TY - GEN

T1 - Almost tight bound for the union of fat tetrahedra in three dimensions

AU - Ezra, Esther

AU - Sharir, Micha

PY - 2007

Y1 - 2007

N2 - We show that the combinatorial complexity of the union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O(n2+ε), for any ε > 0; the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combinedwith the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in IR3, having arbitrary side lengths, is O(n2+ε), for any ε ≥ 0 (again, significantly extending the result of [24]). Our analysis can easily be extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in IR3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.

AB - We show that the combinatorial complexity of the union of n "fat" tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is O(n2+ε), for any ε > 0; the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combinedwith the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in IR3, having arbitrary side lengths, is O(n2+ε), for any ε ≥ 0 (again, significantly extending the result of [24]). Our analysis can easily be extended to yield a nearly-quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in IR3. Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.

UR - http://www.scopus.com/inward/record.url?scp=46749099018&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2007.4389522

DO - 10.1109/FOCS.2007.4389522

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AN - SCOPUS:46749099018

SN - 0769530109

SN - 9780769530109

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

SP - 525

EP - 535

BT - Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007

T2 - 48th Annual Symposium on Foundations of Computer Science, FOCS 2007

Y2 - 20 October 2007 through 23 October 2007

ER -