Parberry, I. A computer assisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory 24 (1991), 101-116.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....cases, a computer proof may be preferable to a traditional one. However, we feel that one point overlooked in [13] namely the question of elegance of computer proofs, should be addressed. Other authors have also addressed the question of computer aided proofs. For some alternative viewpoints, see [3, 4, 5, 7, 11, 12, 14]. 2 What is Elegance While it is impossible to completely nail down the de nition of elegance, we all have some gut feeling about what it is. Paul Erd os was one of the greatest mathematicians of this century, and certainly the most proli c. He put forward the idea that the creator has a Book in ....

Parberry, I. A computer assisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory 24 (1991), 101-116.

....changes. Experiences gathered so far from automatically solved complex algorithmic problems 2 suggest that computer assisted discovery of solutions This paper is a shortened version of [Eus95] 2 e.g. the computer assisted proof of a optimal depth lower bound for nine input sorting networks [Par91]. for non trivial problems involves besides the guidance by abstract principles comprehensive domain specific and strategic knowledge. The intent of the SEAMLESS approach is to enhance the derivation of correct, efficiently executable specifications from declarative specifications by ....

I. Parberry. A computer-assisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory, 24:101--116, 1991.

....of a sorting network can be measured in two different ways: 1. Its depth which is defined as the number of parallel steps that it takes to sort any input, given that in one step disjoint comparison swap operations can take place simultaneously. Current upper and lower bounds are provided in [69]. Table 1 presents these current bounds on depth for n 16. Inputs 1 2 3 4 5 6 7 8 Upper 0 1 3 3 5 5 6 6 Lower 0 1 3 3 5 5 6 6 Inputs 9 10 11 12 13 14 15 16 Upper 7 7 8 8 9 9 9 9 Lower 7 7 7 7 7 7 7 7 Table 3.4: Current upper and lower bounds on the depth of n input sorting networks. 2. Its ....

....However, despite a large amount of computer resource (a few hours on a 64K processors CM2) and a strong bias (the search algorithm was initialized with the first 32 comparators of Green s construction) the best result achieved was a 61 comparator sorter, one more than Green s solution. Parberry [69] addressed the problem of determining the smallest depth for n = 9 and n = 10. His work used an exhaustive search and required 200 hours on a Cray 2. 3.3.3 Implementation A randomized construction procedure (for which a pseudo code is presented in Figure 3.10) is run by each processor element in ....

Ian Parberry. A computer-assisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory, 24:101--116, 1991.

....such network still has hundreds of times the depth of a 2 78 node bitonic sorter. New bounds on the minimum size and depth of (comparison based) sorting networks are provided by Kahale et al. in [129] Small sorting networks of optimal depth are known for n 10, and for optimal size for n 8 [238]. For randomized sorting, first Reif and Valiant obtained optimal randomized sorting on a nonexpander graph, called the cube connected cycle (see Section 4.7) 270] Leighton and Plaxton provided efficient randomized sorting algorithms on the butterfly (the constant is only 7:45) Theorem 2.8 N ....

Parberry, I. A computer assisted optimal depth lower bound for nine-input sorting networks. Math. Syst. Theory. 24, 1991, pp. 101--116. Mathematical Systems Theory, Vol. 24, pp. 101-116, 1991.

....Encouraging results described in this paper, let us expect that a broader field of applications can be tackled by the END model. In this paper, the END model is applied on two difficult real life problems. The first one is the follow up of an established problem since several approaches ([1, 5, 9]) have been used to try to improve some 25 years old results concerning sorting networks [7] Actually, this problem was also at the origin of an early paper [11] in which GAs were used to try to replicate Hillis s experiment for the 16 input problem and in which some ideas of the END model were ....

....of a sorting network can be measured in two different ways: 1. Its depth which is defined as the number of parallel steps that it takes to sort any input, given that in one step disjoint comparison swap operations can take place simultaneously. Current upper and lower bounds are provided in [9], in which the depth for 9 and 10 input sorting networks is proved to be optimal using an algorithm executed on a supercomputer. Table 1 presents these current bounds on depth for sorting networks for n 16. 2. Its length, that is the number of comparison swap used. Optimal sorting networks for ....

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Ian Parberry, "A Computer-Assisted Optimal Depth Lower Bound for NineInput Sorting Networks". In Mathematical Systems Theory, No 24, 1991, pp. 101-116.

....the representation used for genetic encoding of candidate solutions and the interaction between hosts and parasites. A. Sorting Networks A sorting network is a sorting algorithm consisting of a fixed sequence of comparison swap operations. Such algorithms are called homogeneous [5] or oblivious [10] since the sequence of comparisons is exactly the same for all inputs of a given size. A sorting network for input sequences of length n can be represented graphically (see figure 1) as n horizontal lines. Along the horizontal lines, from left to right, appears a number of vertical lines, each one ....

Ian Parberry. A computer-assisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory, 24:101--116, 1991.

....at most n=2 comparators. In fact, Yao s lower bound was established for the depth of certain selection networks, but it also gives the best result currently known for the case of sorting. For small values of n, some computer aided approaches for finding upper and lower bounds have been developed [6, 17]. Finally, significantly better upper bounds are known for comparator networks that sort all but a small fraction of the n input permutations. In particular, Leighton and Plaxton [15] have described a comparator network of size c 1 n lg n and depth c 2 log n that sort all but a superpolynomially ....

I. Parberry. A computer-assisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory, 24:101--116, 1991.

....presents the END model and compares its efficiency to GA for the sorting network problem. This is the follow up of an established problem for which several approaches have been used to try to improve some 25 years old results ( Belew Kammayer, 1993, Drescher, 1994, Hillis, 1992, Levy, 1992, Parberry, 1991] Actually, this problem was also the origin of an early paper [Tufts Juill e, 1994] in which GA were used to try to replicate Hillis experiments ( Hillis, 1992] for the 16 input problem and in which some ideas of the END model were presented. Encouraging results described in this paper show ....

....of a sorting network can be measured in two different ways: 1. Its depth which is defined as the number of parallel steps that it takes to sort any input, given that in one step disjoint comparison swap operations can take place simultaneously. Current upper and lower bounds are provided in [Parberry, 1991]. Table 1 presents these current bounds on depth for n 16. 2. Its length, that is the number of comparison swap used. Optimal sorting networks for n 8 are known exactly and are presented in [Knuth, 1973] along with the most efficient sorting networks to date for 9 n 16. Table 2 presents these ....

Parberry, I. (1991). A computerassisted optimal depth lower bound for nine-input sorting networks. Mathematical Systems Theory, 24:101--116.

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