C. P. Schnorr and A. Shamir, An optimal sorting algorithm for mesh connected computers. Proc. 18th Annual ACM Symposium on Theory of Computing. 1986, pp. 255--263.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....d is a permutation. A trivial way of routing permutations is by sorting on an appended destination field. Sorting algorithms for the MESH are well known, 6, 10, 3] the most efficient one requiring (8. v ff O(N 3 n) routing steps (rss) Many recent articles on routing and sorting on the MESH, [8, 2, 4], assume a MIMD machine and often have average case performance. Although these results cannot readily be compared to our results for SIMD machines, they show a vivid interest in the subject. In case of special permutations more efficient routing schemes are possible: For bit oriented permutations ....

Schnorr, C. P., A. Shamir, An optimal sorting algorithm for MESH-connected com- puters, Proc. 18th ACM Silrnp. on Th. of Comp., 1986, pp 255-263.

....than v. After the recursive sorting, all that matters is how many such numbers are in each quadrant. It is then easy to analyze where those numbers appear in subsequent steps. History. Thompson and Kung in [21] showed that an m m mesh can sort m 2 numbers in O(m) steps. Schnorr and Shamir in [18] showed that an m m mesh can sort m 2 numbers in snakelike order in (3 o(1) m steps. Schimmler s algorithm appeared in [17] it is considerably simpler than the Schnorr Shamir algorithm, although it is not as fast. Similar comments apply to higher dimensional meshes. Unfortunately, it is ....

Claus P. Schnorr, Adi Shamir, An optimal sorting algorithm for mesh-connected computers, in [1] (1986), 255-261.

....to see how a fully threaded implementation performs relative to our current code. Third, we would like to improve on the restriction (1) Currently, the maximum problem size is proportional to (M P ) 3 2 . We conjecture that we can incorporate some ideas of the Revsort mesh sorting algorithm [10]. Our hope is that the resulting algorithm would allow the maximum problem size to be proportional to (M P ) 5 3 yet be implementable. Acknowledgments We thank David Kotz, Wayne Cripps, and Arne Grimstrup for their cooperation in helping us set up experimental testbeds. ....

C. P. Schnorr and A. Shamir. An optimal sorting algorithm for mesh connected computers. In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, pages 255--263, May 1986.

....log n# each, and the theoretically optimal algorithm due to Schnorr and # Partially supported by DFG Forschergruppe Effiziente Nutzung massiv paralleler Systeme , by Esprit Basic Research Action Nr. 7141 (ALCOM II) and by DFG Leibniz Award. y Partially supported by Volkswagenstiftung. Shamir [18] that has a running time of 3n o#n#. Unfortunately, the low order term increases the running time of this algorithm on existing machines significantly. Other, more or less topology independent sorting methods are parallel versions of Quicksort [5, 6] Samplesort [9, 14] and Radixsort [7] ....

C. P. Schnorr, A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Proc. of the 20th ACM-STOC, Vol. 32, pp. 255--263, 1988.

....This indexing may be good, but in many cases it is desirable to have the packets in the more natural row major (column major) order. Furthermore, sorting in snake like order is unsuited as a subroutine for scattering algorithms. In the one packet model considered by Schnorr and Shamir [19], the best known upper bound for row major sorting is higher than for sorting in snakelike row major order. In our model a PU may hold a constant number of packets and packets may be copied. From the results of this paper it follows that in this model, sorting in row major order is not ....

....Sibeyn and Suel [10] came with a deterministic version. These algorithms are considerably more involved then the algorithm of this paper, and have queue sizes around 20. The best uni axial row major sorting algorithm so far appears to be a modification of the algorithm of Schnorr and Shamir [19]. It takes 4 Delta n o(n) steps. The first near optimal algorithm for k k sorting was discovered by Kaufmann and Sibeyn [9] Then in [12] by Kunde and slightly later also in [10] deterministic versions of this randomized algorithm were described. All these algorithms use blocked snake like ....

