8 research outputs found
A Static Optimality Transformation with Applications to Planar Point Location
Over the last decade, there have been several data structures that, given a
planar subdivision and a probability distribution over the plane, provide a way
for answering point location queries that is fine-tuned for the distribution.
All these methods suffer from the requirement that the query distribution must
be known in advance.
We present a new data structure for point location queries in planar
triangulations. Our structure is asymptotically as fast as the optimal
structures, but it requires no prior information about the queries. This is a
2D analogue of the jump from Knuth's optimum binary search trees (discovered in
1971) to the splay trees of Sleator and Tarjan in 1985. While the former need
to know the query distribution, the latter are statically optimal. This means
that we can adapt to the query sequence and achieve the same asymptotic
performance as an optimum static structure, without needing any additional
information.Comment: 13 pages, 1 figure, a preliminary version appeared at SoCG 201
Entropy, Triangulation, and Point Location in Planar Subdivisions
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 H + O(H^{2/3}+1) where H=H(G,D) is a lower bound on the
expected number of point-line comparisons performed by any linear decision tree
for point location in G under the query distribution D. The preprocessing
algorithm runs in O(n log 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.Comment: 19 pages, 4 figures, lots of formula
A static optimality transformation with applications to planar point location
Proceedings of the 27th {ACM} Symposium on Computational Geometry, Paris, France, June 13-15, 2011info:eu-repo/semantics/publishe
Nearly Optimal Expected-Case Planar Point Location
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 one-dimensional keys. In the one-dimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expected-case 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 en- tropy 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
Nearly Optimal Expected-Case Planar Point Location
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 one-dimensional keys. In the one-dimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expected-case 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