TY - GEN
T1 - Convex hull of imprecise points in o(n log n) time after preprocessing
AU - Ezra, Esther
AU - Mulzer, Wolfgang
PY - 2011
Y1 - 2011
N2 - Motivated by the desire to cope with data imprecision [29], we study methods for preprocessing a set of line-segments (or just lines) in the plane such that whenever we are given a set of points, each of which lies on a distinct object, we can compute their convex hull more eficiently than in "standard settings" (that is, without preprocessing). In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P eficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(nα(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(nα(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-o- between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (≤k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Ω(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "stan-dard" convex hull and sorting problems, in which the two problems require Θ(n log n) steps in the worst case (under the algebraic computation tree model).
AB - Motivated by the desire to cope with data imprecision [29], we study methods for preprocessing a set of line-segments (or just lines) in the plane such that whenever we are given a set of points, each of which lies on a distinct object, we can compute their convex hull more eficiently than in "standard settings" (that is, without preprocessing). In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P eficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(nα(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(nα(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-o- between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (≤k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Ω(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "stan-dard" convex hull and sorting problems, in which the two problems require Θ(n log n) steps in the worst case (under the algebraic computation tree model).
KW - Convex hull
KW - Data imprecision
KW - Geometric data structures
KW - Planar arrangements
KW - Randomized constructions
UR - http://www.scopus.com/inward/record.url?scp=79960199478&partnerID=8YFLogxK
U2 - 10.1145/1998196.1998199
DO - 10.1145/1998196.1998199
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AN - SCOPUS:79960199478
SN - 9781450306829
T3 - Proceedings of the Annual Symposium on Computational Geometry
SP - 11
EP - 20
BT - Proceedings of the 27th Annual Symposium on Computational Geometry, SCG'11
T2 - 27th Annual ACM Symposium on Computational Geometry, SCG'11
Y2 - 13 June 2011 through 15 June 2011
ER -