Journal of the Korea Institute of Information and Communication Engineering (한국정보통신학회논문지)

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
권호 표지
DOI QR코드

DOI QR Code

Minimum Cost Range Assignment for the Vertex Connectivity of Graphs

그래프의 정점 연결성에 대한 최소 범위 할당

  • Kim, Jae-Hoon (Department of Computer Engineering, Busan University of Foreign Studies)
  • Received : 2017.07.06
  • Accepted : 2017.08.01
  • Published : 2017.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

For n points $p_i$ on the m-dimensional plane $R^m$ and a fixed range r, consider a set $T_i$ containing points the distances from $p_i$ of which are less than or equal to r. In case m=1, $T_i$ is an interval on a line, it is a circle on a plane when m=2. For the vertices corresponding to the sets $T_i$, there is an edge between the vertices if the two sets intersect. Then this graph is called an intersection graph G. For m=1 G is called a proper interval graph and for m=2, it is called an unit disk graph. In this paper, we are concerned in the intersection graph G(r) when r changes. In particular, we consider the problem to find the minimum r such that G(r)is connected. For this problem, we propose an O(n) algorithm for the proper interval graph and an $O(n^2{\log}\;n)$ algorithm for the unit disk graph. For the dynamic environment in which the points on a line are added or deleted, we give an O(log n) algorithm for the problem.

m차원 평면 $R^m$ 상에 n개의 점들 $p_i$가 주어질 때, 범위 r에 대해서, 점 $p_i$로부터 거리 r이내 점들의 집합 $T_i$를 생각한다. m=1 일 때, $T_i$는 직선상의 구간이고, m=2일 때, $T_i$는 평면상의 원에 해당된다. 집합 $T_i$들을 정점에 대응하고, 두 집합이 교차하는 경우에 대응하는 두 정점 사이에 간선를 연결하면 교차 그래프 G를 얻을 수 있다. m=1일 때, G는 진구간 그래프(proper interval graph), m=2일 때, G는 단위 원판 그래프(unit disk graph)라고 부른다. 본 논문에서는 범위 r이 변화하면 바뀌는 교차 그래프 G(r)에 관심이 있다. 특별히 G(r)가 연결 그래프가 되는 최소 r을 찾는 문제를 다룰 것이다. 이 문제에 대해서 진구간 그래프 G(r)에 대해서 O(n)시간 알고리즘, 단위 원판 그래프 G(r)에 대해서 $O(n^2{\log}\;n)$시간 알고리즘을 제안한다. 직선상의 점들이 추가 되거나 삭제되는 동적 환경 하에서 위 문제를 O(lon n)시간에 해결하는 알고리즘도 제안한다.

Keywords

Acknowledgement

Supported by : Busan University of Foreign Studies

References

  1. M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, New York, NY: Academic Press, 1980.
  2. Terry. A. McKee and F. R. McMorris, Topics in Intersection Graph Theory, Philadelphia, PA: SIAM, 1999.
  3. M. Pal, "Intersection graphs: an introduction," Annals of Pure and Applied Mathematics, vol. 4, no. 1, pp. 43-91, Oct. 2013.
  4. S. Even and R. E. Tarjan, "Network flow and testing graph connectivity," SIAM Journal on Computing, vol. 4, no. 4, pp. 507-518, Oct. 1975. https://doi.org/10.1137/0204043
  5. P. K. Ghosh and M. Pal, "An Efficient algorithm to solve connectivity problem on trapezoid graphs," Journal of Applied Mathematics and Computing, vol. 24, no. 2, pp. 141-154, May 2007. https://doi.org/10.1007/BF02832306
  6. A. Ilic, "Efficient algorithm for the vertex connectivity of trapezoid graph," Information Processing Letters, vol. 113, no. 11, pp. 398-404, May 2013. https://doi.org/10.1016/j.ipl.2013.02.012
  7. J. H. Park, J. S. Choi and H. S. Lim, "Algorithms for finding disjoint path covers in unit interval graphs," Discrete Applied Mathematics, vol. 205, no. 1, pp. 132-149, May 2016. https://doi.org/10.1016/j.dam.2015.12.002
  8. S. Baswana, S. R. Chaudhury, K. Choudhary and S. Khan, "Dynamic DFS in undirected graphs: breaking the O(m) barrier," in Proceedings of the 27th annual ACM-SIAM symposium on Discrete algorithms, Virginia: VA, pp. 730-739, 2016.
  9. L. Roditty and U. Zwick, "A fully dynamic reachability algorithm for directed graphs with an almost linear update time," SIAM Journal on Computing, vol. 45, no. 3, pp. 712-733, Jun. 2016. https://doi.org/10.1137/13093618X
  10. B. M. Kapron, V. King, and B. Mountjoy, "Dynamic graph connectivity in polylogarithmic worst case time," in Proceedings of the 24th annual ACM-SIAM symposium on Discrete algorithms, Louisiana: LA, pp. 1131-1142, 2013.
  11. J. Holm, K. de Lichtenberg, and M. Thorup, "Polylogarithmic deterministic fully dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity," Journal of ACM, vol. 48, no. 4, pp. 723-760, Jul. 2001. https://doi.org/10.1145/502090.502095
  12. J. H. Kim, "Fully dynamic algorithm for the vertex connectivity of interval graphs," Journal of Korea Institute of Information and Communication Engineering, vol. 17, no. 2, pp. 399-406, Mar. 2013.
  13. C. Crespelle, "Fully dynamic representations of interval graphs," in Proceedings of the 35th International Workshop on Graph-Theoretic Concepts in Computer Science, France: FR, pp. 77-87, 2009.