A simple linear algorithm for computing rectilinear 3-centers Michael Hoffmann Computational Geometry 31(2005) 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: algorithms and 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 ( l 1 or l ∞ ) metric.
Problem statement k: positive integer. : congruent closed axis parallel squares of side length 2. k=3 k-radius:
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 I l is the smallest value of P. Right endpoint of I r is the largest value of P.
One-dimensional 3-centers The one-dimensional 3-center decision problem can be solved in linear time. Given
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 the unit square Points from three clusters
Experimental results Points from the unit square Points from three clusters
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).
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