@article{4098, abstract = {To points p and q of a finite set S in d-dimensional Euclidean space Ed are extreme if {p, q} = S ∩ h, for some open halfspace h. Let e2(d)(n) be the maximum number of extreme pairs realized by any n points in Ed. We give geometric proofs of , if n⩾4, and e2(3)(n) = 3n−6, if n⩾6. These results settle the question since all other cases are trivial.}, author = {Edelsbrunner, Herbert and Stöckl, Gerd}, issn = {1096-0899}, journal = {Journal of Combinatorial Theory Series A}, number = {2}, pages = {344 -- 349}, publisher = {Elsevier}, title = {{The number of extreme pairs of finite point-sets in Euclidean spaces}}, doi = {10.1016/0097-3165(86)90075-0}, volume = {43}, year = {1986}, } @article{4099, abstract = {Let S denote a set of n points in the Euclidean plane. A halfplanar range query specifies a halfplane h and requires the determination of the number of points in S which are contained in h. A new data structure is described which stores S in O(n) space and allows us to answer a halfplanar range query in O(nlog2(1+√5)−1) time in the worst case, thus improving the best result known before. The structure can be built in O(n log n) time.}, author = {Edelsbrunner, Herbert and Welzl, Emo}, issn = {1872-6119}, journal = {Information Processing Letters}, number = {5}, pages = {289 -- 293}, publisher = {Elsevier}, title = {{Halfplanar range search in linear space and O(n0.695) query time}}, doi = {10.1016/0020-0190(86)90088-8}, volume = {23}, year = {1986}, } @article{4103, abstract = {Let A be an arrangement of n lines in the plane. Suppose F1,…, Fk are faces in the dissection induced by A and that Fi is a t(Fi)-gon. We give asymptotic bounds on the maximal sum ∑i=1kt(Fi) which can be realized by k different faces in an arrangement of n lines. The results improve known bounds for k of higher order than n(1/2).}, author = {Edelsbrunner, Herbert and Welzl, Emo}, issn = {1096-0899}, journal = {Journal of Combinatorial Theory Series A}, number = {2}, pages = {159 -- 166}, publisher = {Elsevier}, title = {{On the maximal number of edges of many faces in an arrangement}}, doi = {10.1016/0097-3165(86)90078-6}, volume = {41}, year = {1986}, } @article{4104, abstract = {Point location, often known in graphics as “hit detection,” is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let $\mathcal{S}$ denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of $m$ edges. In this paper we exhibit a substantial refinement of the technique of Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594–606] for locating a point in $\mathcal{S}$ based on separating chains. The new data structure, called a layered dag, can be built in $O(m)$ time, uses $O(m)$ storage, and makes possible point location in $O(\log m)$ time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments. © 1986 Society for Industrial and Applied Mathematics}, author = {Edelsbrunner, Herbert and Guibas, Leonidas and Stolfi, Jorge}, issn = {1095-7111}, journal = {SIAM Journal on Computing}, number = {2}, pages = {317 -- 340}, publisher = {SIAM}, title = {{Optimal point location in a monotone subdivision}}, doi = {10.1137/0215023}, volume = {15}, year = {1986}, } @article{4105, abstract = {A finite set of lines partitions the Euclidean plane into a cell complex. Similarly, a finite set of $(d - 1)$-dimensional hyperplanes partitions $d$-dimensional Euclidean space. An algorithm is presented that constructs a representation for the cell complex defined by $n$ hyperplanes in optimal $O(n^d )$ time in $d$ dimensions. It relies on a combinatorial result that is of interest in its own right. The algorithm is shown to lead to new methods for computing $\lambda $-matrices, constructing all higher-order Voronoi diagrams, halfspatial range estimation, degeneracy testing, and finding minimum measure simplices. In all five applications, the new algorithms are asymptotically faster than previous results, and in several cases are the only known methods that generalize to arbitrary dimensions. The algorithm also implies an upper bound of $2^{cn^d } $, $c$ a positive constant, for the number of combinatorially distinct arrangements of $n$ hyperplanes in $E^d $. © 1986 Society for Industrial and Applied Mathematics}, author = {Edelsbrunner, Herbert and O'Rourke, Joseph and Seidel, Raimund}, issn = {1095-7111}, journal = {SIAM Journal on Computing}, number = {2}, pages = {341 -- 363}, publisher = {SIAM}, title = {{Constructing arrangements of lines and hyperplanes with applications}}, doi = {10.1137/0215024}, volume = {15}, year = {1986}, } @article{4106, abstract = {Let B be a set of nb black points and W a set of nw, white points in the Euclidean plane. A line h is said to bisect B (or W) if, at most, half of the points of B (or W) lie on any one side of h. A line that bisects both B and W is called a ham-sandwich cut of B and W. We give an algorithm that computes a ham-sandwich cut of B and W in 0((nh+nw) log (min {nb, nw}+ 1)) time. The algorithm is considerably simpler than the previous most efficient one which takes 0((nb + nw) log (nb + nw)) time.