R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. JCSS, 47:501--548, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....subcubes. The task of comparing subtrees mapped to the same variable generally has to be performed by a standard sorting algorithm. This can be done by a randomized sorting algorithm [VB81, ALMN90] within the same time bound O( log p) The best known deterministic sorting algorithm [CP90] exceeds this time bound by a factor of O( log log p) 2 Fundamental Concepts and Notation We first give a short description of our model of computation. A network of processors is a set of processors, interconnected by bidirectional communication links. Each processor has the capabilities ....

R. Cypher and C.G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 193--203, 1990.

....are given by a deterministic algorithm based on Balance Sort. The lower bounds for the PRAM interconnection hold for any type of PRAM. The lower bounds for the f(x) x case require the comparison model of computation. The current best deterministic bound for T (H) on a hypercube comes from [CyP]. In the hypercube case for f(x) log x, the term in the sorting time involving T (H) is thus possibly nonoptimal by an O( log log H) factor; however, the algorithm is optimal when N is large (N H (log H) log log H ) or small (N = O(H) Note that if we are allowed to use a randomized ....

.... establishes the solutions for the f(x) x case in Theorem 1: Lemma 9 When f(x) x , the algorithm sorts in deterministic time T (H) On a hypercube, the best known value of T (H) is O(log H log log H) if precomputation is allowed and O(log H (log log H) with no precomputation [CyP]. On an EREW PRAM, we have T (H) O(log H) Col] Lemmas 8 and 9 complete the proof for the upper bounds of Theorem 1. The lower bounds for all the terms not involving T (H) were shown in [ViSb] In fact, we can go even farther for the P HMM model and show that the algorithm is uniformly optimal ....

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Robert E. Cypher & C. Greg Plaxton, "Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers," Journal of Computer and System Sciences 47 (1993), 501--548.

....redundancy information to make recovery possible in the event of a disk failure [CLG, Sch] Subsequently to our work, Aggarwal and Plaxton [AgP] developed another optimal deterministic external sorting algorithm. Their algorithm is based on the hypercube sorting method of Cypher and Plaxton [CyP] which has higher constant factors. Barve, Grove, and Vitter [BGV] developed a practical sorting method based on merge sort, which does probably better than disk striping even for moderate values of D. The algorithm, however, is not theoretically optimal for some values of N , D, and B. This ....

Robert E. Cypher & C. Greg Plaxton, "Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers," Journal of Computer and System Sciences 47 (1993), 501--548.

....time algorithm was given in [7] as a subroutine of a batched planar point location algorithm. Furthermore, a randomized O(log n) expected time scheme for multisearching was given in [10] Since searching is related to sorting and there is a deterministic O(log n log log n) time sorting algorithm [4], the question was open, if there exists an algorithm for the multisearch problem that runs faster than O(log 2 n) This paper gives a step in the right direction, by presenting an algorithm with time complexity O(log n(log log n) 3 ) for a n processor hypercube. It further presents as an ....

....a n log n processor hypercube, which leads to algorithms for the batched planar point location problem and for the triangulation of a simple polygon with the same resource bounds. Our multisearch algorithm is more of theoretical than of practical interest, because it uses the sorting algorithm of [4] as a subroutine. However, any practical improvement to sorting would immediately make our algorithm more practical. The paper is organized as follows. In Section 2 we review the denition of a hypercube interconnection network and some basic algorithms for this parallel machine. Then in Section 3 ....

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R. Cypher and C.G. Plaxton. Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers, ACM Proceedings of the 22th Annual ACM Symposium on Theory of Computing, pp. 193 203, 1990.

.... space complexities listed in Table 3 for manipulating linear quadtrees with path encoding on a hypercube are obtained from [2] by using standard PRAM simulation on a hypercube, as described by Nassimi and Sahni ( 13] together with Cypher and Plaxton s deterministic hypercube sorting algorithm ([4]) Follows from [2] by standard PRAM simulation on a hypercube as described in [12] together with [4] This operation is trivial for pointer based quadtrees, and listed for completeness only. The hypercube time complexity assumes O(1) time instruction broadcast (as, e.g. on the Connection ....

