為了確立伺服器和客戶端之間的角色，因此我們提出集中式的最小 k-支配集 合的自我穩定演算法。進一步為了讓每個節點有自己不同的支配需求，而延伸出 最小多支配集合的問題，而提出使用集中式模型的最小多支配集合自我穩定演算 法。如此一來只要由集合中的支配節點負責扮演伺服器的角色，不僅可以節省成 本，並且可增加整體網路的可靠度。
在實驗部份，我們模擬出傳統找最小支配集合的演算法及我們所提出的自我 穩定演算法、先前賽局的方法實作[17]，並且實作出了四種拓樸邏輯(UDG、ER model、BA model、WS model)，當作執行時的拓樸邏輯，如此一來即可在不同 方法之間做效能比較。實驗結果顯示，無論在何種拓樸邏輯中，k 為 1~5 時，我 們的方法所需的平均支配者個數都是最少的，代表我們的方法效能最好，但如果 k 再繼續增加，我們的方法的效果顯然較不明顯，而且要到達穩定的時間會更長，
是以時間來換取效能的方法。而其它方法在不同拓樸邏輯下，效果各有優缺點，
因此每個演算法其實會有較適合自己運作的拓樸邏輯。
未來可以把最小支配集合問題延伸為探討最大獨立集合的問題，來和[4]，
[7]，[8]，[25] 做比較，差別在於希望集合中的節點越多越好。再來若能在網路 中找一條路徑把支配集合中的節點相連，形成網路的骨幹，即為連通的支配集合 (connected dominating set)[26]，[27]，[28]，如此一來，若想把資料傳給網路中的 節點，我們只要傳給骨幹中的任一節點，它再廣播出去，其它接收到資料的支配 節點再廣播，就能確保網路中所有節點都能收到資料。最後若給予每個節點一個 權重值，之後找出支配集合，使得支配集合內的節點權重值總合越大或越小越 好，依不同需求來設計。由於還沒有人提出針對找最小多支配集合的方法，往後 若有人提出即可參考我們的方法做比較。
65
參考文獻
[1] F. Harary and T. W. Haynes, “Double domination in graphs,” Ars Combinatoria, vol. 55, pp.201–213, Apr. 2000.
[2] L. Jia, R. Rajaraman, and T. Suel, “An efficient distributed algorithm for constructing small dominating sets,” Distributed Computing, vol. 15, no. 4, pp.
193–205, 2002.
[3] E. W. Dijkstra, “Self-stabilizing systems in spite of distributed control,”
Comm.ACM, vol. 17, no. 11, pp. 643–644, Nov. 1974.
[4] S. M. Hedetniemi, S. Hedetniemi, D. P. Jacobs, and P. K. Srimani,
“Self-stabilizing algorithms for minimal dominating sets and maximal independent sets,” Computers & Mathematics with Applications, vol. 46, no. 5-6, pp. 805–811, Sep. 2003.
[5] Z. Xu, S. T. Hedetniemi, W. Goddard, and P. K. Srimani, “A synchronous self-stabilizing minimal domination protocol in an arbitrary network graph,” In Proceedings of IWDC, LNCS 2918, pp. 26–32, 2003.
[6] H. Kakugawa and T. Masuzawa, “A self-stabilizing minimal dominating set algorithm with safe convergence,” in Int’l Parallel and Distributed Processing Symposium, Apr. 2006.
[7] V. Turau, “Linear self-stabilizing algorithms for the independent and dominating set problems using an unfair distributed scheduler,” Inform. Process.
Lett., vol. 103, no. 3, pp. 88–93, 2007.
[8] W. Goddard, S. T. Hededtniemi, D. P. Jacobs, P. K. Srimani, and Z. Xu,
“Self-stabilizing graph protocols,” Inform. Process. Lett., vol. 18, no. 1, pp.
189–199, 2008.
66
[9] S. Kamei , H. Kakugawa, “A self-stabilizing approximation algorithm for the distributed minimum k-domination,” IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences, no. 5, pp. 1109–1116, 2005.