}, author = {Edelsbrunner, Herbert and Waupotitsch, Roman}, issn = {1095-855X}, journal = {Journal of Symbolic Computation}, number = {2}, pages = {171 -- 178}, publisher = {Elsevier}, title = {{Computing a ham-sandwich cut in two dimensions}}, doi = {10.1016/S0747-7171(86)80020-7}, volume = {2}, year = {1986}, } @article{4107, abstract = {A set of m planes dissects E3 into cells, facets, edges and vertices. Letting deg(c) be the number of facets that bound a cellc, we give exact and asymptotic bounds on the maximum of ∈cinCdeg(c), if C is a family of cells of the arrangement with fixed cardinality.}, author = {Edelsbrunner, Herbert and Haussler, David}, issn = {1872-681X}, journal = {Discrete Mathematics}, number = {C}, pages = {139 -- 146}, publisher = {Elsevier}, title = {{The complexity of cells in 3-dimensional arrangements}}, doi = {10.1016/0012-365X(86)90008-7}, volume = {60}, year = {1986}, } @article{4108, abstract = {We propose a uniform and general framework for defining and dealing with Voronoi diagrams. In this framework a Voronoi diagram is a partition of a domainD induced by a finite number of real valued functions onD. Valuable insight can be gained when one considers how these real valued functions partitionD ×R. With this view it turns out that the standard Euclidean Voronoi diagram of point sets inR d along with its order-k generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.}, author = {Edelsbrunner, Herbert and Seidel, Raimund}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {1}, pages = {25 -- 44}, publisher = {Springer}, title = {{Voronoi diagrams and arrangements}}, doi = {10.1007/BF02187681}, volume = {1}, year = {1986}, } @article{4109, abstract = {Rectangle location search in d dimensions is finding the d-dimensional axis-parallel box of a non-overlapping collection C that contains a query point. A new data structure is proposed that requires optimal space and 0(logd|C|) time for a search. The significance of this data structure in practical applications is substantiated by empirical examinations of its behaviour.}, author = {Edelsbrunner, Herbert and Haring, Günter and Hilbert, D}, issn = {1460-2067}, journal = {Computer Journal}, number = {1}, pages = {76 -- 82}, publisher = {Oxford University Press}, title = {{Rectangular point location in d-dimensions with applications}}, doi = {10.1093/comjnl/29.1.76}, volume = {29}, year = {1986}, } @article{4110, abstract = {For H a set of lines in the Euclidean plane, $A(H)$ denotes the induced dissection, called the arrangement of H. We define the notion of a belt in $A(H)$, which is bounded by a subset of the edges in $A(H)$, and describe two algorithms for constructing belts. All this is motivated by applications to a host of seemingly unrelated problems including a type of range search and finding the minimum area triangle with the vertices taken from some finite set of points.}, author = {Edelsbrunner, Herbert and Welzl, Emo}, issn = {1095-7111}, journal = {SIAM Journal on Computing}, number = {1}, pages = {271 -- 284}, publisher = {SIAM}, title = {{Constructing belts in two-dimensional arrangements with applications}}, doi = {10.1137/0215019}, volume = {15}, year = {1986}, } @article{4321, abstract = {The fire-bellied toads Bombina bombina and B. variegata differ extensively in biochemistry, morphology, and behavior. We use a survey of five diagnostic enzyme loci across the hybrid zone near Cracow in Southern Poland to estimate the dispersal rate, selection pressures, and numbers of loci which maintain this zone. The enzyme clines coincide closely with each other and with morphological and mitochondrial DNA clines. Although the zone lies on a broad transition between environments suitable for bombina and variegata, the close concordance of diverse characters, together with increased aberrations and mortality in hybrids, suggest that the zone is maintained largely by selection against hybrids. There are strong “linkage disequilibria” between each pair of (unlinked) enzyme loci (R̄ = 0.129 [2-unit support limits: 0.119–0.139]). These are probably caused by gene flow into the zone, and they give an estimate of dispersal (σ = 890 [790–940] m gen−½). The clines are sharply stepped, with most of the change occurring within 6.15 (5.45–6.45) km, but with long tails of introgression on either side. This implies that the effective selection pressure on each enzyme marker (due largely to disequilibrium with other loci) is s* = 0.17 (0.159–0.181) at the center but that the selection acting directly on the enzyme loci is weak or zero (se < 0.0038). The stepped pattern implies a barrier to gene flow of 220 (48–415) km. This would substantially delay neutral introgression but would have little effect on advantageous alleles; the two taxa need not evolve independently. Strong selection is needed to maintain such a barrier: hybrid populations must have their mean fitness reduced by a factor of 0.65 (0.60–0.77). This selection must be spread over a large number of loci to account for the concordant patterns and the observed cline widths (N = 300 [80–2,000]).