This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Adaptive Binary Sorting Schemes and Associated Interconnection Networks
June 1994 (vol. 5 no. 6)
pp. 561-572

Many routing problems in parallel processing, such as concentration and permutationproblems, can be cast as sorting problems. In this paper, we consider the problem ofsorting on a new model, called an adaptive sorting network. We show that any sequenceof n bits can be sorted on this model in O(lg/sup 2/ n) bit-level delay using O(n) constantfanin gates. This improves the cost complexity of K.E. Batcher's binary sorters (1968) bya factor of O(lg/sup 2/ n) while matching their sorting time. The only other network thatcan sort binary sequences in O(n) cost is the network version of columnsort algorithm,but this requires excessive pipelining. In addition, using binary sorters, we constructpermutation networks with O(n lg n) bit-level cost and O(lg/sup 3/ n) bit-level delay.These results provide the asymptotically least-cost practical concentrators andpermutation networks to date. We note, of course, that the well-known AKS sortingnetwork has O(lg n) sorting time and O(n lg n) cost, but the constants hidden in thesecomplexities are so large that our complexities outperform those of the AKS sortingnetwork until n becomes extremely large.

[1] M. Ajtai, J. Komlos, and E. Szemeredi, "AnO(nlogn) sorting network,"Combinatorica, vol. 3, pp. 1-19, 1983.
[2] N. Alon, "Eigenvalues and expanders,"Combinatorica, vol. 6, pp. 83-96, 1986.
[3] K. E. Batcher, "Sorting networks and their applications," inProc. AFIPS Spring Joint Conference, 1968, pp. 307-314.
[4] V. Benes,Mathematical Theory of Connecting Networks and Telephone Traffic. New York: Academic, 1965.
[5] T.H. Cormen, C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms, McGraw-Hill, Cambridge, Mass., 1990.
[6] T. H. Cormen, "Concentrator switches for routing messages in parallel computers," M.S. thesis, Massachuselts Inst. of Technol., 1986.
[7] B. Douglass and A. Yavuz Oruç, "Self-routing and route balancing in connection networks," Tech. Rep. UMIACS-TR-90-32, CS-TR-2421, Inst. for Advanced Comput. Studies, Univ. of Maryland, College Park, MD, 1990.
[8] M. Dowd, Y. Perl, L. Rudolph, and M. Saks, "The balanced sorting network," inProc. ACM Princ. Distrib. Comput., Aug. 1983, pp. 161-172.
[9] M. Dowd, Y. Perl, L. Rudolph, and M. Saks, "The periodic balanced sorting network,"J. ACM, vol. 36, pp. 738-757, Oct. 1989.
[10] O. Gabber and Z. Galil, "Explicit constructions of linear sized superconcentrators,"J. Comput. Syst. Sci., vol. 22, pp. 407-420, 1981.
[11] C. Y. Jan and A. Yavuz Oruç, "Fast self-routing permutation switching on an asympotically minimum cost network," Tech. Rep. UMIACS-TR-91-127, CS-TR-2753, Inst. for Advanced Comput. Studies, Univ. of Maryland, College Park, MD, 1991.
[12] D. E. Knuth,The Art of Computer Programming, Vol. 3, Reading, MA: Addison-Wesley, 1973.
[13] D. Koppelman and A. Yavuz Oruç, "A self-routing permutation network,"J. Parallel Distributed Comput., no. 10, pp. 140-151, 1990.
[14] T. Leighton, "Tight bounds on the complexity of parallel sorting,"IEEE Trans. Comput., vol. C-34, no. 4, pp. 344-354, Apr. 1985.
[15] M. Lu and A. Yavuz Oruç, "Efficient networks for realizing point-to-point assignments in parallel processors," Tech. Rep. UMIACS-TR 92-63,CS-TR-2910, Inst. for Advanced Comput. Studies, Univ. of Maryland, College Park, MD, 1992.
[16] G. A. Margulis, "Explicit constructions of concentrators,"Problems Inform. Transmission, vol. 9, no. 4, pp. 325-332, 1973.
[17] D. E. Muller and F. P. Preparata, "Bounds to complexities of networks for sorting and switching,"J. ACM, vol. 22, no. 2, Apr. 1975.
[18] D. Nassimi and S. Sahni, "Parallel algorithms to set up the Bens network,"IEEE Trans. Comput., vol. C-31, pp. 148-154, Feb. 1982.
[19] I. Parberry, "Current progress on efficient sorting networks," Tech. Rep. CS-89-30, Dept. of Comput. Sci., Pennsylvania State Univ., University Park, PA, 1989.
[20] M. S. Paterson, "Improved sorting networks withO(lgn) depth,"Algorithmica, vol. 5, pp. 75-92, 1990.
[21] M. S. Pinsker, "On the complexity of a concentrator," inProc. 7th Int. Teletraffic Congress, 1973, pp. 318/1-318/4.
[22] N. Pippenger, "Superconcentrators,"SIAM J. Computing, vol. 6, pp. 298-304, 1977.
[23] D. Richards, "Parallel sorting: A bibliography,"ACM SIGACT News, vol. 18, pp. 28-48, Summer 1986.
[24] L. Rudolph, "A robust sorting network,"IEEE Trans. Comput., vol. C-34, no. 4, pp. 326-335, Apr. 1985.
[25] D. C. Van Voorhis, "An economical construction for sorting networks," inProc. AFIPS Nat. Comput. Conf., vol. 43, pp. 921-926, 1974.
[26] I. Wegener,The Complexity of Boolean Functions. New York: Wiley, 1987.

Index Terms:
Index Termssorting; parallel algorithms; communication complexity; binary sequences; adaptive binary sorting schemes; interconnection networks; routing problems; parallel processing; concentration; permutation problems; sorting problems; cost complexity; permutation networks; AKS sorting network
Citation:
M.V. Chien, A. Yavuz Oruc, "Adaptive Binary Sorting Schemes and Associated Interconnection Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 5, no. 6, pp. 561-572, June 1994, doi:10.1109/71.285603
Usage of this product signifies your acceptance of the Terms of Use.