COCOON 1997: Computing and Combinatorics pp 304-310

# Optimal algorithms for circle partitioning

• Kuo-Hui Tsai
• Da-Wei Wang
Session 9: Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1276)

## Abstract

Given a set of n points F on a circle and an integer k, we would like to find a size k subset of F such that these points are “evenly distributed” on the circle. We define two different criteria to capture the intuitive notion of evenness. Let U be a set of points on a circle and let u1, u2, u3,..., uk be the order of the points in U visited clockwise starting from u1. Denote by d; the distance between ui-1(mod k) and ui. Let min(U) = min(di) denote the minimum distance between every pair of adjacent points and similarly max(U) = max(di) denote the maximum distance between every pair of adjacent points. We feel that a set U is evenly distributed if min(U) is the largest among all the size k subset or max(U) is the smallest among all the size k subset. We call the former problem maxmin point location problem and the latter minmax point location problem.

For both problems we find O(n) time algorithms to find the optimal solutions, if the points on the circle are sorted. The basic idea of the algorithm is to open the circle at some point and treat the circle as a line. By applying the Frederickson's algorithm for tree partitioning, we first find an optimal solution for the corresponding line case (which is a special case of tree), then carefully relocate points in the optimal solution for the line case to get the optimal solution for the original problem. Both problems have applications for resource allocation on Synchronous Ethernet, a protocol designed for real-time applications over Ethernet.

## References

1. 1.
Wen-Liar Hsu and Kuo-Hui Tsai. Linear time algorithms on circular-arc graphs. IPL, 40:123–129, 1991.Google Scholar
2. 2.
Da-Wei Wang Kuo-Hui Tsai and Ming-Ta Ko. On point location problems on a circle with applications to synchronous etheret. In Proceedings of the Fourteenth IASTED International Conference, pages 168–177, 1996.Google Scholar
3. 3.
Kuo-Hui Tsai, C.H. Chang, J.M. Ho, D.W.Wang, M.W.Li, and K.F. Huang. Synchronous ethernet — a network supporting real time communications. In Proceeding of the First Workshop on Real-time and Media System, pages 29–33, Taipei, Taiwan, R.O.C., July 1995.Google Scholar
4. 4.
Kuo-Hui Tsai, C.H. Chang, and M..W.Li. On supporting real-time channels over ethernet. In Proceeding of the Second Workshop on Real-time and Media System, Taipei, Taiwan, R.O.C., July 1996.Google Scholar
5. 5.
Kuo-Hui Tsai and D. T. Lee. k best cuts for circular arc graphs. In Proceedings of the 5th International Symposium of Algorithms and Computation pages 550–558. Springer-Verlag, 1994.Google Scholar