在這篇論文裡，我們主要是研究部分排序(也稱為一般化排序)的問題，以及初始命名的問題。部分排序的問題，是將前k小的排序，並按照最大到最小或最小到最大排序。初始命名的問題，是n個multiprocessor 系統上，對n個multiprocessor分發一個唯一的識別數，這個問題可以被認為是一個全部排序問題的特例。我們提出一些演算法，解決這些問題。主要結果是給與這些演算法準確的分析。
在傳統模型上，我們修改插入排序和快速排序，解決部分排序的問題。我們的分析，得出在這種兩種部分排序演算法之間的整個比賽過程，並且顯示部分插入排序算法獲得領先從 $k = 1$ (開始)直到$kleqfrac{3}{5}sqrt{n}$，在那之後，部分快速排序演算法將開始領先到底。
我們也延長部分排序，在單一跳躍無線網路衝突察覺(WNCD)模型應用上的問題。 此種擴展符合無線網路的趨勢，並且是『分割並征服』的研究基礎。由重複利用發現最大值演算法，我們提出一種部分排序演算法，並且證明它的平均數時間複雜性是Theta (k+log(n-k))。
對於初始命名問題來說，在WNCD 模型上，我們能直接利用排序演算法為解決它。不過，這些排序演算法，比不上建造一棵劃分樹的方法。 我們的研究，顯示劃分樹法要求2.88 n 平均時間間隙。在重建和分析這種方法之後，我們改進其結果，從2.88 n降到2.46 n。

In this dissertation, we focus on the study of the partial sorting (generalized sorting) problem and the initialization problem. The partial sorting problem is
to find the first k smallest (or largest) elements among n input elements and to report them in nondecreasing (or nonincreasing). The initialization problem on a multiprocessor system is to assign each of n input elements a unique identification number, from 1 to n. This problem can be regarded as a special case of the sorting problem in which all input elements have the same value. We propose
some algorithms for solving these problems. The main result is to give precise analysis for these algorithms.
On the traditional model, we modify two algorithms, based on insertion sort and quicksort, to solve the partial sorting problem. Our analysis figures out the whole race between the two partial sorting algorithms and shows that the partial insertion sort algorithm obtains the leading position from k = 1 (the beginning) until k 3
5pn. After that, the partial quicksort algorithm will take the leading position on the way to the end.
We also extend the partial sorting problem on the Single-Hop wireless network with collision detection (WNCD) model. The extension fits in with the wireless trend and may be a foundation for studying divide-and-conquer. With the repeat
maximum finding scheme, we propose a partial sorting algorithm and prove that its average time complexity is (k + log (n − k)). For the initialization problem on the WNCD model, we can invoke the sorting algorithms directly for solving it. However, those sorting algorithms would not be better than the method of building a partition tree. We show that the partition tree method requires 2.88n time slots in average. After reconstructing and analyzing the method, we improve the result from 2.88n to 2.46n.