.... are obtained from [2] by using standard PRAM simulation on a hypercube, as described by Nassimi and Sahni ( 13] together with Cypher and Plaxton s deterministic hypercube sorting algorithm ( 4] Follows from [2] by standard PRAM simulation on a hypercube as described in [12] together with [4]. This operation is trivial for pointer based quadtrees, and listed for completeness only. The hypercube time complexity assumes O(1) time instruction broadcast (as, e.g. on the Connection Machine) 5 In this paper, we study two problem areas which remained unsolved in the previous ....

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R. Cypher and C. G. Plaxton, "Deterministic sorting in nearly logarithmic time on a hypercube and related computers," to appear in Proc. ACM Symposium on Theory of Computing, 1990.

....n) where n is the number of points in the set. 1 Multiprocessor models have also been used to obtain faster solutions. Using a hypercube with O(n) processors, O(log n(log log n) 2 ) time has been obtained in [6] by using a fast sorting algorithm with O(log n log log n) time described in [4, 5]. We have also used a divide and conquer strategy to solve the two set closest pair problem with time complexity O(n log d Gamma1 n) on a sequential machine, where d is the space dimension. 2 The 2D case Let P, Q two sets of points on the plane, given by their Cartesian coordinates, such that ....

R. Cypher and C.G. Planton, "Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers", Proceedings of the Twenty-Second Annual ACM Symposium of Theory of Computing, p.p. 93-203, 1990

.... sorted array X Y can be generated, using O(1) space per processor in time O(n log n oe(n) For instance, this can be applied to CRCW P RAM (with oe(n) equal to 1) or to a hypercube (with oe(n) equal to log n using Batcher s sort [4] or (log log n) 2 using the asymptotically faster method of [9]) Of course, as in section 3.2.3, the whole array X Y of n 2 elements cannot exist at any one time in space of O(1) per processor. The array will again be produced as a sequence of segments with O(n) elements in each segment, every segment sorted, and every element in a segment less than or ....

R.E. Cypher and C.G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. In Proceedings of 15th Annual ACM Symposium on Theory of Computing, pages 193--203, 1990.

....runs in asymptotically optimal O(N) time. On the r dimensional hypercube the algorithm has asymptotic complexity O(r 2 ) which is the same as that of Batcher oddeven merge sorting algorithm on the hypercube [2] Although there are asymptoticallyfaster sorting algorithms for the hypercube [6], they are not practically useful for reasonable number (less than 2 20 ) of keys [18] We note, however, that there are randomized algorithms which perform better on hypercubic networks than the Batcher algorithm in practice [5] Adaptation of such approaches for product networks appears to be ....

R. Cypher and C. G. Plaxton, "Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers," Journal of Computer and System Sciences, vol. 47, pp. 501--548, Dec. 1993.

.... of size m that can sort n numbers in time sort(n; m) is n universal with slowdown O(sort(n; m) e.g. this means that the shuffle exchange network and the cube connected cycles network, each of size n, are n universal with slowdown O(logn Delta (log log n) 2 ) using the algorithm presented in [3]. A more general approach is the concept of dynamic simulations (also sometimes called emulations with redundancy) Such simulations allow that single guest processors are simulated at several host processors, and that the set of simulating processors may vary during the simulation. A model of ....

R. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. J. Comput. Syst. Sci., 47:501--548, 1993.

....independent of the sorting algorithm, the multibutterfly is an efficient and highly fault tolerant routing network. 1. 2 Other related results Prior to this work, the fastest deterministic algorithm for sorting N keys on an N node multibutterfly was the Sharesort algorithm of Cypher and Plaxton [CP90] This algorithm was designed to run on the butterfly network, or on any other hypercubic network (e.g. the shuffleexchange network and the hypercube) Since the multibutterfly network contains a butterfly network, it applies to multibutterflies as well (but doesn t take advantage of the ....

