7,911 research outputs found
Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations are Equivalent
We show that Delaunay triangulations and compressed quadtrees are equivalent
structures. More precisely, we give two algorithms: the first computes a
compressed quadtree for a planar point set, given the Delaunay triangulation;
the second finds the Delaunay triangulation, given a compressed quadtree. Both
algorithms run in deterministic linear time on a pointer machine. Our work
builds on and extends previous results by Krznaric and Levcopolous and Buchin
and Mulzer. Our main tool for the second algorithm is the well-separated pair
decomposition(WSPD), a structure that has been used previously to find
Euclidean minimum spanning trees in higher dimensions (Eppstein). We show that
knowing the WSPD (and a quadtree) suffices to compute a planar Euclidean
minimum spanning tree (EMST) in linear time. With the EMST at hand, we can find
the Delaunay triangulation in linear time.
As a corollary, we obtain deterministic versions of many previous algorithms
related to Delaunay triangulations, such as splitting planar Delaunay
triangulations, preprocessing imprecise points for faster Delaunay computation,
and transdichotomous Delaunay triangulations.Comment: 37 pages, 13 figures, full version of a paper that appeared in SODA
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The impact of traffic localisation on the performance of NoCs for very large manycore systems
The scaling of semiconductor technologies is leading to processors with increasing numbers of cores. The adoption of Networks-on-Chip (NoC) in manycore systems requires a shift in focus from computation to communication, as communication is fast becoming the dominant factor in processor performance. In large manycore systems, performance is predicated on the locality of communication. In this work, we investigate the performance of three NoC topologies for systems with thousands of processor cores under two types of localised traffic. We present latency and throughput results comparing fat quadtree, concentrated mesh and mesh topologies under different degrees of localisation. Our results, based on the ITRS physical data for 2023, show that the type and degree of localisation of traffic significantly affects the NoC performance, and that scale-invariant topologies perform worse than flat topologies
Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently
In this paper we define the notion
of a probabilistic neighborhood in spatial data: Let a set of points in
, a query point , a distance metric \dist,
and a monotonically decreasing function be
given. Then a point belongs to the probabilistic neighborhood of with respect to with probability f(\dist(p,q)). We envision
applications in facility location, sensor networks, and other scenarios where a
connection between two entities becomes less likely with increasing distance. A
straightforward query algorithm would determine a probabilistic neighborhood in
time by probing each point in .
To answer the query in sublinear time for the planar case, we augment a
quadtree suitably and design a corresponding query algorithm. Our theoretical
analysis shows that -- for certain distributions of planar -- our algorithm
answers a query in time with high probability
(whp). This matches up to a logarithmic factor the cost induced by
quadtree-based algorithms for deterministic queries and is asymptotically
faster than the straightforward approach whenever .
As practical proofs of concept we use two applications, one in the Euclidean
and one in the hyperbolic plane. In particular, our results yield the first
generator for random hyperbolic graphs with arbitrary temperatures in
subquadratic time. Moreover, our experimental data show the usefulness of our
algorithm even if the point distribution is unknown or not uniform: The running
time savings over the pairwise probing approach constitute at least one order
of magnitude already for a modest number of points and queries.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-44543-4_3
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
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