291 research outputs found
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Sublinear Explicit Incremental Planar Voronoi Diagrams
A data structure is presented that explicitly maintains the graph of a
Voronoi diagram of point sites in the plane or the dual graph of a convex
hull of points in three dimensions while allowing insertions of new
sites/points. Our structure supports insertions in
expected amortized time, where suppresses polylogarithmic terms.
This is the first result to achieve sublinear time insertions; previously it
was shown by Allen et al. that amortized combinatorial
changes per insertion could occur in the Voronoi diagram but a sublinear-time
algorithm was only presented for the special case of points in convex position.Comment: 14 pages, 10 figures. Presented ant JCDCGGG 201
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