TY - GEN

T1 - On the union of cylinders in three dimensions

AU - Ezra, Esther

PY - 2008

Y1 - 2008

N2 - We show that the combinatorial complexity of the union of n infinite cylinders in ℝ3, having arbitrary radii, is O(n 2+ε), for any ε > 0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir [3], who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir [3], in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearly-quadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in [3]. Finally, we extend our technique to the case of "cigars" of arbitrary radii (that is, Minkowski sums of line-segments and balls), and show that the combinatorial complexity of the union in this case is nearly-quadratic as well. This problem has been studied in [3] for the restricted case where all cigars are (nearly) equal-radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders, and is significantly simpler than the proof presented in [3].

AB - We show that the combinatorial complexity of the union of n infinite cylinders in ℝ3, having arbitrary radii, is O(n 2+ε), for any ε > 0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir [3], who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir [3], in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearly-quadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in [3]. Finally, we extend our technique to the case of "cigars" of arbitrary radii (that is, Minkowski sums of line-segments and balls), and show that the combinatorial complexity of the union in this case is nearly-quadratic as well. This problem has been studied in [3] for the restricted case where all cigars are (nearly) equal-radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders, and is significantly simpler than the proof presented in [3].

UR - http://www.scopus.com/inward/record.url?scp=57949086064&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.25

DO - 10.1109/FOCS.2008.25

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AN - SCOPUS:57949086064

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 179

EP - 188

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -