Dynamic ham-sandwich cuts in the plane

  • Authors:

  • 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;Vincent Yeung

  • Affiliations:

  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA 02139, USA;Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA;School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA 02139, USA;Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA 02139, USA;Department of Computer Science, Tufts University, Medford, MA 02155, USA;Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02139, USA;Département d'informatique, Université Libre de Bruxelles, ULB CP212, 1050 Brussels, Belgium;Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA 02139, USA;Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA;Department of Computer Science, Tufts University, Medford, MA 02155, USA;Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA 02139, USA

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2009

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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^+^@e) amortized time and O(n^4^/^3^+^@e) 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(klog^4n) 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(klog^4n) 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.