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A theorem on the relation between BSRk and BSR+
Information Processing Letters 71 (1999) 71–73
A theorem on the relation between BSRk and BSR+ Limin Xiang ∗ , Kazuo Ushijima 1 Department of Computer Science and Communication Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Received 1 May 1999 Communicated by S.G. Akl
Abstract PRAM is the most popular model of parallel computation. Of its three variants most commonly used, CREW is more powerful than EREW, and CRCW is the most powerful. BSR is another PRAM model, which is more powerful than CRCW. BSRk and BSR+ are models extended from BSR, and in this paper a theorem is shown on the relation between BSRk and BSR+ . 1999 Elsevier Science B.V. All rights reserved. Keywords: Model of parallel computing; Broadcasting with selective reduction; Parallel algorithms
1. Introduction For two models A and B of parallel computation, the statement that model A is more powerful than model B, denoted by B ⊂ A, means that [19] with the same number of processors, • all problems can be solved on A in no more time than on B, and • some problems can be solved on A in less time than on B. Thus, for the three well known PRAM models EREW, CREW and CRCW, EREW ⊂ CREW ⊂ CRCW. In [3], another PRAM model BSR (Broadcasting with Selective Reduction) was presented, which is more powerful than CRCW. For the selection of BSR, one criterion is permitted only (BSR is denoted by BSR1 in this paper.) In [5,6], BSR was extended as BSRk (k > 2, is a constant integer) so that k criteria are permitted for the selection, and in [19], BSR was ∗ Corresponding author. Email: [email protected]. 1 Email: [email protected].
extended as BSR+ in which an arbitrary logic expression is permitted for the selection. It is known that for k>2 EREW ⊂ CREW ⊂ CRCW ⊂ BSR1 ⊂ BSRk ⊂ BSRk+1 ⊂ BSR+ [3,5,6,19]. In this paper, we will show a theorem on the relation between BSRk and BSR+ . 2. The theorem Definitions of BSR1 , BSRk and BSR+ in detail as well as parallel algorithms on them for many applications can be found in the literature [1–19]. Here, we give only the three kinds of Broadcast Instructions for BSR1 , BSRk and BSR+ , and omit the explanation for them. A Broadcast Instruction of BSR1 is denoted by uj := < di |ti σ lj ,
0020-0190/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 9 9 ) 0 0 0 8 6 - 1
72
L. Xiang, K. Ushijima / Information Processing Letters 71 (1999) 71–73
a Broadcast Instruction of BSRk is denoted by ^ (h) (h) (h) ti σ lj , uj := < di 16h6k
and a Broadcast Instruction of BSRk is denoted by uj := < di (1) (1) (2) (2) (c) (c) ℘ ti σ (1) lj , ti σ (2) lj , . . . , ti σ (c) lj . Lemma 1. Let D be a set of data and {D1 , D2 , . . . , Ds } be a partition on D, then, >, 6=} and σ (m) is the NOT of σ (m) . Thus, we have a partition {D1 , D2 , . . . , Ds } on D, where,
Dr = di |Gr (i, j ) for r = 1, 2, . . . , s, and D = di |℘ (y1, y2 , . . . , yk ) . By Lemma 1, the Broadcast Instruction of BSR+ above can be implemented by the following instructions on BSRk : for r = 1 to s do urj := e(<); /∗ e(<) is the neutral element of < ∗ /; (r) uj :=
L. Xiang, K. Ushijima / Information Processing Letters 71 (1999) 71–73
G1 ∧ G3 = FALSE,
and
G2 ∧ G3 = FALSE. By Theorem 2, the solution on BSR3 can be: Number1 := 0; Number2 := 0; Number3 := 0; X 1| x[i] = a ∧ y[i] = b ; Number1 := X 1| x[i] 6= a ∧ y[i] = b ∧ z[i] = c ; Number2 := X 1| x[i] = a ∧ y[i] 6= b ∧ z[i] = c ; Number3 :=
[8] [9] [10]
[11]
Number := Number1 + Number2 + Number3 .
