한국인터넷방송통신학회논문지 (The Journal of the Institute of Internet, Broadcasting and Communication)

  • 제12권3호
  • /
  • Pages.51-61
  • /
  • 2012
  • /
  • 2289-0238(pISSN)
  • /
  • 2289-0246(eISSN)
한국인터넷방송통신학회 (The Institute of Internet, Broadcasting and Communication)
권호 표지
DOI QR코드

DOI QR Code

방향 그래프의 Prim 최소신장트리 알고리즘

A Prim Minimum Spanning Tree Algorithm for Directed Graph

  • 최명복 (강릉원주대학교 멀티미디어공학과) ;
  • 이상운 (강릉원주대학교 멀티미디어공학과)
  • Choi, Myeong-Bok (Dept. of Multimedia Engineering, Gangnung-Wonju National University Wonju Campus) ;
  • Lee, Sang-Un (Dept. of Multimedia Engineering, Gangnung-Wonju National University Wonju Campus)
  • 투고 : 2012.04.10
  • 심사 : 2012.06.08
  • 발행 : 2012.06.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

본 논문에서는 무방향 그래프의 최소신장트리 (Minimum Spanning Tree, MST) 알고리즘인 Prim MST 알고리즘으로 방향 그래프의 최소신장트리 (DMST)를 구하는 알고리즘을 제안하였다. 먼저, 무방향 그래프와 방향 그래프의 차이점을 반영하여 각 노드에서 유출되는 호들 중 최소 가중치를 가진 호 (Minimum Weight Arc, MWA)를 선택하는 Prim DMST 알고리즘을 제안하였다. 다음으로 Prim DMST 알고리즘과 DMST의 대표적인 Chu-Liu/Edmonds DMST 알고리즘을 실제 3개 그래프에 적용하여 DMST를 찾지 못하는 단점을 보였다. 마지막으로 항상 DMST를 찾을 수 있는 알고리즘으로 Prim DMST를 변형시킨 진보된 Prim DMST 알고리즘을 제안하였다. Prim DMST 알고리즘은 각 노드의 유출 호들 중 MWA를 선택하는 방법이다. 반면에 진보된 Prim DMST 알고리즘은 각 노드의 유출 호들과 유입 호들 중 일치하는 호들을 선택하는 방법을 택하였으며, 만약에 일치하는 호가 없을 경우 각 노드의 유출 호들 중 MWA를 선택하는 방법이다. 제안된 알고리즘을 17개의 다양한 그래프에 적용한 결과, 항상 Chu-Liu/Edmonds DMST 알고리즘과 동일한 DMST를 찾는데 성공하였다. 또한, Chu-Liu/Edmonds DMST 알고리즘과 같이 사이클을 제거하기 위한 복잡한 계산을 하지 않아도 되며, Prim DMST 알고리즘 보다 수행속도를 크게 단축시킬 수 있었다.

This paper suggests an algorithm that obtains Directed Graph Minimum Spanning Tree (DMST), using Prim MST algorithm which is Minimum Spanning Tree (MST) of undirected graph. At first, I suggested the Prim DMST algorithm that chooses Minimum Weight Arc(MWA) from out-going nodes from each node, considering differences between undirected graph and directed graph. Next, I proved a disadvantage of Prim DMST algorithm and Chu-Liu/Edmonds DMST (typical representative DMST) of not being able to find DMST, applying them to 3 real graphs. Last, as an algorithm that can always find DMST, an advanced Prim DMST is suggested. The Prim DMST algorithm uses a method of choosing MWA among out-going arcs of each node. On the other hand, the advanced Prim DMST algorithm uses a method of choosing a coinciding arc from the out-going and in-going arcs of each node. And if there is no coinciding arc, it chooses MWA from the out-going arcs from each node. Applying the suggested algorithm to 17 different graphs, it succeeded in finding the same DMST as that found by Chu-Liu/Edmonds DMST algorithm. Also, it does not require such a complicated calculation as that of Chu-Liu/Edmonds DMST algorithm to delete the cycle, and it takes less time for process than Prim DMST algorithm.

