TY - JOUR

T1 - On the zone of a circle in an arrangement of lines

AU - Nivasch, Gabriel

N1 - Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2015/11

Y1 - 2015/11

N2 - Let L be a set of n lines in the plane, and let C be a convex curve in the plane, like a circle or a parabola. The zone of C in L, denoted Z(C,L), is defined as the set of all faces in the arrangement A(L) that are intersected by C. Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of Z(C,L) is at most O(nα(n)), where α is the inverse Ackermann function, by translating the sequence of edges of Z(C,L) into a sequence S that avoids the subsequence ababa. Whether the worst-case complexity of Z(C,L) is only linear is a longstanding open problem.In this paper we provide evidence that, if C is a circle or a parabola, then the zone of C has at most linear complexity: We show that a certain configuration of segments with endpoints on C is impossible. As a consequence, the Hart-Sharir sequences, which are essentially the only known way to construct ababa-free sequences of superlinear length, cannot occur in S.Hence, if it could be shown that every family of superlinear-length, ababa-free sequences must eventually contain all Hart-Sharir sequences, that would settle the zone problem for a circle/parabola.

AB - Let L be a set of n lines in the plane, and let C be a convex curve in the plane, like a circle or a parabola. The zone of C in L, denoted Z(C,L), is defined as the set of all faces in the arrangement A(L) that are intersected by C. Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of Z(C,L) is at most O(nα(n)), where α is the inverse Ackermann function, by translating the sequence of edges of Z(C,L) into a sequence S that avoids the subsequence ababa. Whether the worst-case complexity of Z(C,L) is only linear is a longstanding open problem.In this paper we provide evidence that, if C is a circle or a parabola, then the zone of C has at most linear complexity: We show that a certain configuration of segments with endpoints on C is impossible. As a consequence, the Hart-Sharir sequences, which are essentially the only known way to construct ababa-free sequences of superlinear length, cannot occur in S.Hence, if it could be shown that every family of superlinear-length, ababa-free sequences must eventually contain all Hart-Sharir sequences, that would settle the zone problem for a circle/parabola.

KW - Arrangement

KW - Davenport-Schinzel sequence

KW - Inverse Ackermann function

KW - Zone

UR - http://www.scopus.com/inward/record.url?scp=84947795230&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2015.06.032

DO - 10.1016/j.endm.2015.06.032

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AN - SCOPUS:84947795230

SN - 1571-0653

VL - 49

SP - 221

EP - 231

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

ER -