TY - JOUR

T1 - Entropy, triangulation, and point location in planar subdivisions

AU - Collette, Séxbastien

AU - Dujmović, Vida

AU - Iacono, John

AU - Langerman, Stefan

AU - Morin, Pat

PY - 2012/7

Y1 - 2012/7

N2 - A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is (Mathematical Equation Presented) + O( (Mathematical Equation Presented) 1/2 + 1) where (Mathematical Equation Presented)(G, D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in Gunder the query distribution D. The preprocessing algorithm runs in O(nlog n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.

AB - A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D, is (Mathematical Equation Presented) + O( (Mathematical Equation Presented) 1/2 + 1) where (Mathematical Equation Presented)(G, D) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in Gunder the query distribution D. The preprocessing algorithm runs in O(nlog n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.

KW - Computational geometry

KW - Minimum-entropy triangulation

KW - Point location

UR - http://www.scopus.com/inward/record.url?scp=84864863491&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864863491&partnerID=8YFLogxK

U2 - 10.1145/2229163.2229173

DO - 10.1145/2229163.2229173

M3 - Article

AN - SCOPUS:84864863491

VL - 8

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 3

M1 - 29

ER -