키워드

참고문헌

  1. Wikipedia, http://en.wikipedia.org/wiki/Minimum_ spanning_tree, Wikimedia Foundation Inc., 2007.
  2. O. Borůvka, "O Jistem Problemu Minimalnim," Prace Mor. Prrodved. Spol. V Brne (Acta Societ. Natur. Moravicae), Vol. III, No. 3, pp. 37-58, 1926.
  3. J. Nesetril, E. Milkov'a, and H. Nesetrilov'a, "Otakar Boruvka on Minimum Spanning Tree Problem (Translation of the both 1926 Papers, Comments, History)," DMATH: Discrete Mathematics, Vol. 233, 2001.
  4. R. C. Prim, "Shortest Connection Networks and Some Generalisations," Bell System Technical Journal, Vol. 36, pp. 1389-1401, 1957. https://doi.org/10.1002/j.1538-7305.1957.tb01515.x
  5. J. B. Kruskal, "On the Shortest Spanning Subtree and The Traveling Salesman Problem," Proceedings of the American Mathematical Society, Vol. 7, pp. 48-50, 1956. https://doi.org/10.1090/S0002-9939-1956-0078686-7
  6. P. A. Jensen, "Operations Research Models and Methods," John Wiley and Sons, 2003.
  7. S. Boyd, "Applications of Combinational Optimization: Optimal Paths and Trees," http://www.site.uottawa.ca/-sylvia/csi5166 web /5166tespg26to61.pdf, School of Information Technology and Engineering (SITE), University of Ottawa, Canada, 2005.
  8. C. Duhamel, L. Gouveia, P. Moura, and M. C. Souza, "Models and Heuristics for a Minimum Arborescence Problem," Research Report LIMOS/RR-04-13, http://www.isima.fr/limos/publi/ RR-04-13.pdf, 2004.
  9. S. J. Yang, "The Directed Minimum Spanning Tree Problem," http://www.ce.rit.edu/-sjyeec/dmst.html, 2000.
  10. A. Schrijver, "Advanced Graph Theory and Combinatorial Optimization," Dept. of Mathematics, University of Amsterdam, Netherlands, http://citeseer.ist.psu.edu/schrijver01advanced.html, 2001.
  11. N. Bezroukov, "Minimum Spanning Trees," Softpanorama, http://www.softpanorama.org/ Algorithms/Digraphs/mst.shtml, 2006.
  12. Wikipedia, "Tree(Graph Theory)," http://en.wikipedia.org/wiki/Tree_graph_Theory/, 2007.
  13. R. Krishnam, B and Raghavachari, "The Directed Minimum-Degree Spanning Tree Problem," FSTTCS 2001, LNCS 2245, pp. 232-243, 2001.
  14. T. UNO, "An Algorithm for Enumerating all Directed Spanning Trees in a Directed Graph," International Symposium on Algorithm and Computation(ISAAC96), pp. 166-173, 1996.
  15. M. Llewellyn, "COP 3503: Computer Science II - Introduction to Graphs," http://www.cs.ucf.edu/courses/cop3503/summer04, 2004.
  16. K. Ikeda, "Mathematical Programming," Dept. Information Science and Intelligent Systems, The University of Tokushima, http:// www-b 2.is.tokushima-u.ac.jp/-ikeda/suuri/`maxflow /MaxflowApp. shtml. 2005.
  17. WWL. Chen, "Discrete Mathematics," Department of Mathematics, Division of ICS, Macquarie University, Australia, http://www.maths.mq.edu.au/-wchen/lndmfolder/ lndm.html, 2003.
  18. R. Wenger, "CIS 780: Analysis of Algorithms," http://www.cse.ohio-state.edu/-wenger/cis780/ shortest_path.pdf, 2004.
  19. J. R. N. Forbes, "CPS196.2 Robotics: Graphs and Matrices," http://www.cs.duke.edu/courses/ cps196.2/fall03/pdf/graphs_and_ matrices.pdf
  20. C. List, "MT303: Operations Research Graphs & Networks - Handout 3," Department of Mathematics, National University of Ireland, http://www,maths.may.ie/clist/teching/netw_hando ut_3 .pdf, 2003.
  21. H. K. Park, C. H. Lee, "Embedded System Implementation of Tree Routing Structure for Ubiquitous Sensor Network," Journal of the Korea Academia-Industrial cooperation Society, v.11 no.10, pp.4531-4535, October 2011.
  22. W. J. Wang, C. S. Han, "Bond Graph Modeling, Analysis and Control of Dual Stage System," Journal of the Korea Academia-Industrial cooperation Society, v.13, no.4, pp.1453-1459, April 2012. https://doi.org/10.5762/KAIS.2012.13.4.1453