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Schnorr, C.P., A. Shamir, `An Optimal Sorting Algorithm for Mesh Connected Computers,' Proc. 18th Symposium on Theory of Computing, pp. 255--263, ACM, 1986.

.... that information that has a considerable impact on the sorting may remain unknown to the PUs in a corner of the mesh for almost 2 Delta n steps: Lemma 8 Sorting on a MIMD mesh in the one packet model requires at least 3 Delta n Gamma O( p n) steps for sorting in (snake like) row major order [94, 46]. Whereas these results are almost trivial, much deeper methods are required to prove lower bounds for all indexing schemes: Lemma 9 There is no indexing scheme with respect to which sorting in the one packet model can be performed in fewer than 2:27 Delta n steps [23] Thompson and Kung [107] ....

....are required to prove lower bounds for all indexing schemes: Lemma 9 There is no indexing scheme with respect to which sorting in the one packet model can be performed in fewer than 2:27 Delta n steps [23] Thompson and Kung [107] presented the first O(n) sorting algorithm. Schnorr and Shamir [94] then developed the first optimal algorithm. Theorem 13 Sorting on MIMD meshes in the one packet model with respect to the snake like row major indexing can be performed in 3 Delta n O(n 3=4 ) steps [107, 62, 94] On a m Theta n mesh the algorithm takes m 2 Delta n o(n m) steps. ....

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Schnorr, C.P., A. Shamir, `An Optimal Sorting Algorithm for Mesh Connected Computers,' Proc. 18th Symposium on Theory of Computing, pp. 255--263, ACM, 1986.

....relies on the TranspositionSort (Quinn 1987) procedure to get things sorted. TranspositionSort is the basic linear array sort. It works similar to the uniprocessor BubbleSort: comparing then exchanging (where necessary) alternating pairs of items. An improvement of ShearSort is called RevSort (Schnorr Shamir 1986). It is identical to ShearSort, except CyclicSort is used to sort rows instead of Transposition Sorts. This algorithm finishes in p N(log log N 3) steps and the complexity of the each step is equivalent to each step of ShearSort. CyclicSort is a descendant of the TranspositionSort, and ....

Schnorr, C. P. & Shamir, A. (1986), An Optimal Sorting Algorithm for Mesh Connected Computers, in `Proceedings of the 18th Annual ACM Symposium on Theory of Computing ', ACM.

....of two over meshes without buses. Moreover, our algorithm has a queue size of two. In this context, we point out that the 3n Gamma 3 step off line scheme for routing on the standard mesh described by Annexstein and Baumslag [37] as well as the 3n o(n) sorting algorithm of Schnorr and Shamir [38], achieve a queue size of one because two packets can be exchanged across an edge in a single step. Since we do not allow two arbitrary processors connected to a common bus to exchange two packets in a single step, it seems difficult to design any algorithm with queue size one that uses the buses ....

C. P. Schnorr and A. Shamir, An optimal sorting algorithm for mesh-connected computers, In Proc. 18th ACM Symposium on Theory of Computing, May 1986, 255--263.

.... observation is that, if G is a connected graph, it is always possible to obtain an algorithm for PG 2 with complexity S 2 (N ) O(N ) To do so we simply emulate the 2 dimensional grid in PG 2 with constant dilation and congestion [2] Then, the O(N ) algorithm presented by Schnorr and Shamir [9] for sorting N 2 keys on two dimensional N Theta N grid can be emulated by PG 2 with complexity O(N ) leading to S 2 (N ) O(N ) Hence, any arbitrary N r node r dimensional product network can sort with complexity O(r 2 N ) 5 Application to Specific Networks In this section we ....