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 Preliminaries and Previous works . . . . . . . . . . . . . . . . 4
2.1 Partial Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Single-Hop Wireless Networks( WNCD) . . . . . . . . . . . . . . 7
2.3 Notations and the Previous Result for Maximum Finding . . . . 12
2.4 Nakano and Olariu’s Algorithm with the Layer Concept . . . . . 14
Chapter 3 The Partial Sorting Problem on the Traditional Model . . . . . 20
3.1 Partial Insertion Sort Algorithm(PISA) . . . . . . . . . . . . . . 20
3.2 Analysis for PISA . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 The Partial Quicksort Algorithm . . . . . . . . . . . . . . . . . . 32
3.4 Analysis of the Partial Quicksort Algorithm . . . . . . . . . . . . 32
3.5 Simulations of Partial Sorting . . . . . . . . . . . . . . . . . . . . 41
3.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . 43
Chapter 4 The Partial Sorting Problem on Single-Hop Wireless Networks . 46
4.1 Sorting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Analysis of the Sorting Algorithm . . . . . . . . . . . . . . . . . . 49
4.3 The Partial Sorting Algorithm . . . . . . . . . . . . . . . . . . . . 54
4.4 Analysis of the Partial Sorting Algorithm . . . . . . . . . . . . . . 55
4.5 Simulations of the Partial Sorting Algorithm . . . . . . . . . . . 63
4.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 5 The Initialization Problem on Single-Hop Wireless Networks . . 67
5.1 Analysis of Nakano and Olariu’s Algorithm with the Layer Concept 68
5.2 The Classification of the Nodes in the Partition Tree . . . . . . . 74
5.3 CRBP Algorithm Based on Conflict Reduction . . . . . . . . . . . 77
5.4 Analysis of CRBP Algorithm . . . . . . . . . . . . . . . . . . . . 79
5.5 Simulations of CRBP Algorithm . . . . . . . . . . . . . . . . . . . 85
5.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . 86
Chapter 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

[1] S. H. Bae, S. J. Lee, W. Su, and M. Gerla, “Performance evaluation of the
on-demand multicast routing protocol in multihop wireless networks,” IEEE
Network, Vol. 14, pp. 70–77, 2000.
[2] N. Bambos and J. M. Rulnick, “Mobile power management for wireless com-
munication networks,” Wireless Networks, Vol. 3, pp. 3–14, 1997.
[3] K. Batcher, “Sorting networks and their applications,” Proc. of AFIPS
Spring Joint Comput. Conf., Vol. 32, pp. 307–314, 1968.
[4] G. Baudet and D. Stevenson, “Optimal sorting algorithms for parallel com-
puters,” IEEE Transactions on Computers, No. 1, pp. 84–87, 1978.
[5] R. S. Bhuvaneswaran, J. L. Bordim, J. Cui, N. Ishii, and K. Nakano, “An
energy-efficient initialization protocol for wireless sensor networks,” IEICE
Transactions on Fundamentals, No. 2, pp. 447–454, Feb. 2002.
[6] R. S. Bhuvaneswaran, J. L. Bordim, J. Cui, and K. Nakano, “Fundamental
protocols for wireless sensor networks,” IEICE Transactions on Fundamen-
tals, No. 11, pp. 2479–2488, Nov. 2002.
[7] D. Bitton, H. Boral, D. Dewitt, and K. Wilkinson, “Parallel algorithms
for the execution of relational database operations,” ACM Transactions on
Database Systems, Vol. 8, No. 3, pp. 324–353, 1983.
[8] D. Bitton, D. J. DeWitt, and D. K. Hsiao, “A taxonomy of parallel sorting,”
Computing Surveys, Vol. 16, No. 3, pp. 287–318, Sep. 1988.
[9] J. L. Bordim, J. Cui, T. Hayashi, K. Nakano, and S. Olariu, “Energy-efficient
initialization protocols for ad-hoc radio networks,” IEICE Transactions on
Fundamentals, No. 9, pp. 1796–1803, Sep. 2000.
[10] J. L. Bordim, J. Cui, N. Ishii, and K. Nakano, “Doubly-logarithmic energy-
efficient initializaiton protocol for single-hop radio network,” IEICE Trans-
actions on Fundamentals, No. 5, pp. 967–976, May 2002.
[11] J. I. Capetanakis, The multiple access broadcast channel: Protocol and ca-
pacity considerations. PhD thesis, Mass. Inst. Tech., Aug. 1977.
[12] J. I. Capetanakis, “Tree algorithms for packet broadcast channels,” IEEE
Transactions on Information Theory, Vol. 25, No. 5, pp. 505–515, May 1979.
[13] J. Carle and J. F. Myoupo, “Collision detection based-deterministic protocol
for dynamic initialization of radio networks,” Proc. ISCA 13th Int. Conf. on
Parallel and Distributed Computing Systems (PDCS-2000), Las Vegas, USA,
pp. 159–164, 2000.
[14] W. M. Chen and H. K. Hwang, “Analysis in distribution of two randomized
algorithms for finding the maximum in a broadcast communication model,”
Journal of Algorithms, Vol. 46, No. 2, pp. 140–177, 2003.
