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1997 Combinatorial properties of one-dimensional arrangements
Frédéric Cazals
Experiment. Math. 6(1): 87-94 (1997).


Motivated by problems from computer graphics and robotics---namely, ray tracing and assembly planning---we investigate the combinatorial structure of arrangements of segments on a line and of arcs on a circle. We show that there are, respectively, $1\times 3\times5\times\hbox{\mathversion{normal}$\cdots$}\times(2n{-}1)$ and $(2n) !$/$n!$ such arrangements; that the probability for the $i$-th endpoint of a random arrangement to be an initial endpoint is $(2n{-}i)$/$(2n{-}1)$ or $\half$, respectively; and that the average number of segments or arcs the $i$-th endpoint is contained in are $(i{-}1)(2n{-}i)$/$(2n{-}1)$ or $(n{-}1)$/$2$, respectively. The constructions used to prove these results provide sampling schemes for generating random inputs that can be used to test programs manipulating arrangements.

We also point out how arrangements are classically related to Catalan numbers and the ballot problem.


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Frédéric Cazals. "Combinatorial properties of one-dimensional arrangements." Experiment. Math. 6 (1) 87 - 94, 1997.


Published: 1997
First available in Project Euclid: 13 March 2003

zbMATH: 0881.05006
MathSciNet: MR1464584

Primary: 52A37
Secondary: 05A05 , 05C75 , 68R05

Keywords: algorithms , combinatorics , computational geometry , data structures

Rights: Copyright © 1997 A K Peters, Ltd.


Vol.6 • No. 1 • 1997
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