}, author = {Szymura, Jacek and Barton, Nicholas H}, issn = {1558-5646}, journal = {Evolution; International Journal of Organic Evolution}, pages = {1141 -- 1159}, publisher = {Society for the Study of Evolution}, title = {{Genetic analysis of a hybrid zone between the fire-bellied toads Bombina bombina and B. variegata, near Cracow in Southern Poland}}, doi = {10.1111/j.1558-5646.1986.tb05740.x}, volume = {40}, year = {1986}, } @article{4323, abstract = {It is noted that the sibling competition model for the evolution of sex and recombination, as it has been developed so far, involves truncation selection. After briefly reviewing aspects of the development and behaviour of such models an analytical treatment is presented which involves additive selection. Additive selection, as compared with truncation selection, decreases the advantage of sex to such an extent that it is unlikely that sibling competition could overcome its intrinsic two-fold cost, although it could still be important in promoting family variability produced by other mechanisms, such as polyandry.}, author = {Barton, Nicholas H and Post, R.J.}, issn = {1095-8541}, journal = {Journal of Theoretical Biology}, number = {4}, pages = {381 -- 387}, publisher = {Elsevier}, title = {{Sibling competition and the advantage of mixed families}}, doi = {10.1016/S0022-5193(86)80033-9}, volume = {120}, year = {1986}, } @article{4324, abstract = {The maintenance of polygenic variation through a balance between mutation and stabilizing selection can be approximated in two ways. In the ‘Gaussian’ approximation, a normal distribution of allelic effects is assumed at each locus. In the ‘House of Cards’ approximation, the effect of new mutations is assumed to be large compared with the spread of the existing distribution. These approximations were developed to describe models where alleles may have a continuous range of effects. However, previous analyses of models with only two alleles have predicted an equilibrium variance equal to that given by the ‘House of Cards’ approximation. These analyses of biallelic models have assumed that, at equilibrium, the population mean is at the optimum. Here, it is shown that many stable equilibria may coexist, each giving a slight deviation from the optimum. Though the variance is given by the ‘House of Cards’ approximation when the mean is at the optimum, it increases towards a value of the same order as that given by the ‘Gaussian’ approximation when the mean deviates from the optimum. Thus, the equilibrium variance cannot be predicted by any simple model, but depends on the previous history of the population.}, author = {Barton, Nicholas H}, issn = {1469-5073}, journal = {Genetical Research}, number = {3}, pages = {209 -- 216}, publisher = {Cambridge University Press}, title = {{The maintenance of polygenic variation through a balance between mutation and stabilising selection}}, doi = {10.1017/S0016672300023156}, volume = {47}, year = {1986}, } @article{4111, abstract = {This paper describes an optimal solution for the following geometric search problem defined for a set P of n points in three dimensions: Given a plane h with all points of P on one side and a line ℓ in h, determine a point of P that is hit first when h is rotated around ℓ. The solution takes O(n) space and O(log n) time for a query. By use of geometric transforms, the post-office problem for a finite set of points in two dimensions and certain two-dimensional point location problems are reduced to the former problem and thus also optimally solved.}, author = {Edelsbrunner, Herbert and Maurer, Hermann}, issn = {1872-6119}, journal = {Information Processing Letters}, number = {1}, pages = {39 -- 47}, publisher = {Elsevier}, title = {{Finding extreme-points in 3-dimensions and solving the post-office problem in the plane}}, doi = {10.1016/0020-0190(85)90107-3}, volume = {21}, year = {1985}, } @article{4112, abstract = {The batched static version of a searching problem asks for performing a given set of queries on a given set of objects. All queries are known in advance. The batched dynamic version of a searching problem is the following: given a sequence of insertions, deletions, and queries, perform them on an initially empty set. We will develop methods for solving batched static and batched dynamic versions of searching problems which are in particular applicable to decomposable searching problems. The techniques show that batched static (dynamic) versions of searching problems can often be solved more efficiently than by using known static (dynamic) data structures. In particular, a technique called “streaming” is described that reduces the space requirements considerably. The methods have also a number of applications on set problems. E.g., the k intersecting pairs in a set of n axis-parallel hyper-rectangles in d dimensions can be reported in O (nlogd−1n + k) time using only O(n) space.