[15] H. H. Chern and H. K. Hwang, “Phase changes in random m-ary search trees
and generalized quicksort,” Random Structures and Algorithms, Vol. 19,
pp. 316–358, 2001.
[16] H. H. Chern and H. K. Hwang, “Transitional behaviors of the average cost
of quicksort with median-of-(2t+1),” Algorithmica, Vol. 29, pp. 44–69, 2001.
[17] C. F. Chiasserini and R. R. Rao, “A distributed power management policy
for wireless ad-hoc networks,” Proc. of IEEE Wireless Communications and
Networking Conference (WCNC 2000), Chicago, USA, Sep., pp. 23–28, 2000.
[18] C. F. Chiasserini and R. R. Rao, “Energy efficient battery management,”
IEEE JSAC Wireless Series, Vol. 19, No. 7, pp. 1235–1245, 2001.
[19] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to
Algorithms. USA: MIT Press/McGraw-Hill, 2 ed., 2001.
[20] R. Dechter and L. Kleinrock, “Broadcast communications and distributed
algorithms,” IEEE Transactions on Computers, Vol. 35, No. 3, pp. 210–219,
Mar. 1986.
[21] J. Dongarra and S. Sullivan, “The top 10 algorithms,” Computing in Science
& Engineering, 2000.
[22] T. Elbatte and A. Ephremides, “Joint scheduling and power control for wire-
less ad hoc networks,” IEEE Transactions on Wireless Communications,
Vol. 3, No. 1, pp. 74–85, 2004.
[23] S. Fujita and M. Yamashita, “A nonoblivious bus access scheme yields an
optimal partial sorting algorithm,” Journal of Parallel and Distributed Com-
puting, Vol. 34, No. 1, pp. 111–116, 1996.
[24] P. J. Grabner and H. Prodinger, “An asymptotic study of a recursion occur-
ring in the analysis of an algorithm on broadcast communication,” Informa-
tion Processing Letters, Vol. 65, pp. 89–93, 1998.
[25] P. J. Grabner and H. Prodinger, “Sorting algorithms for broadcast commu-
nications: Mathematical analysis,” Theoretical Computer Science, Vol. 289,
pp. 51–67, 2002.
[26] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematicss: A
Foundation for Computer Science. USA: Addison-Wesley Publishing Com-
pany, 2 ed., 1994.
[27] D. R. Helman, D. A. Bader, and J. JaJa, “A randomized parallel sorting
algorithm with an experimental study,” Journal of Parallel and Distributed
Computing, Vol. 52, No. 1, pp. 1–23, 1998.
[28] P. Hennequin, “Combinatorial analysis of quicksort algorithm,” RAIRO In-
formatique Theorique et Applications, Vol. 23, pp. 317–333, 1989.
[29] P. Hennequin, Analyse en moyenne d’algorithme, tri rapide et arbres de
recherche. PhD thesis, 1991.
[30] C. A. R. Hoare, “Find, (algorithm 65),” Communications of the ACM, Vol. 4,
pp. 321–332, 1961.
[31] C. A. R. Hoare, “Quicksort,” Computer Journal, Vol. 5, pp. 10–15, 1962.
[32] J. H. Huang and L. Kleinrock, “Distributed selectsort sorting algorithm on
broadcast communication,” Parallel Computing, Vol. 16, pp. 183–190, 1990.
[33] J. P. Hubaux, T. Gross, J. Y. Le, and M. Vetterti, “Performance evaluation
of the on-demand multicast routing protocol in multihop wireless networks,”
IEEE Communications Magazine, Vol. 39, pp. 118–124, 2001.
[34] H. K. Hwang, “Second phase changes in random m-ary search trees and
generalized quicksort: convergence rates,” Annals of Probability, Vol. 31,
No. 2, pp. 609–629, 2003.
