TY - GEN

T1 - On the overlay of envelopes in four dimensions

AU - Kolturi, Vladlen

AU - Sharir, Micha

PY - 2002

Y1 - 2002

N2 - We show that the complexity of the overlay of two envelopes of arrangements of n semi-algebraic surfaces or surface patches of constant description complexity in four dimensions is 0(n4-1/|s/2+ϵ), for any ϵ > 0, where s is a constant related to the maximal degree of the surfaces. This is the first non-trivial (sub-quartic) bound for this problem, and for s = 1,2 it almost matches the near-cubic lower bound. We discuss several applications of this result, including (i) an improved bound for the complexity of the region enclosed between two envelopes in four dimensions, (ii) an improved bound for the complexity of the space of all hyperplane transversals of a collection of simply-shaped convex sets in 4-space, (iii) an improved bound for the complexity of the space of all line transversals of a similar collection of sets in 3-space, and (iv) improved bounds for the complexity of the union of certain families of objects in four dimensions. The analysis technique we introduce is quite general, and has already proved useful in unrelated contexts.

AB - We show that the complexity of the overlay of two envelopes of arrangements of n semi-algebraic surfaces or surface patches of constant description complexity in four dimensions is 0(n4-1/|s/2+ϵ), for any ϵ > 0, where s is a constant related to the maximal degree of the surfaces. This is the first non-trivial (sub-quartic) bound for this problem, and for s = 1,2 it almost matches the near-cubic lower bound. We discuss several applications of this result, including (i) an improved bound for the complexity of the region enclosed between two envelopes in four dimensions, (ii) an improved bound for the complexity of the space of all hyperplane transversals of a collection of simply-shaped convex sets in 4-space, (iii) an improved bound for the complexity of the space of all line transversals of a similar collection of sets in 3-space, and (iv) improved bounds for the complexity of the union of certain families of objects in four dimensions. The analysis technique we introduce is quite general, and has already proved useful in unrelated contexts.

UR - http://www.scopus.com/inward/record.url?scp=0012527692&partnerID=8YFLogxK

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AN - SCOPUS:0012527692

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 810

EP - 819

BT - Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

PB - Association for Computing Machinery

T2 - 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

Y2 - 6 January 2002 through 8 January 2002

ER -