Short rewriting, and geometric explanations related to the active bijection, for: Extension-lifting bijections for oriented matroids, by S. Backman, F. Santos, C.H. Yuen, arXiv:1904.03562v2 (October 29, 2023)

For an oriented matroid M, and given a generic single element extension and a generic single element lifting of M, the main result of [1] provides a bijection between bases of M and certain reorientations of M induced by the extension-lifting. This note is intended to somehow clarify and precise the geometric setting for this paper in terms of oriented matroid arrangements and oriented matroid programming, to describe and prove the main bijective result in a short simple way, and to show how it consists of combining two direct bijections and a central bijection, which is the same as a special case - practically uniform - of the bounded case of the active bijection [5, 6]. (The relation with the active bijection is addressed in [1] in an indirect and more complicated way.

The Complexity of Order Type Isomorphism

The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets $X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to $Y$ for which every $(d{+}1)$-tuple $(a_1,a_2,\ldots,a_{d+1})$ of $X$ and the corresponding tuple $(f(a_1),f(a_2),\ldots,f(a_{d+1}))$ in $Y$ have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an $O(n^d)$ algorithm for this task, thereby improving upon the $O(n^{\lfloor{3d/2}\rfloor})$ algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on Discrete Algorithms (SODA14