[35] H. K. Hwang and R. Neininger, “Phase change of limit laws in the quick-
sort recurrences under varying toll functions,” SIAM Journal on Computing,
Vol. 31, No. 6, pp. 1687–1722, 2002.
[36] H. K. Hwang and T. H. Tsai, “Quickselect and dickman function,” Combi-
natorics, Probability and Computing, Vol. 11, No. 4, pp. 353–371, 2002.
[37] J. JaJa, “A perspective on quicksort,” Computing in Science & Engineering,
2000.
[38] D. E. Knuth, The Art of Computer Programming. Volume 3: Sorting and
Searching. Addison-Wesley Publishing Company, 1973.
[39] L. Kronsjo, Algorithms: Their Complexity and Efficiency. John Wiley &
Sons, 1987.
[40] S. Levitan, “Algorithms for broadcast protocol multiprocessor,” Proc. of 3rd
International Conference on Distributed Computing Systems, pp. 666–671,
1982.
[41] S. P. Levitan and C. C. Foster, “Finding an extremum in a network,” Proc.
of 1982 International Symposium on Computer Architechure, pp. 321–325,
1982.
[42] P. J. Lin, “A survey of recent research trends and experimental guidelines
in ad-hoc wireless networks,” Master’s thesis, Feng Chia University, 2002.
[43] J. M. Marberg and E. Gafni, “Sorting and selection in multi-channel broad-
cast networks,” Proc. 1985 Internat. conference on Parallel Processing,
pp. 846–850, 1985.
[44] C. U. Martel, “Maximum finding on a multi access broadcast network,”
Information Processing Letters, Vol. 52, pp. 7–13, 1994.
[45] C. U. Martel and M. Moh, “Optimal prioritized conflict resolution on a
multiple access channel,” IEEE Transactions on Computers, Vol. 40, No. 10,
pp. 1102–1108, Oct. 1991.
[46] C. U. Martel, W. M. Moh, and T. S. Moh, “Dynamic prioritized conflict
resolution on multiple access broadcast networks,” IEEE Transactions on
Computers, Vol. 45, No. 9, pp. 1074–1079, 1996.
[47] C. Martinez, “Partial quicksort,” Proc. of the 6th ACM-SIAM Workshop on
Algorithm Engineering and Experiments and the 1st ACM-SIAM Workshop
on Analytic Algorithmics and Combinatorics, pp. 224–228, 2004.
[48] P. Mathys and P. Flajolet, “q-ary collision resolution algorithms in random-
access systems with free or blocked channel access,” IEEE Transactions on
Information Theory, Vol. 31, No. 2, pp. 217–243, 1985.
[49] J. Menon, “A study of sort algorithm for multiprocessor database machines,”
Proc. of the Twelfth International Conference on Very Large Data Bases,
pp. 197–206, 1986.
[50] A. Micic and I. Stojmenovic, “A hybrid randomized initialization protocol
for tdma in single-hop wireless networks,” Proceedings of the International
Parallel and Distributed Processing Symposium (IPDPS.02), pp. 147–154,
2002.
[51] W. M. Moh, C. U. Martel, and T. S. Moh, “A dynamic solution to prioritized
conflict resolution on a multiple access broadcast channel,” Proc. of 1993
International Conference on Parallel and Distributed Systems, pp. 414–418,
1993.
[52] K. Nakano and S. Olariu, “Energy-efficient initialization protocols for single-
hop radio networks with no collision detection,” IEEE Transactions on Par-
allel and Distributed Systems, Vol. 11, No. 8, pp. 851–863, Aug. 2000.
[53] K. Nakano and S. Olariu, “Randomized initialization protocols for ad-hoc
networks,” IEEE Transactions on Parallel and Distributed Systems, Vol. 11,
No. 7, pp. 749–759, July 2000.
[54] K. Nakano and S. Olariu, “A survey on leader election protocols for radio
networks,” Proc. International Symposium on Parallel Architectures, Algo-
rithms, and Networks (I-SPAN), pp. 71–76, 2002.
