par Abbott, Timothy G.;Burr, Michael A. M.A.;Chan, Timothy M.;Demaine, Erik D. 
;Demaine, Martin L.;Hugg, John J.;Kane, Daniel D.;Langerman, Stefan ;Nelson, Jelani J.;Rafalin, Eynat E.;Seyboth, Kathryn K.;Yeung, Vincent V.
Référence Computational geometry, 42, 5, page (419-428)
Publication Publié, 2009-07
;Demaine, Martin L.;Hugg, John J.;Kane, Daniel D.;Langerman, Stefan ;Nelson, Jelani J.;Rafalin, Eynat E.;Seyboth, Kathryn K.;Yeung, Vincent V.Référence Computational geometry, 42, 5, page (419-428)
Publication Publié, 2009-07
Article révisé par les pairs
| Résumé : | 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(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 (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 ((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. |
