Paper by Erik D. Demaine

Reference:

Timothy G. Abbott, Michael A. Burr, Timothy M. Chan, Erik D. Demaine, Martin L. Demaine, John Hugg, Daniel Kane, Stefan Langerman, Jelani Nelson, Eynat Rafalin, Kathryn Seyboth, and Vincent Yeung, “Dynamic Ham-Sandwich Cuts in the Plane”, Computational Geometry: Theory and Applications, volume 42, number 5, July 2009, pages 419–428. Special issue of selected papers from the 17th Canadian Conference on Computational Geometry, 2005.

Abstract:

We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham-sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n^1/3+ε) amortized time and O(n^4/3+ε) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(log n) time and ham-sandwich queries in O(k log⁴ n) time. In addition, if each point set has convex peeling depth k, then we can maintain the decomposition automatically using O(k log n) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(k log⁴ n) time plus the O((k b) polylog (k b)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.

Comments:

This paper is also available from ScienceDirect.

Length:

The paper is 15 pages.

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Related papers:

DynamicHamSandwich_CCCG2005 (Dynamic Ham-Sandwich Cuts of Convex Polygons in the Plane)