[55] K. Nakano and S. Olariu, “Uniform leader election protocols in radio net-
works,” IEEE Transactions on Parallel and Distributed Systems, Vol. 13,
No. 5, pp. 516–526, May 2002.
[56] K. Nakano, S. Olariu, and J. L. Schwing, “Broadcast-efficient protocols for
mobile radio networks,” IEEE Transactions on Parallel and Distributed Sys-
tems, Vol. 10, No. 12, pp. 1276–1289, Dec. 1999.
[57] K. Nakano, S. Olariu, and A. Zomaya, “Energy-efficient routing in the broad-
cast communication model,” IEEE Transactions on Parallel and Distributed
Systems, Vol. 13, No. 2, pp. 1201–1210, Dec. 2002.
[58] K. Nakano, S. Olariu, and A. Y. Zomaya, “Energy-efficient permutation
routing protocols in radio networks,” IEEE Transactions on Parallel and
Distributed Systems, Vol. 12, No. 6, pp. 544–557, June 2001.
[59] T. Ozaki, J. B. Kim, and T. Suda, “Bandwidth-efficient multicast routing
for multihop, ad hoc wireless networks,” Proc. of the IEEE INFOCOM 2001,
Anchorage, AL, Apr., pp. 1182–1191, 2001.
[60] M. H. P. Yi and L. Keqin, “Efficient and scalable quicksort on a linear array
with a reconfigurable pipelined bus system,” Future Generation Computer
Systems, Vol. 13, No. 6, pp. 501–513, 1998.
[61] A. Panholzer and H. Prodinger, “A generating functions approach for the
analysis of grand averages for multiple quickselect,” Random Structures Al-
gorithms, Vol. 13, pp. 189–209, 1998.
[62] H. Prodinger, “How to select a loser,” Discrete Mathematics, Vol. 120,
pp. 149–159, 1993.
[63] H. Prodinger, “Multiple quickselect - hoare’s find algorithm for several ele-
ments,” Information Processing Letters, Vol. 56, No. 3, pp. 123–129, 1995.
[64] S. Rajasekaran, “An optimal parallel algorithm for sorting multisets,” In-
formation Processing Letters, Vol. 67, No. 3, pp. 141–143, 1998.
[65] R. Ramanathan and J. Redi, “A brief overview of ad-hoc networks: Chal-
lenges and directions,” IEEE Communications Magazine, Vol. 40, pp. 20–22,
2002.
[66] K. V. S. Ramarao, “Distributed sorting on local area network,” IEEE Trans-
actions on Computers, Vol. C-37, No. 2, pp. 239–243, Feb. 1988.
[67] E.M. Royer and C. K. Toh, “A brief overview of ad-hoc networks: Challenges
and directions,” IEEE Personal Communications Magazine, Vol. 6, No. 2,
pp. 46V–55, 1999.
[68] E.M. Royer and C. K. Toh, “A brief overview of ad-hoc networks: Challenges
and directions,” IEEE Personal Communications, Vol. 8, No. 1, pp. 16–28,
2001.
[69] A. K. Salkintzis and C. Chamzas, “An in-band power-saving protocol for
mobile data networks,” IEEE Transactions on Communications, Vol. 46,
pp. 1194–1205, 1998.
[70] R. Sanchez, J. Evans, and G. Minden, “Networking on the battlefield: chal-
lenges in highly dynamic multihop wireless networks,” Vol. 2, 1999.
[71] R. Sedgewick, “The analysis of quicksort programs,” Acta Informatica, 1976.
[72] R. Sedgewick, Quicksort. New York: Garland Publishing, 1978.
[73] R. Sedgewick, Algorithms in C. USA: Addison-Wesley Publishing Company,
3 ed., 1998.
[74] R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms.
USA: Addison-Wesley Publishing Company, 1996.
[75] N. Sharma and P. Tagle, “Performance of fault-tolerant sorting network for
atm switching,” Performance Evaluation, Vol. 34, No. 2, pp. 91–107, 1998.
