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Abstract. A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with...
Abstract. Let G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be infH l(H) / a(H) and define the isoperimetric constant h () to be infG h (G)...
Abstract. The volume of the n -dimensional polytope n(x):= {y Rn : yi 0 and y1 + + yi x1 + + xi for all 1 i n } for arbitrary x:=(x1, . . ., xn) with xi 0 for all i defines a polynomial in variables xi which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two different...
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