Dynamic ham-sandwich cuts in the plane
Abbott, Timothy G.
Burr, Michael A.
Chan, Timothy M.
Demaine, Erik D.
Demaine, Martin L.
Yeung, VincentNote: Order does not necessarily reflect citation order of authors.
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CitationAbbott, Timothy G., Michael A. Burr, Timothy M. Chan, Erik D. Demaine, Martin L. Demaine, John Hugg, Daniel Kane, et al. 2009. “Dynamic Ham-Sandwich Cuts in the Plane.” Computational Geometry 42 (5) (July): 419–428. doi:10.1016/j.comgeo.2008.09.008.
AbstractWe 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(n1/3+ε) amortized time and O(n4/3+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(logn) time and ham-sandwich queries in O(klog4n) time. In addition, if each point set has convex peeling depth k , then we can maintain the decomposition automatically using O(klogn) 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(klog4n) time plus the O((kb)polylog(kb)) 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.
Citable link to this pagehttp://nrs.harvard.edu/urn-3:HUL.InstRepos:13890811
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