4 research outputs found
The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data
We present a new multi-dimensional data structure, which we call the skip
quadtree (for point data in R^2) or the skip octree (for point data in R^d,
with constant d>2). Our data structure combines the best features of two
well-known data structures, in that it has the well-defined "box"-shaped
regions of region quadtrees and the logarithmic-height search and update
hierarchical structure of skip lists. Indeed, the bottom level of our structure
is exactly a region quadtree (or octree for higher dimensional data). We
describe efficient algorithms for inserting and deleting points in a skip
quadtree, as well as fast methods for performing point location and approximate
range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in
the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30
Fully Retroactive Approximate Range and Nearest Neighbor Searching
We describe fully retroactive dynamic data structures for approximate range
reporting and approximate nearest neighbor reporting. We show how to maintain,
for any positive constant , a set of points in indexed by time
such that we can perform insertions or deletions at any point in the timeline
in amortized time. We support, for any small constant ,
-approximate range reporting queries at any point in the timeline
in time, where is the output size. We also show how to
answer -approximate nearest neighbor queries for any point in the
past or present in time.Comment: 24 pages, 4 figures. To appear at the 22nd International Symposium on
Algorithms and Computation (ISAAC 2011
Algorithms for Triangles, Cones & Peaks
Three different geometric objects are at the center of this dissertation: triangles, cones and peaks.
In computational geometry, triangles are the most basic shape for planar subdivisions.
Particularly, Delaunay triangulations are a widely used for manifold applications in engineering, geographic information systems, telecommunication networks, etc.
We present two novel parallel algorithms to construct the Delaunay triangulation of a given point set.
Yao graphs are geometric spanners that connect each point of a given set to its nearest neighbor in each of cones drawn around it.
They are used to aid the construction of Euclidean minimum spanning trees
or in wireless networks for topology control and routing.
We present the first implementation of an optimal -time sweepline algorithm to construct Yao graphs.
One metric to quantify the importance of a mountain peak is its isolation.
Isolation measures the distance between a peak and the closest point of higher elevation.
Computing this metric from high-resolution digital elevation models (DEMs) requires efficient algorithms.
We present a novel sweep-plane algorithm that can calculate the isolation of all peaks on Earth in mere minutes