Abstract
We show that the total number of faces bounding any one cell in an arrangement ofn (d−1)-simplices in ℝd isO(n d−1 logn), thus almost settling a conjecture of Pach and Sharir. We present several applications of this result, mainly to translational motion planning in polyhedral environments. We than extend our analysis to derive other results on complexity in arrangements of simplices. For example, we show that in such an arrangement the total number of vertices incident to the same cell on more than one “side” isO(n d−1 logn). We, also show that the number of repetitions of a “k-flap,” formed by intersectingd−k given simplices, along the boundary of the same cell, summed over all cells and allk-flaps, isO(n d−1 log2n). We use this quantity, which we call theexcess of the arrangement, to derive bounds on the complexity ofm distinct cells of such an arrangement.
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Work on this paper by the first author has been partially supported by National Science Foundation Grant CCR-92-11541. Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-90-J-1284, by National Science Foundation Grants CCR-89-01484 and CCR-91-22103, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F.—the German-Israeli Foundation for Scientific Reseach and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.
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Aronov, B., Sharir, M. Castles in the air revisited. Discrete Comput Geom 12, 119–150 (1994). https://doi.org/10.1007/BF02574371
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DOI: https://doi.org/10.1007/BF02574371