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 193--203, May 1990.

....pattern, the sort is quite efficient on most parallel machines, and it is often used as the sort to which other sorts are compared. In addition to Bitonic sort, there are several other sorting algorithms that have oblivious routing patterns. Out of these both columnsort [23] and smoothsort [12] are reasonably practical when the number of keys is much larger than the number of processors (for p processors, columnsort requires p 3 keys to run most efficiently) Columnsort has been implemented on the CM 5 with running times that were not as fast as some of other sorts on the CM 5 ....

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 193--203, May 1990.

....we know that any deterministic sorting algorithm on a network can also be used for deterministic permutation routing. There are, e.g. routing algorithms based on AKS sorting [1] that can be implemented on a constant degree network with runtime (log n) 19] Furthermore, the sorting result in [7] implies that any permutation can be routed deterministically on a hypercube, shuffle exchange, Throughout the paper, the terms with high probability and w.h.p. mean with probability at least 1 n where 0 is an arbitrary constant. 2 cube connected cyles and butterfly of size n ....

R. Cypher, C.G. Plaxton. Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers. Journal of Computer and System Sciences 47, pp. 501-548, 1993.

....lower bounds also extends to certain restricted classes of non oblivious sorting algorithms on hypercubic machines and multi dimensional meshes. However, our lower bound argument does not allow the copying of elements by the algorithm. Thus, the Sharesort sorting algorithm of Cypher and Plaxton [26], which achieves a running time of O(lg n lg lg n) with preprocessing) on any of the hypercubic machines, is not subject to our lower bound. Nonetheless, we believe that our present results are already interesting in their own right, and that they may constitute an important step towards more ....

....an arbitrary amount of internal computation, and can send one element, plus an arbitrary amount of auxiliary information, to one of its neighbors. While this class covers a fairly wide range of sorting algorithms, it does unfortunately not include the Sharesort algorithm of Cypher and Plaxton [26], which makes copies of some of the elements. 2.6.2 Multi Dimensional Meshes We can also extend our lower bounds to some restricted classes of sorting algorithms on multi dimensional meshes. In [121] Wanka describes the following natural extension of the class of ascend algorithms to ....

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R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. JCSS, 47:501--548, 1993. 185

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R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. JCSS, 47:501--548, 1993.

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R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. Journal of Computer and Systems Sciences, 47:501--548, 1993.

....Smoothsort runs on a hypercube in O(log(n=p) log 3 p) time when n pq and in O( n=p) log 3=2 p= log 1=2 (n=pq) time when n pq. Recently, Cypher and Plaxton presented an algorithm called Sharesort that runs in O(logn(log log n) 2 ) time on cube type computers with p = n processors [6]. In terms of worst case analysis, this is the first result to improve upon the O(log 2 n) time bitonic sort for cube type computers with p = n processors. For n p this algorithm can be easily generalized to obtain an O( n=p) log n log log n log log(n=p) time sorting algorithm. Sharesort ....

....processors available greatly exceeds the number of records that are being permuted. Thus the planning stage can use the excess processors to calculate a plan very efficiently. In the original description of Sharesort, an O( n=p) log n log log n) time algorithm for shared key sorting was presented [6]. The remainder of this paper is devoted to obtaining new solutions to the shared key sorting problem that have improved running times. These new algorithms for shared key sorting will in turn yield new algorithms for sorting on cube type computers that improve upon all previous deterministic ....

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Robert Cypher and C. Greg Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. In Proc. 22nd Annual Symposium on Theory of Computing, 1990. to appear.

