A simple linear algorithm for computing rectilinear 3-centers Michael Hoffmann Computational Geometry 31(2005)150-165 Reporter: Lincong Fang Nov 3,2005.

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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).

Thank you!