Algorithmica

, Volume 8, Issue 1–6, pp 177–194

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

• M. S. Chang
• C. Y. Tang
• R. C. T. Lee
Article

Abstract

Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E r k ) whereE r k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE r 17 . We also prove that ¦E r k ¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n 2) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)0.5 m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n 1.5 log0.5 n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n 2 +n 1.5 log0.5 n).

Key words

Matching Computational geometry Bottleneck optimization problem Relative neighborhood graph

References

1. [1]
Burkard, R. E., Derigs, U.,Assignment and Matching Problems: Solution Methods with FORTRAN Programs, Lecture Notes in Economics and Mathematical Systems, Vol. 184, 1980.Google Scholar
2. [2]
Gabow, H. N., A Scaling Algorithm for Weighted Matching on General Graphs,Proc. 26th Annual IEEE Symp. on Foundations of Computer Science (1985), pp. 90–100.Google Scholar
3. [3]
Gabow, H. N., and Tarjan, R. E., Algorithms for Two Bottleneck Optimization Problems,J. Algorithms,9 (1988), 411–417.
4. [4]
Katajainen, J., and Nevalainen, O., Computing Relative Neighborhood Graphs in the Plane,Pattern Recognition,19 (1986), 221–228.
5. [5]
Lawler, E. L.,Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York 1976.
6. [6]
Toussaint, G. T., The Relative Neighborhood Graph of a Finite Planar Set,Pattern Recognition,12 (1980), 261–268.

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

• M. S. Chang
• 1
• C. Y. Tang
• 2
• R. C. T. Lee
• 3
• 4
1. 1.Institute of Computer Science and Information EngineeringNational Chung Cheng UniversityChiayi, TaiwanRepublic of China
2. 2.Institute of Computer ScienceNational Tsing Hua UniversityHsinchu, TaiwanRepublic of China
3. 3.National Tsing Hua UniversityHsinchuTaiwan
4. 4.Academia SinicaTaipeiTaiwan, Republic of China