}, author = {Edelsbrunner, Herbert and Overmars, Mark}, issn = {1090-2678}, journal = {Journal of Algorithms}, number = {4}, pages = {515 -- 542}, publisher = {Elsevier}, title = {{Batched dynamic solutions to decomposable searching problems}}, doi = {10.1016/0196-6774(85)90030-6}, volume = {6}, year = {1985}, } @article{4113, abstract = {Let S denote a set of n points in the Euclidean plane. A subset S′ of S is termed a k-set of S if it contains k points and there exists a straight line which has no point of S on it and separates S′ from S−S′. We let fk(n) denote the maximum number of k-sets which can be realized by a set of n points. This paper studies the asymptotic behaviour of fk(n) as this function has applications to a number of problems in computational geometry. A lower and an upper bound on fk(n) is established. Both are nontrivial and improve bounds known before. In particular, is shown by exhibiting special point-sets which realize that many k-sets. In addition, is proved by the study of a combinatorial problem which is of interest in its own right.}, author = {Edelsbrunner, Herbert and Welzl, Emo}, issn = {1096-0899}, journal = {Journal of Combinatorial Theory Series A}, number = {1}, pages = {15 -- 29}, publisher = {Elsevier}, title = {{On the number of line separations of a finite set in the plane}}, doi = {10.1016/0097-3165(85)90017-2}, volume = {38}, year = {1985}, } @article{4114, abstract = {Proportional link linkage (PLL) clustering methods are a parametric family of monotone invariant agglomerative hierarchical clustering methods. This family includes the single, minimedian, and complete linkage clustering methods as special cases; its members are used in psychological and ecological applications. Since the literature on clustering space distortion is oriented to quantitative input data, we adapt its basic concepts to input data with only ordinal significance and analyze the space distortion properties of PLL methods. To enable PLL methods to be used when the numbern of objects being clustered is large, we describe an efficient PLL algorithm that operates inO(n 2 logn) time andO(n 2) space}, author = {Day, William and Edelsbrunner, Herbert}, issn = {1432-1343}, journal = {Journal of Classification}, number = {2-3}, pages = {239 -- 254}, publisher = {Springer}, title = {{Investigation of Proportional Link Linkage Clustering Methods}}, doi = {10.1007/BF01908077}, volume = {2}, year = {1985}, } @article{4115, abstract = {A polygon in the plane is convex if it contains all line segments connecting any two of its points. Let P and Q denote two convex polygons. The computational complexity of finding the minimum and maximum distance possible between two points p in P and q in Q is studied. An algorithm is described that determines the minimum distance (together with points p and q that realize it) in O(logm + logn) time, where m and n denote the number of vertices of P and Q, respectively. This is optimal in the worst case. For computing the maximum distance, a lower bound Ω(m + n) is proved. This bound is also shown to be best possible by establishing an upper bound of O(m + n).}, author = {Edelsbrunner, Herbert}, issn = {1090-2678}, journal = {Journal of Algorithms}, number = {2}, pages = {213 -- 224}, publisher = {Academic Press}, title = {{Computing the extreme distances between two convex polygons}}, doi = {10.1016/0196-6774(85)90039-2}, volume = {6}, year = {1985}, } @article{4116, abstract = {A straight line that intersects all members of a set S of objects in the real plane is called a transversal of S. Geometric transforms are described that reduce transversal problems for various types of objects to convex hull problems for points. These reductions lead to efficient algorithms for finding transversals which are also described. Applications of the algorithms are found in computer graphics: “Reproduce the line displayed by a collection of pixels”, and in statistics: “Find the line that minimizes the maximum distance from a collection of (weighted) points in the plane”.}, author = {Edelsbrunner, Herbert}, issn = {0304-3975}, journal = {Theoretical Computer Science}, number = {1}, pages = {55 -- 69}, publisher = {Elsevier}, title = {{Finding Transversals for Sets of Simple Geometric-Figures}}, doi = {10.1016/0304-3975(85)90005-2}, volume = {35}, year = {1985}, } @article{4120, abstract = {Let P be a set of n points in the Euclidean plane and let C be a convex figure. We study the problem of preprocessing P so that for any query point q, the points of P in C+q can be retrieved efficiently. If constant time sumces for deciding the inclusion of a point in C, we then demonstrate the existence of an optimal solution: the algorithm requires O(n) space and O(k + log n) time for a query with output size k. If C is a disk, the problem becomes the wellknown fixed-radius neighbour problem, to which we thus provide the first known optimal solution.}, author = {Chazelle, Bernard and Edelsbrunner, Herbert}, issn = {1095-855X}, journal = {Journal of Symbolic Computation}, number = {1}, pages = {47 -- 56}, publisher = {Elsevier}, title = {{Optimal solutions for a class of point retrieval problems}}, doi = {10.1016/S0747-7171(85)80028-6}, volume = {1}, year = {1985}, }