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We give a simple lazy randomized incremental algorithm to compute &#8804;<italic>k</italic>-levels in arrangements of <italic>x</italic>-monotone Jordan curves in the plane, and in arrangements of planes in three-dimensional space. If each pair of curves intersects in at most <italic>s</italic> points, the expected running time of the algorithm is(More)
In TSP with neighborhoods (TSPN) we are given a collection S of regions in the plane, called neighborhoods, and we seek the shortest tour that visits all neighborhoods. Until now constant-factor approximation algorithms have been known only for cases where the neighborhoods are of approximately the same size. In this paper we present the first polynomial(More)
Snap rounding is a method for converting arbitrary-precision arrangements of segments into fixed-precision representation. We present an algorithm for snap rounding with running time O((n + I) log n), where I is the number of intersections between the input segments. In the worst case, our algorithm is an order of magnitude more efficient than the best(More)
A box-tree is a \ifasci so-called \emph{bounding-volume hierarchy} \else bounding-volume hierarchy \fi 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(More)
In this paper we study several instances of the problem of determining the maximum number of topologically distinct two-dimensional images that three-dimensional scenes can induce. To bound this number, we investigate arrangements of curves and of surfaces that have a certain sparseness property. Given a collection of n algebraic surface patches of constant(More)
Inmanyapplications itisimportant thatonecanview asceneat different levels of detail. More precisely, onewould like to visualize the part of the scene that is close at a high level of detail, andthepart that is faraway atalowlevel of detail. We proposea hierarchy of detail levels for a polyhedral terrain that allows this: given a view point, it is possible(More)
In this paper we show how to traverse a subdivision and to report all cells, edges and vertices, without making use of mark bits in the structure or a stack. We do this by performing a depth-rst search on the subdivision, using local criteria for deciding what is the next cell to visit. Our method is extremely simple and provably correct. The algorithm has(More)
We prove that, for any constant &#949;&gt;0, the complexity of the vertical decomposition of a set of <italic>n</italic> triangles in three-dimensional space is <italic>O</italic>(<italic>n</italic><supscrpt>2+&#949;</supscrpt>+<italic>K</italic>), where <italic>K</italic> is the complexity of the arrangement of the triangles. For a single cell the(More)