[12]
References
[13]
[1] S.G. Akl, Parallel Computation: Models and Methods, Prentice-Hall, Upper Saddle River, NJ, 1997. [2] S.G. Akl, L. Chen, Efficient parallel algorithms on proper circular arc graphs, IEICE Trans. Inform. Systems (Special Issue on Architecture, Algorithms and Networks for Massively Parallel Computing) E79-D (8) (1996) 1015–1020. [3] S.G. Akl, G.R. Guenther, Broadcasting with selective reduction, in: Proc. IFIP 11th World Comput. Congress, San Francisco, 1989, pp. 515–520. [4] S.G. Akl, G.R. Guenther, Applications of broadcasting with selective reduction to the maximal sum subsegment problem, Internat. J. High Speed Comput. 3 (1991) 107–119. [5] S.G. Akl, I. Stojmenovi´c, Multiple criteria BSR: An implementation and applications to computational geometry problems, in: Proc. 27th Hawaii International Conference on System Sciences, Maui, Hawaii, Vol. II, January 1994, pp. 159–168. [6] S.G. Akl, I. Stojmenovi´c, Broadcasting with selective reduction: A powerful model of parallel computation, in: A.Y. Zomaya (Ed.), Parallel and Distributed Computing Handbook, McGraw-Hill, New York, 1996, pp. 192–222. [7] L. Bergogne, C. Cerin, Applications of BSR model of computation for subsegment problems, in: Proc. 4th International
[14]
[15]
[16] [17]
[18]
[19]
73
Conference on High Performance Computing, 1998, pp. 126– 131. L. Chen, Partitioning graphs into Hamiltonian ones, Parallel Comput. 22 (4) (1996) 607–618. L. Chen, Optimal bucket sorting and overlap representations, Parallel Algorithms Appl. 10 (1997) 249–269. L. Chen, Optimal circular arc representations: properties, recognition, and construction, J. Comput. Systems Sci. 56 (3) (1998) 320–331. L.P. Gewali, I. Stojmenovi´c, Computing external watchman routes on PRAM, BSR, and interconnection models of parallel computation, Parallel Process. Lett. 4 (1994) 83–93. L.F. Lindon, S.G. Akl, An optimal implementation of broadcasting with selective reduction, IEEE Trans. Parallel Distrib. Systems 4 (3) (1993) 256–269. R.A. Melter, I. Stojmenovi´c, Solving city block metric and digital geometry problems on the BSR model of parallel computation, J. Math. Imaging Vision 5 (1995) 119–127. D. Semé, J.-F. Myoupo, A parallel solution of the sequence alignment problem using BSR model, in: Proc. Internat. Conference on Parallel and Distributed Computing, 1997, pp. 357–362. I. Stojmenovi´c, Constant time BSR solutions to parenthesis matching, tree decoding, and tree reconstruction from its traversals, IEEE Trans. Parallel Distrib. Systems 7 (2) (1996) 218–224. L. Xiang, K. Ushijima, ANSV problem on BSRs, Inform. Process. Lett. 65 (3) (1998) 135–138. L. Xiang, K. Ushijima, Decoding and drawing an arbitrary tree from its pre-post sequence, in: Proc. PDCS’98, IASTED/ACTA Press, 1998, pp. 182–185. L. Xiang, K. Ushijima, Decoding and drawing on BSR for a binary tree from its i − p sequence, Parallel Process. Lett. 9 (1) (1999) 103–110. L. Xiang, K. Ushijima, S.G. Akl, I. Stojmenovi´c, An efficient implementation for the BROADCAST instruction of BSR+ , IEEE Trans. Parallel Distrib. Systems, to appear.
A theorem on the relation between BSRk and BSR+ Limin Xiang ∗ , Kazuo Ushijima 1 Department of Computer Science and Communication Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Received 1 May 1999 Communicated by S.G. Akl
Abstract PRAM is the most popular model of parallel computation. Of its three variants most commonly used, CREW is more powerful than EREW, and CRCW is the most powerful. BSR is another PRAM model, which is more powerful than CRCW. BSRk and BSR+ are models extended from BSR, and in this paper a theorem is shown on the relation between BSRk and BSR+ . 1999 Elsevier Science B.V. All rights reserved. Keywords: Model of parallel computing; Broadcasting with selective reduction; Parallel algorithms
1. Introduction For two models A and B of parallel computation, the statement that model A is more powerful than model B, denoted by B ⊂ A, means that [19] with the same number of processors, • all problems can be solved on A in no more time than on B, and • some problems can be solved on A in less time than on B. Thus, for the three well known PRAM models EREW, CREW and CRCW, EREW ⊂ CREW ⊂ CRCW. In [3], another PRAM model BSR (Broadcasting with Selective Reduction) was presented, which is more powerful than CRCW. For the selection of BSR, one criterion is permitted only (BSR is denoted by BSR1 in this paper.) In [5,6], BSR was extended as BSRk (k > 2, is a constant integer) so that k criteria are permitted for the selection, and in [19], BSR was ∗ Corresponding author. Email: [email protected]. 1 Email: [email protected].