[76] S. T. Sheu and T. F. Sheu, “A bandwidth allocation/sharing/ extension
protocol for multimedia over ieee 802.11 ad hoc wireless lans,” IEEE Journal
on Selected Areas in Communications, Vol. 19, No. 10, pp. 2065–2081, 2001.
[77] S. H. Shiau and C. B. Yang, “A fast maximum finding algorithm on broadcast
communication,” Information Processing Letters, Vol. 60, pp. 81–96, 1996.
[78] S. H. Shiau and C. B. Yang, “The layer concept and conflicts on broadcast
communication,” Journal of Chang Jung Christian University, Vol. 2, No. 1,
pp. 37–46, June 1998.
[79] S. H. Shiau and C. B. Yang, “A fast sorting algorithm on broadcast commu-
nications,” Proc. of Second International Conference on Parallel Computing
Systems(PCS99), Ensenada, Baja California, Mexico, pp. 87–92, 1999.
[80] S. H. Shiau and C. B. Yang, “The generalization of quicksort,” Proc. of
2000 Workshop on Internet and Distributed Systems(WIND’2000), Taiwan,
R.O.C., pp. 94–99, 2000.
[81] S. H. Shiau and C. B. Yang, “A fast initialization algorithm for single-
hop wireless networks,” IEICE Transactions on Communications, No. 11,
pp. 4285–4292, 2005.
[82] S. H. Shiau and C. B. Yang, “Generalization of sorting in single hop wireless
networks,” Accepted by IEICE Transactions on Information and Systems,
2006.
[83] C. Y. Tang and M. J. Chiu, “Distributed sorting on the serially connected
local area networks,” Proc. of 1989 Singapore International Conference on
Networks, pp. 458–462, 1989.
[84] S. S. Tseng, Parallel Sorting Algorithms. PhD thesis, Institute of Computer
Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC, 1984.
[85] B. S. Tsybakov and V. A. Mikhailov, “Free synchronous packet access in a
broadcast channel with feedback,” Problemy Peredachi Informatsii, Vol. 14,
No. 4, pp. 32–59, 1978.
[86] D. E. Willard, “Log-logarithmic protocols for resolving ethernet and
semaphore conflicts,” Proc. of 16th Annual ACM Symposium on Theory of
Computing, pp. 512–521, 1984.
[87] D. E. Willard, “Log-logarithmic selection resolution protocols in a multiple
access channel,” SIAM Journal on Computing, Vol. 15, pp. 468–477, 1986.
[88] R. Wong, R. Topor, and H. Shen, “A parallel sort-balance mutual range-
join algorithm on hypercube computers,” Microprocessors And Microsys-
tems, Vol. 22, No. 3, 1998.
[89] C. B. Yang, “Reducing conflict resolution time for solving graph prob-
lems in broadcast communications,” Information Processing Letters, Vol. 40,
pp. 295–302, 1991.
[90] C. B. Yang, “Computational geometry on the broadcast communication
model,” Journal of Information Science and Engineering, Vol. 15, pp. 383–
395, May 1999.
[91] C. B. Yang, R. C. T. Lee, and W. T. Chen, “Finding minimum spanning
trees based upon single-channel broadcast communications,” Proc. of Inter-
national Computer Symposium 1988, Taipei, Taiwan, pp. 1451–1456, 1988.
[92] C. B. Yang, R. C. T. Lee, and W. T. Chen, “Parallel graph algorithms
based upon broadcast communications,” IEEE Transactions on Computers,
Vol. 39, No. 12, pp. 1468–1472, Dec. 1990.
[93] C. B. Yang, R. C. T. Lee, and W. T. Chen, “Conflict-free sorting algorithm
broadcast under single-channel and multi-channel broadcast communication
models,” Proc. of International Conference on Computing and Information,
pp. 350–359, 1991.
[94] H. Yasuura, N. Takagi, and S. Yajima, “The parallel enumeration sort-
ing scheme for VLSI,” IEEE Transactions on Computers, Vol. 31, No. 12,
pp. 1192–1201, 1982.