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A simple linear algorithm for computing rectilinear 3-centers. Michael Hoffmann Computational Geometry 31(2005)150-165. Reporter: Lincong Fang Nov 3,2005. Outline. About the author Problem statement Previous works One-dimensional 3-centers Planar rectilinear 3-centers
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A simple linear algorithm for computing rectilinear 3-centers Michael Hoffmann Computational Geometry 31(2005)150-165 Reporter: Lincong Fang Nov 3,2005
Outline • About the author • Problem statement • Previous works • One-dimensional 3-centers • Planar rectilinear 3-centers • Experimental results • Conclusions
About the author • Institute for Theoretical Computer Science, ETH Zürich, Switzerland. • Senior researcher. • Scientific interests: algorithmsand data structures, in particular from computational geometry, software design and combinatorial game theory.
Problem statement • K-center problem: given demand points, locate k facilities, such that for any point the nearest facility is as close as possible. • Rectilinear k-center problem: distance between points is measured according to the rectilinear (l1or l∞) metric.
Problem statement k-radius: k: positive integer. : congruent closed axis parallel squares of side length 2 . k=3
Previous works • Z.Drezner, On the rectangular p-center problem, gave a linear time algorithm for k=2. • M.Sharir, E.Welzl, Rectilinear and polygonal p-piercing and p-center problems, gave a linear time algorithm for k≤3. • M.Blum, R.W.Floyd, V.Pratt, R.L.Rivest, R.E.Tarjan, Time bounds for selection.
One-dimensional 3-centers Left endpoint of Il is the smallest value of P. Right endpoint of Iris the largest value of P.
One-dimensional 3-centers Given The one-dimensional 3-center decision problem can be solved in linear time.
One-dimensional 3-centers Knowing the smallest feasible radius, the optimal radius can be solved in linear time. :the smallest feasible radius from :the predecessor of
One-dimensional 3-centers Algorithm 1 Input: of n real numbers. Output: 3-covering for . • While • Compute and
One-dimensional 3-centers • Test feasibility of • If is feasible • If is infeasible • Solve the problem brute-force.
One-dimensional 3-centers • The one-dimensional 3-center problem can be solved in linear time. • Complexity:
Planar rectilinear 3-centers Type 1: Type 2:
Planar rectilinear 3-centers Type 2 can be computed in linear time: Then it can be compute just like one-dimension.
Planar rectilinear 3-centers Type 1
Planar rectilinear 3-centers Algorithm 2
Planar rectilinear 3-centers Type 1 can be computed in linear time.
Experimental results Points from three clusters Points from the unit square
Experimental results Points from three clusters Points from the unit square
Conclusions • A new linear time algorithm for the rectilinear 3-center problem. • Two heuristics to improve its performance in practice. • The implementation appeared as part of the CGAL since Release2.1(January 2000).