Abstract
Let κ be a set of n non-intersecting objects in 3-space. A depth order of κ, if exists, is a linear order<of the objects in κ such that if K, L ε κ and K lies vertically below L then K<L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If κ is a set of n triangles whose xy-projections are all ‘fat’, then a depth order for κ can be computed in time O(n log6 n). (ii) If κ is a set of n convex and simply-shaped objects whose xy-projections are all ‘fat’ and their sizes are within a constant ratio from one another, then a depth order for κ can be computed in time O(nλ s 1/2) (n) log4 n), where s is the maximum number of intersections between the xy-projections of the boundaries of any pair of objects in κ.
Work on this paper by the first author has been supported by National Science Foundation Grant CCR-93-01259 and an NYI award. Work on this paper by the third author has been supported by NSF Grant CCR-91-22103, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.
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Agarwal, P.K., Katz, M.J., Sharir, M. (1994). Computing depth orders and related problems. In: Schmidt, E.M., Skyum, S. (eds) Algorithm Theory — SWAT '94. SWAT 1994. Lecture Notes in Computer Science, vol 824. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58218-5_1
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DOI: https://doi.org/10.1007/3-540-58218-5_1
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