extended as BSR+ in which an arbitrary logic expression is permitted for the selection. It is known that for k>2 EREW ⊂ CREW ⊂ CRCW ⊂ BSR1 ⊂ BSRk ⊂ BSRk+1 ⊂ BSR+ [3,5,6,19]. In this paper, we will show a theorem on the relation between BSRk and BSR+ . 2. The theorem Definitions of BSR1 , BSRk and BSR+ in detail as well as parallel algorithms on them for many applications can be found in the literature [1–19]. Here, we give only the three kinds of Broadcast Instructions for BSR1 , BSRk and BSR+ , and omit the explanation for them. A Broadcast Instruction of BSR1 is denoted by uj := < di |ti σ lj ,
0020-0190/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 9 9 ) 0 0 0 8 6 - 1
72
L. Xiang, K. Ushijima / Information Processing Letters 71 (1999) 71–73
a Broadcast Instruction of BSRk is denoted by ^ (h) (h) (h) ti σ lj , uj := < di 16h6k
and a Broadcast Instruction of BSRk is denoted by uj := < di (1) (1) (2) (2) (c) (c) ℘ ti σ (1) lj , ti σ (2) lj , . . . , ti σ (c) lj . Lemma 1. Let D be a set of data and {D1 , D2 , . . . , Ds } be a partition on D, then
Dr = di |Gr (i, j ) for r = 1, 2, . . . , s, and D = di |℘ (y1, y2 , . . . , yk ) . By Lemma 1, the Broadcast Instruction of BSR+ above can be implemented by the following instructions on BSRk : for r = 1 to s do urj := e(<); /∗ e(<) is the neutral element of < ∗ /; (r) uj :=
L. Xiang, K. Ushijima / Information Processing Letters 71 (1999) 71–73
G1 ∧ G3 = FALSE,
and
G2 ∧ G3 = FALSE. By Theorem 2, the solution on BSR3 can be: Number1 := 0; Number2 := 0; Number3 := 0; X 1| x[i] = a ∧ y[i] = b ; Number1 := X 1| x[i] 6= a ∧ y[i] = b ∧ z[i] = c ; Number2 := X 1| x[i] = a ∧ y[i] 6= b ∧ z[i] = c ; Number3 :=
[8] [9] [10]
[11]
Number := Number1 + Number2 + Number3 .
[12]
References
[13]
[1] S.G. Akl, Parallel Computation: Models and Methods, Prentice-Hall, Upper Saddle River, NJ, 1997. [2] S.G. Akl, L. Chen, Efficient parallel algorithms on proper circular arc graphs, IEICE Trans. Inform. Systems (Special Issue on Architecture, Algorithms and Networks for Massively Parallel Computing) E79-D (8) (1996) 1015–1020. [3] S.G. Akl, G.R. Guenther, Broadcasting with selective reduction, in: Proc. IFIP 11th World Comput. Congress, San Francisco, 1989, pp. 515–520. [4] S.G. Akl, G.R. Guenther, Applications of broadcasting with selective reduction to the maximal sum subsegment problem, Internat. J. High Speed Comput. 3 (1991) 107–119. [5] S.G. Akl, I. Stojmenovi´c, Multiple criteria BSR: An implementation and applications to computational geometry problems, in: Proc. 27th Hawaii International Conference on System Sciences, Maui, Hawaii, Vol. II, January 1994, pp. 159–168. [6] S.G. Akl, I. Stojmenovi´c, Broadcasting with selective reduction: A powerful model of parallel computation, in: A.Y. Zomaya (Ed.), Parallel and Distributed Computing Handbook, McGraw-Hill, New York, 1996, pp. 192–222. [7] L. Bergogne, C. Cerin, Applications of BSR model of computation for subsegment problems, in: Proc. 4th International
[14]
[15]
[16] [17]
[18]
[19]
73
Conference on High Performance Computing, 1998, pp. 126– 131. L. Chen, Partitioning graphs into Hamiltonian ones, Parallel Comput. 22 (4) (1996) 607–618. L. Chen, Optimal bucket sorting and overlap representations, Parallel Algorithms Appl. 10 (1997) 249–269. L. Chen, Optimal circular arc representations: properties, recognition, and construction, J. Comput. Systems Sci. 56 (3) (1998) 320–331. L.P. Gewali, I. Stojmenovi´c, Computing external watchman routes on PRAM, BSR, and interconnection models of parallel computation, Parallel Process. Lett. 4 (1994) 83–93. L.F. Lindon, S.G. Akl, An optimal implementation of broadcasting with selective reduction, IEEE Trans. Parallel Distrib. Systems 4 (3) (1993) 256–269. R.A. Melter, I. Stojmenovi´c, Solving city block metric and digital geometry problems on the BSR model of parallel computation, J. Math. Imaging Vision 5 (1995) 119–127. D. Semé, J.-F. Myoupo, A parallel solution of the sequence alignment problem using BSR model, in: Proc. Internat. Conference on Parallel and Distributed Computing, 1997, pp. 357–362. I. Stojmenovi´c, Constant time BSR solutions to parenthesis matching, tree decoding, and tree reconstruction from its traversals, IEEE Trans. Parallel Distrib. Systems 7 (2) (1996) 218–224. L. Xiang, K. Ushijima, ANSV problem on BSRs, Inform. Process. Lett. 65 (3) (1998) 135–138. L. Xiang, K. Ushijima, Decoding and drawing an arbitrary tree from its pre-post sequence, in: Proc. PDCS’98, IASTED/ACTA Press, 1998, pp. 182–185. L. Xiang, K. Ushijima, Decoding and drawing on BSR for a binary tree from its i − p sequence, Parallel Process. Lett. 9 (1) (1999) 103–110. L. Xiang, K. Ushijima, S.G. Akl, I. Stojmenovi´c, An efficient implementation for the BROADCAST instruction of BSR+ , IEEE Trans. Parallel Distrib. Systems, to appear.