....running time is improved to O(lg n lg (3) n) and then to O(lg n lg (4) n) in Sections 2.3 and 2.4, respectively. An O(lg n lg n) algorithm is presented is Section 2.5. This improvement in the running time is obtained at the expense of using a non uniform variant of the Sharesort algorithm [5] that requires a certain amount of preprocessing. Finally, in Section 2.6 we show how to avoid the non uniformity introduced in Section 2.5. Thus, our best asymptotic result is a uniform selection algorithm with a running time of O(lg n lg n) 2 Selection by successive approximation 2.1 ....

....some y with ffi=2 y ffi. Sparse enumeration sort implies that T (ffi; y) O(ffi) and hence T (d; 0) O(d lg lg d) O(lg n lg (3) n) 2.4 An O(lg n lg (4) n) algorithm We can improve the time bound achieved in Section 2. 3 by making use of the Sharesort algorithm of Cypher and Plaxton [5]. Several variants of that algorithm exist; in particular, detailed descriptions of two versions of Sharesort may be found in [5] Both of these variants are designed to sort n records on an n processor hypercubic network. The first algorithm runs in O(lg n(lg lg n) 3 ) time and the second ....

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R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. Journal of Computer and System Sciences, 47:501--548, 1993.

....hypercube algorithm to obtain: Theorem 5 There is a selection algorithm for any p processor bounded degree hypercubic machine that uses O( log p) 1 ff=2 Delta (log log p) 2 ) steps in the worst case. Remark: This algorithm makes use of the Sharesort sorting algorithm of Cypher and Plaxton [6]. Sharesort is a normal hypercube algorithm which, for n = p, runs in O( log p) Delta (log log p) 2 ) time on any hypercubic machine. For n p, this immediately implies an upper bound of O( n=p) Delta (log p) Delta (log log p) 2 ) for any hypercubic machine. 8 Concluding Remarks It is ....

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. JCSS, 47:501--548, 1993.

....sampling, which was previously used by Cole and Yap [5] to solve the selection problem in an idealized model of computation called the parallel comparision model. The algorithms also use as subroutines sorting algorithms for hypercubic networks due to Nassimi and Sahni [8] and Cypher and Plaxton [6]. 1.1 Hypercubic networks A hypercube contains n = 2 d nodes, each of which has a distinct d bit label (d must be a nonnegative integer) A 1 Laboratoire de l Informatique du Parall elisme, Institut IMAG, Ecole Normale Sup erieure de Lyon, 46, All ee d Italie, 69364 Lyon Cedex 07, France. ....

.... fastest previously known algorithm for solving the selection problem on a hypercubic network is due to Plaxton and runs in O(log n log log n) time on an n node network [9] The selection problem can also be solved in O(log n log log n) using the hypercubic sorting algorithm of Cypher and Plaxton [6]. In [9] Plaxton also showed that any deterministic algorithm for solving the selection problem on a p processor hypercubic network requires Omega Gammaq n=p) lg lg p lg p) time in the worst case. Since the selection problem can be solved in linear time sequentially [3] the lower bound ....

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R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. JCSS, 47:501--548, 1993.

....The running time is improved to O(lg n lg (3) n) and then to O(lg n lg (4) n) in Sections 2.3 and 2.4, respectively. Finally, an O(lg n lg n) algorithm is presented is Section 2.5. This last improvement is made at the expense of using a non uniform variant of the Sharesort algorithm [5] that requires a certain amount of preprocessing. 2.1 Approximate selection In this section, we develop an efficient subroutine for approximate selection based on the parallel comparison model algorithm of Cole and Yap [4] There are two major differences. First, we use Nassimi and Sahni s sparse ....

....some y with ffi=2 y ffi. Sparse enumeration sort implies that T (ffi; y) O(ffi) and hence T (d; 0) O(d lg lg d) O(lg n lg (3) n) 2.4 An O(lg n lg (4) n) algorithm We can improve the time bound achieved in Section 2. 3 by making use of the Sharesort algorithm of Cypher and Plaxton [5]. Several variants of that algorithm exist; in particular, detailed descriptions of two versions of Sharesort may be found in [5] Both of these variants are designed to sort n records on an n processor hypercubic network. The first algorithm runs in O(lg n(lg lg n) 3 ) time and the second ....

