ≤ n, the least median of squares regression line is a line y = ax + b for which the median of the squared residuals is a minimum over all choices of a and b. An algorithm is described that computes such a line in O(n 2) time and O(n) memory space, thus improving previous upper bounds on the problem. This algorithm is an application of a general method built on top of the topological sweep of line arrangements.}, author = {Edelsbrunner, Herbert and Souvaine, Diane}, issn = {1537-274X}, journal = {Journal of the American Statistical Association}, number = {409}, pages = {115 -- 119}, publisher = {American Statistical Association}, title = {{Computing least median of squares regression lines and guided topological sweep}}, doi = {10.1080/01621459.1990.10475313}, volume = {85}, year = {1990}, } @article{4065, abstract = {We prove that given n⩾3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n−9 sides, and with not more than 3n−6 distinct slopes. Furthermore, we construct sets that require 6n−9 sides and 3n−6 slopes for n⩾3. The upper bound on the number of slopes implies a new bound on a recently studied transversal problem.}, author = {Edelsbrunner, Herbert and Robison, Arch and Shen, Xiao}, issn = {1872-681X}, journal = {Discrete Mathematics}, number = {2}, pages = {153 -- 164}, publisher = {Elsevier}, title = {{Covering convex sets with non-overlapping polygons}}, doi = {10.1016/0012-365X(90)90147-A}, volume = {81}, year = {1990}, } @article{4066, abstract = {We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m 2/3 n logn +n 2); we can calculatem such cells specified by a point in each, in worst-case timeO(m 2/3 n log3 n+n 2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m 2/3 n logn+n 2), but this number is onlyO(m 3/5– n 4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m 3/4– n 3/4+3 +m) log2 n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m 3/4– n 3/4+3 +m] log2 n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m 2/3 n d/3 logn+n d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.}, author = {Edelsbrunner, Herbert and Guibas, Leonidas and Sharir, Micha}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {197 -- 216}, publisher = {Springer}, title = {{The complexity of many cells in arrangements of planes and related problems}}, doi = {10.1007/BF02187785}, volume = {5}, year = {1990}, } @inproceedings{4067, abstract = {This paper proves an O(m 2/3 n 2/3+m+n) upper bound on the number of incidences between m points and n hyperplanes in four dimensions, assuming all points lie on one side of each hyperplane and the points and hyperplanes satisfy certain natural general position conditions. This result has application to various three-dimensional combinatorial distance problems. For example, it implies the same upper bound for the number of bichromatic minimum distance pairs in a set of m blue and n red points in three-dimensional space. This improves the best previous bound for this problem.}, author = {Edelsbrunner, Herbert and Sharir, Micha}, booktitle = {Proceedings of the International Symposium on Algorithms}, isbn = {978-3-540-52921-7}, location = {Tokyo, Japan}, pages = {419 -- 428}, publisher = {Springer}, title = {{A hyperplane Incidence problem with applications to counting distances}}, doi = {10.1007/3-540-52921-7_91}, volume = {450}, year = {1990}, } @article{4068, abstract = {LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations (i1i2in−1in)(inin−1i2i1) is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n–2 geometric permutations. This upper bound is tight.}, author = {Edelsbrunner, Herbert and Sharir, Micha}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {35 -- 42}, publisher = {Springer}, title = {{The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2}}, doi = {10.1007/BF02187778}, volume = {5}, year = {1990}, } @article{4069, abstract = {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.}, author = {Edelsbrunner, Herbert}, issn = {1439-6912}, journal = {Combinatorica}, number = {3}, pages = {251 -- 260}, publisher = {Springer}, title = {{An acyclicity theorem for cell complexes in d dimension}}, doi = {10.1007/BF02122779}, volume = {10}, year = {1990}, } @article{4070, abstract = {Let S be a set of n closed intervals on the x-axis. A ranking assigns to each interval, s, a distinct rank, p(s)∊ [1, 2,…,n]. We say that s can see t if p(s)