[10] T. C. Huang, J. C. Lin, C. Y. Chen, and C. P. Wang, “A self-stabilizing algorithm for finding a minimal 2-dominating set assuming the distributed demon model,” Computers and Mathematics with Applications, Vol. 54, pp.
350-356, 2007.
[11] T. C. Huang, C. Y. Chen, and C. P. Wang, “A linear-time self-stabilizing algorithm for the minimal 2-dominating set in general networks,” Journal of Information Science and Engineering, Vol. 24, pp. 175-187, 2008.
[12] N. Guellati and H. Kheddouci, “A survey on self-stabilizing algorithms for independence, domination, coloring, and matching in graphs,” J.Parallel Distrib.
Comput., vol. 70, pp. 406–415, 2010.
[13] E. W. Dijkstra, “Guarded commands, nondeterminacy, and formal derivation of programs,” Comm. ACM, vol. 18, no. 8, pp. 453–457, Aug.1975.
[14] V. Srivastava, J. Neel, A. B. MacKenzie, R. Menon, L. A. DaSilva,J. E. Hicks, J.
H. Reed, and R.P. Gilles, “Using game theory to analyze wireless ad hoc networks,” IEEE Communications Surveys & Tutorials, vol. 7, no. 4, pp. 46–56, 2005.
[15] M. Felegyhazi, J. P. Hubaux and L. Buttyan, “Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks,” IEEE Transactions on Mobile Computing, volume 5, number 5, May 2006.
[16] K. Zhu, D. Niyato and P. Wang, “Network selection in heterogeneous wireless networks: evolution with incomplete information,” in Proc. IEEE Wireless
67
Commun. and Netw. Conf. (WCNC), pp. 1–6, 2002.
[17] Li-Hsing Yen and Zong-Long Chen, “Game-theoretic approach to
self-stabilizing distributed formation of minimal multi-dominating sets,”
submitted for journal publication, 2013.
[18] E. J. Cockayne, R. M. Dawes, and S. T. Hedetniemi, “Total domination in graphs,” Networks, vol.10, no. 3, pp. 211–219, 1980.
[19] W. Goddard, S. T. Hedetniemi, D. P. Jacobs, and P. K. Srimani, “A self-stabilizing distributed algorithm for minimal total domination in an arbitrary system graph,” in Proc. 17th Int’l Parallel and Distributed Processing Symp., Apr. 2003.
[20] B. Liang and Z. Haas, “Virtual backbone generation and maintenance in ad hoc network mobility management,” in Proceedings of the 2000 IEEE INFOCOM, March 2000.
[21] B.N. Clark, C.J. Colbourn, and D.S. Johnson, “Unit Disk Graphs,” Discrete Math., vol. 86, pp. 165–177, 1990.
[22] P.-J. Wan, K. Alzoubi, and O. Frieder, “Distributed construction of connected dominating set in wireless ad hoc networks,” Proc. IEEE INFOCOM, vol. 3, pp.
1597–1604, June 2002.
[23] Watts, D. J. and Strogatz, S. H., “Collective dynamics of ‘small-world’
networks”, Nature, vol. 393, 440–442, 1998.
[24] Barabási, A. L. and Albert, R. “Emergence of scaling in random networks,”
Science, vol. 286, no. 5439, pp. 509–512, October 1999.
[25] T. Moscibroda and R. Wattenhofer, “Efficient computation of maximal independent sets in unstructured multi-hop radio networks,” Proc. of the IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS), 2004.
68
[26] X. Cheng, M. Ding, and D. Chen, “An approximation algorithm for connected dominating set in ad hoc networks,” Proc. of International Workshop on Theoretical Aspects of Wireless Ad Hoc, Sensor, and Peer-to-Peer Networks (TAWN), 2004.
[27] Y. Li, M. T. Thai, F. Wang, C.-W. Yi, P. Wan, and D.-Z. Du, “On greedy construction of connected dominating sets in wireless networks,” J. Wireless Commun. and Mobile Computing, Special Issue on Ad Hoc Networks, vol. 5, no.
88, pp. 927–932.
[28] P.-J. Wan, K. M. Alzoubi, O. Frieder, “Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks”, to appear in ACM Mobile Networks and Applications, 2002.