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http://dx.doi.org/10.7236/JIWIT.2012.12.3.51

A Prim Minimum Spanning Tree Algorithm for Directed Graph  

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)
Publication Information
The Journal of the Institute of Internet, Broadcasting and Communication / v.12, no.3, 2012 , pp. 51-61 More about this Journal
Abstract
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.
Keywords
Minimum Spanning Tree; MST; Directed Graph Minimum Spanning Tree; DMST; Prim MST Algorithm;
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Times Cited By KSCI : 2  (Citation Analysis)
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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.   DOI
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.   DOI
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.   DOI   ScienceOn