The complexity of point configurations
نویسندگان
چکیده
Goodman, J.E. and R. Pollack, The complexity of point configurations, Discrete Applied Mathematics 31 (1991) 167-180. There are several natural ways to extend the notion of the order of points on a line to higher dimensions. This article focuses on three of them-combinatorial type, order type, and isotopy class-and surveys work done in recent years on the efficient encoding of order types and on complexity questions relating to all three classifications. 1. The combinatorial type of a configuration Let us consider first a configuration of points in the plane: S= {Pi, . . . ,P,}. If we project the points of S onto a directed line, this will induce an ordering on S. But of course we may get a different ordering if we project onto a different line. So let us allow the directed line to rotate continuously, say in a counterclockwise direction. This gives a periodic sequence of permutations, which we call the circular sequence associated to the configuration P,, . . . , P,, [24]. For the configuration shown in Fig. 1, for example, we get the sequence a(S): . . . 12345~ 23,4s 1325425 13524’3553124=53421 35432154’3245231 . . . * Supported in part by NSF grant DMS-85-01492, NSA grant MDA904-89-H-2038, PSC-CUNY grant 666426, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF grant STCSS-09648. ** Supported in part by NSF grants DMS-85-01947 and CCR-89-01484, NSA grant MDA904-89H-2030. and DIMACS. 0166-218X/91/$03.50
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عنوان ژورنال:
- Discrete Applied Mathematics
دوره 31 شماره
صفحات -
تاریخ انتشار 1991