To overview the gene content of the important pathogen Phytophthora infestans, large-scale cDNA and genomic sequencing was performed. A set of 75,757 high-quality expressed sequence tags (ESTs) from P. infestans was obtained from 20 cDNA libraries representing a broad range of growth conditions, stress responses, and developmental stages. These included libraries from P. infestans-potato and -tomato interactions, from which 963 pathogen ESTs were identified. To complement the ESTs, onefold coverage of the P. infestans genome was obtained and regions of coding potential identified. A unigene set of 18,256 sequences was derived from the EST and genomic data and characterized for potential functions, stage-specific patterns of expression, and codon bias. Cluster analysis of ESTs revealed major differences between the expressed gene content of mycelial and spore-related stages, and affinities between some growth conditions. Comparisons with databases of fungal pathogenicity genes revealed conserved elements of pathogenicity, such as class III pectate lyases, despite the considerable evolutionary distance between oomycetes and fungi. Thirty-seven genes encoding components of flagella also were identified. Several genes not anticipated to occur in oomycetes were detected, including chitin synthases, phosphagen kinases, and a bacterial-type FtsZ cell-division protein. The sequence data described are available in a searchable public database.

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation of n sites in the plane is presented. The change reduces its O(n log n) expected running time to O(n log log n) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well for n ~ 216, the range of the experiments. It is conjectured that the average number of edges it creates--a good measure of its effficiency--is no more than twice optimal for n less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in the Lp metric for 1 < p -< oo.

A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram of n independent and identically distributed points. The method is applied to derive exact asymptotic bounds on the expected number of simplices of the Voronoi diagram of points chosen from the uniform distribution on the interior of a d-dimensional ball; it is shown that in this case, the complexity of the diagram is 0(n) for fixed d. An algorithm for constructing the Voronoi diagram is presented and analyzed using the new method. The algorithm is shown to require only ©(n) time on average for random points from a ci-ball assuming a real-RAM model of computation with a constant-time floor function. This algorithm is asymptotically faster than any previously known and optimal in the average-case sense.

A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram of n independent and identically distributed points. The method is applied to derive exact asymptotic bounds on the expected number of simplices of the Voronoi diagram of points chosen from the uniform distribution on the interior of a d-dimensional ball; it is shown that in this case, the complexity of the diagram is 0(n) for fixed d. An algorithm for constructing the Voronoi diagram is presented and analyzed using the new method. The algorithm is shown to require only ©(n) time on average for random points from a ci-ball assuming a real-RAM model of computation with a constant-time floor function. This algorithm is asymptotically faster than any previously known and optimal in the average-case sense.

The vascular tissue of higher plants consists of specialized cells that differ from all other cells with respect to their shape and size, their organellar composition, their extracellular matrix, the type of their plasmodesmata, and their physiological functions. Intact and pure vascular tissue can be isolated easily and rapidly from leaf blades of common plantain (Plantago major), a plant that has been used repeatedly for molecular studies of phloem transport. Here, we present a transcriptome analysis based on 5,900 expressed sequence tags (ESTs) and 3,247 independent mRNAs from the Plantago vasculature. The vascular specificity of these ESTs was confirmed by the identification of well-known phloem or xylem marker genes. Moreover, reverse transcription-polymerase chain reaction, macroarray, and northern analyses revealed genes and metabolic pathways that had previously not been described to be vascular specific. Moreover, common plantain transformation was established and used to confirm the vascular specificity of a Plantago promoter-b-glucuronidase construct in transgenic Plantago plants. Eventually, the applicability and usefulness of the obtained data were also demonstrated for other plant species. Reporter gene constructs generated with promoters from Arabidopsis (Arabidopsis thaliana) homologs of newly identified Plantago vascular ESTs revealed vascular specificity of these genes in Arabidopsis as well. The presented vascular ESTs and the newly developed transformation system represent an important tool for future studies of functional genomics in the common plantain vasculature.

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It is shown that the expected number of vertices of the convex hull of n points drawn from a uniform distribution over the interior of a (/-dimensional polytope is 0(log d_1 n). A similar lower bound is derived for a large class of polytopes. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

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