Stabbing c-oriented polygons

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Abstract

Given a point and a set of objects in 2-space, the point's stabbing number is the number of objects in the set enclosing it. We introduce the notion of c-oriented objects, that is, objects whose edges are oriented in only a constant number of previously defined directions. We devise time-optimal algorithms for determining the stabbing numbers of points with respect to a set of c-oriented polygons: Given a mixed set of points and polygons of size n we show how to determine the stabbing numbers of all points in O(n log n) time. For a static set of polygons we are able to answer stabbing number queries in O(log n) time. For the second problem the same time bound was achieved previously, but only with much higher space- and preprocessing-costs.

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This work was carried out under a DAAD (Deutscher Akademischer Austauschdienst) Grant, while visiting Mc-Master University.

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