[Article contains additional citation context not shown here]

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. Journal of Computer and System Sciences, 47:501--548, 1993.

....existence of an O(log n) deterministic sorting algorithm for the hypercube and related computers remains open. The conference version of this paper listed a number of extensions and applications of the Sharesort algorithm, and stated that the details would appear in the full version of the paper [7]. Instead, these additional results will appear in a separate paper. ....

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. Journal of Computer and System Sciences, 47:501--548, 1993.

....algorithm is non adaptive in the sense that it can be described solely in terms of oblivious routing and compare interchange operations; there is no queueing. The very 4 high probability version is adaptive because it makes use of the Sharesort algorithm of Cypher and Plaxton as a subroutine [9]. Note that the permutation routing problem, in which each processor has a packet of information to send to another processor, and no two packets are destined to the same processor, is trivially reducible to the sorting problem. The idea is to sort the packets based on their destination ....

....any input d vector probability at least 1 Gamma (d) The scheme of x7 can also be used to prove Theorem 10.1 below with the function as defined in Theorem 10.2. In this case, we can dramatically decrease the failure probability by making use of the Sharesort algorithm of Cypher and Plaxton [9]. Sharesort is a polynomial time uniform hypercubic sorting algorithm with worst case running time O(d Delta lg 2 d) 9] Note that Sharesort runs in O(d) time on O(d= lg 2 d) cubes. Hence, we can modify the scheme of x7 by cutting off the sorting recurrence at Theta(d= lg 2 d) cubes ....

[Article contains additional citation context not shown here]

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. Journal of Computer and System Sciences, 47:501--548, 1993.

....Our lower bound also extends to certain restricted classes of non oblivious sorting algorithms on hypercubic machines and multi dimensional meshes. However, our lower bound argument does not allow the copying of elements by the algorithm. Thus, the Sharesort sorting algorithm of Cypher and Plaxton [7], which achieves a running time of O(lg n lg lg n) with preprocessing) on any of the hypercubic machines, is not subject to our lower bound. Nonetheless, we believe that our present results are already interesting in their own right, and that they may constitute an important step towards more ....

....lower bounds to more general classes of non oblivious sorting algorithms on the hypercube. Of particular 11 interest in this respect would be the class of normal comparison based sorting algorithms, or any other natural class of algorithms including the Sharesort algorithm of Cypher and Plaxton [7]. Another possible direction for future research would be to consider other restricted classes of sorting networks. Finally, it is an open problem whether our lower bound technique can be applied to selection networks. ....

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. JCSS, 47:501--548, 1993.

....within any particular processor be sorted. There has been a great deal of previous research related to the problem of sorting on the hypercube and related networks, under a variety of different models of computation. For an overview of the hypercube sorting literature, the reader is referred to [4]. The time bounds for the merging and sorting algorithms described in this section do not apply to the 1 port model of computation that we have been considering up to this point. Instead, we will make use of a restricted form of the less realistic (lg p) port model, in which a processor can send ....

....the moderately large coefficients in Section 5 could be improved with only minor modifications to the algorithm. 13 Finally, we note that a substantial theoretical advance has been made on the problem of hypercube sorting subsequent to the work described in this paper. Namely, Cypher and Plaxton [4] have developed a hypercube sorting algorithm with running time O(lg n(lg lg n) 2 ) assuming one item per processor) On the other hand, the pipelined merging approach described in this paper continues to provide strictly the fastest known algorithms for merging on the pipelined hypercube, and ....

[Article contains additional citation context not shown here]

R. E. Cypher and C. G. Plaxton. Deterministic sorting in nearly logarithmic time on the hypercube and related computers. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 193--203, May 1990.

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R.Cypher and G. Plaxton, Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers, Proc. of 22nd ACM Symp. on Theory of Computing, 1990, pp. 193-203.

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