374 research outputs found
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
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
Distance-Sensitive Planar Point Location
Let be a connected planar polygonal subdivision with edges
that we want to preprocess for point-location queries, and where we are given
the probability that the query point lies in a polygon of
. We show how to preprocess such that the query time
for a point~ depends on~ and, in addition, on the distance
from to the boundary of~---the further away from the boundary, the
faster the query. More precisely, we show that a point-location query can be
answered in time , where
is the shortest Euclidean distance of the query point~ to the
boundary of . Our structure uses space and
preprocessing time. It is based on a decomposition of the regions of
into convex quadrilaterals and triangles with the following
property: for any point , the quadrilateral or triangle
containing~ has area . For the special case where
is a subdivision of the unit square and
, we present a simpler solution that achieves a
query time of . The latter solution can be extended to
convex subdivisions in three dimensions
Adaptive Planar Point Location
We present a self-adjusting point location structure for convex subdivisions. Let n be the number of vertices in a convex subdivision S. Our structure for S uses O(n) space and processes any online query sequence sigma in O(n + OPT) time, where OPT is the minimum time required by any linear decision tree for answering point location queries in S to process sigma. The O(n + OPT) time bound includes the preprocessing time. Our result is a two-dimensional analog of the static optimality property of splay trees. For connected subdivisions, we achieve a processing time of O(|sigma| log log n + n + OPT)
Dynamic Distribution-Sensitive Point Location
We propose a dynamic data structure for the distribution-sensitive point
location problem. Suppose that there is a fixed query distribution in
, and we are given an oracle that can return in time the
probability of a query point falling into a polygonal region of constant
complexity. We can maintain a convex subdivision with vertices
such that each query is answered in expected time, where OPT
is the minimum expected time of the best linear decision tree for point
location in . The space and construction time are . An
update of as a mixed sequence of edge insertions and deletions
takes amortized time. As a corollary, the randomized incremental
construction of the Voronoi diagram of sites can be performed in expected time so that, during the incremental construction, a nearest
neighbor query at any time can be answered optimally with respect to the
intermediate Voronoi diagram at that time.Comment: To appear in Proceedings of the International Symposium of
Computational Geometry, 202
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