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41st Annual Symposium on Foundations of Computer Science
Nearly optimal expectedcase planar point location
Redondo Beach, California
November 12November 14
ISBN: 0769508502
ASCII Text  x  
S. Arya, T. Malamatos, D.M. Mount, "Nearly optimal expectedcase planar point location," 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 208, 41st Annual Symposium on Foundations of Computer Science, 2000.  
BibTex  x  
@article{ 10.1109/SFCS.2000.892108, author = {S. Arya and T. Malamatos and D.M. Mount}, title = {Nearly optimal expectedcase planar point location}, journal ={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, volume = {0}, year = {2000}, issn = {02725428}, pages = {208}, doi = {http://doi.ieeecomputersociety.org/10.1109/SFCS.2000.892108}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2013 IEEE 54th Annual Symposium on Foundations of Computer Science TI  Nearly optimal expectedcase planar point location SN  02725428 SP EP A1  S. Arya, A1  T. Malamatos, A1  D.M. Mount, PY  2000 KW  probability; search problems; computational geometry; trees (mathematics); nearly optimal expectedcase planar point location; expected search time; planar polygonal subdivision; subdivision; search structure; optimal binary search tree; planar point location; data structure; polygonal subdivision; polygonal cells; convex cells VL  0 JA  2013 IEEE 54th Annual Symposium on Foundations of Computer Science ER   
We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for onedimensional keys. In the onedimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expectedcase search time, and further there exist search trees achieving expected search times of at most H+2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H+o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.
Index Terms:
probability; search problems; computational geometry; trees (mathematics); nearly optimal expectedcase planar point location; expected search time; planar polygonal subdivision; subdivision; search structure; optimal binary search tree; planar point location; data structure; polygonal subdivision; polygonal cells; convex cells
Citation:
S. Arya, T. Malamatos, D.M. Mount, "Nearly optimal expectedcase planar point location," focs, pp.208, 41st Annual Symposium on Foundations of Computer Science, 2000
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