J. E. Goodman, R. Pollack, and B. Sturmfels. Coordinate representation of order types requires exponential storage. In Proc. 21st Annu. ACM Sympos. Theory Comput., pages 405410, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the existential theory of the reals. Bienstock [8] proved that any given arrangement of p pseudolines can be forced to occur in every crossing minimal drawing of an appropriate graph, with O(p 3 ) edges and 5p(p 1) crossings by using some results of Goodman, Pollack, and Sturmfels [17] on arrangements whose straight line realizations require vertex coordinates with exponentially many bits. Therefore, assuming that the coordinates must be stored and that drawing a graph consumes time proportional to the size of the drawing, this yields that there is no polynomialtime algorithm ....

J. E. Goodman, R. Pollack and B. Sturmfels, Coordinate representation of order types requires exponential storage, Proc. 22nd STOC (1989) 405-410.

....dense point sets in IR d , which have spread O(n 1=d ) Valtr and others [33, 59, 60, 61, 62] have established several combinatorial results for dense point sets that improve corresponding bounds for arbitrary point sets. For other combinatorial and algorithmic results related to spread, see [15, 23, 37, 41, 42, 47]. In Section 2, we prove that the Delaunay triangulation of any set of n points in IR 3 with spread has complexity O( 3 ) In particular, the Delaunay triangulation of any dense point set in IR 3 has only linear complexity. This bound is tight in the worst case for all = O( p n) and ....

J. E. Goodman, R. Pollack, and B. Sturmfels. Coordinate representation of order types requires exponential storage. Proc. 21st Annu. ACM Sympos. Theory Comput., 405-410, 1989.

....consider only straight edge constructions: 1) join two points to generate a line and 2) intersect two lines to generate a point. 1 As previously stated, these are not the only way to generate new points; however, almost every geometric algorithm uses these two. Goodrich, Pollack, and Sturmfels [14] describe arrangements of points and lines whose realizations on the integer grid must have exponential bit complexity. Their technique uses a geometric implementation of repeated squaring. In fact any arithmetic operation can be implemented given a straight edge and a nite collection of ....

J. E. Goodman, R. Pollack, and B. Sturmfels. Coordinate representation of order types requires exponential storage. In Proc. 21st Annu. ACM Sympos. Theory Comput., pages 405-410, 1989.

....representable points can vary from 2 Gamma3P to 2 GammaP . There is clearly considerable unevenness in the distribution of representable points. 2.5 Relation to Other Work Before proceeding to the proof, let us compare this result to other work. Mnev [9] and Goodman, Pollack and Sturmfels [6] examine the problem of finding coordinate representations for order types. An order type is A B C Figure 2: Order Type vs. Combinatorial Structure an assignment of an orientation for every triple of points A, B, and C: the orientation is positive if triangle ABC is counter clockwise, negative ....

Jacob E. Goodman, Richard Pollack, Bernd Sturmfels. Coordinate Representation of Order Types Requires Exponential Storage. 21st Annual ACM Symposium on the Theory of Computing, ACM Press, May 1989.

....consider only straight edge constructions: 1) join two points to generate a line and 2) intersect two lines to generate a point. 1 As previously stated, these are not the only way to generate new points; however, almost every geometric algorithm uses these two. Goodrich, Pollack, and Sturmfels [14] describe arrangements of points and lines whose realizations on the integer grid must have exponential bitcomplexity. Their technique uses a geometric implementation of repeated squaring. In fact any arithmetic operation can be implemented given a straight edge and a finite collection of ....

J. E. Goodman, R. Pollack, and B. Sturmfels. Coordinate representation of order types requires exponential storage. In Proc. 21st Annu. ACM Sympos. Theory Comput., pages 405--410, 1989.

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J. E. Goodman, R. Pollack, and B. Sturmfels. Coordinate representation of order types requires exponential storage. In Proc. 21st Annu. ACM Sympos. Theory Comput., pages 405410, 1989.

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