....to Specific Networks In this section we obtain the time complexity of sorting for several product networks in the literature. To do so, we obtain upper bounds for the value of S 2 (N ) for each network. Using this value in Theorem 1 will yield the desired running time. Grid: Schnorr and Shamir [9] have shown that it is possible to sort N 2 keys in a N 2 node 2 dimensional grid in O(N ) time steps. This value of S 2 (N ) implies that our algorithm will take O(r 2 N ) time steps to sort N r keys in a N r node r dimensional grid. If the number of dimensions r is bounded, this ....

C. P. Schnorr and A. Shamir, "An Optimal Sorting Algorithm for Mesh Connected Computers," in Proceedings of the 18th Annual ACM Symposium on Theory of Computing, (Berkeley, CA), pp. 255--263, May 1986.

....of sorting using the multiway merge sorting algorithm presented for several product networks in the literature. To do so, we obtain upper bounds for the values of S 2 (N) and R(N) for each network. Using these values in Theorem 1 will yield the desired running time. 18 Grid: Schnorr and Shamir [29] have shown that it is possible to sort N 2 keys on a N 2 node 2 dimensional grid in 3N o(N) time steps. It is also trivial to show that the time to perform a permutation on the N node linear array is at most R(N) N Gamma 1. These values of S 2 (N) and R(N) imply that our algorithm will ....

C. P. Schnorr and A. Shamir, "An Optimal Sorting Algorithm for Mesh Connected Computers, " in Proceedings of the 18th Annual ACM Symposium on Theory of Computing, (Berkeley, CA), pp. 255--263, May 1986.

....any step in the computation, a single packet plus an unbounded amount of header information may be transmitted across each directed edge. It is assumed that a comparison exchange operation between adjacent packets can be performed in a single step. For this model of the mesh, Schnorr and Shamir [102] showed an upper bound of 3n o(n) for sorting into row major order. They also proved a lower bound of 3n Gamma o(n) independently discovered by Kunde [54] The same proof technique has also been used to show lower bounds for arbitrary indexing schemes [54] the best general lower bound is ....

....j mod n 2 Gamma2fi . We point out that the idea of deterministically spreading elements of similar rank and destination over the network is not really new. In particular, similar techniques are used in the Columnsort algorithm of Leighton [64] and the 3n o(n) algorithm of Schnorr and Shamir [102], as well as in several of Kunde s algorithms (e.g. see [57, 59] However, none of these papers elaborates on the close relationship between these techniques and the ideas used in many randomized algorithms. 90 In the following, we attempt to explain this relationship, and to design a general ....

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C. P. Schnorr and A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Proceedings of the 18th ACM Symposium on Theory of Computing, pages 255--263, May 1986.

....order which is fast for all sizes of the mesh. This algorithm additionally has to be uni axial, that is, it cannot use horizontal and vertical links simultaneously. None of the known sorting algorithms appears to perform well for small size meshes. Stricker [10] points the algorithm RevSort of [9] as an exception, but this algorithm requires more than 2 1 = 2 Delta log log n 3 Delta n steps for sorting in snakelike order and is not uni axial. Our algorithm borrows some ideas from the recursive algorithm of Lang et al. 4] Among other things, we show that n Theta n array can be ....

....fit into the routing algorithm. We present two scattering algorithms. The first one is useful only for large values of s. The second can be used even for small s. 6. 1 Fast scattering for large s The scattering algorithm of this section is inspired by the sorting algorithm of Schnorr and Shamir [9]. It takes only 2 Delta s O(s 7=10 ) time, and scatters the packets as good as sorting the s Theta s squares in column major order. Each s Theta s square is divided into subsquares of size s oe Theta s oe , for some oe 1=2. By a row of subsquares we mean the set of s 1 Gammaoe ....

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Schnorr, C.P., A. Shamir, `An Optimal Sorting Algorithm for Mesh Connected Computers,' Proc. 18th Symposium on Theory of Computing, pp. 255-263, ACM, 1986.

....sorting is often a key step in other mesh algorithms. Several practical O( p N) time algorithms to sort on a p N Theta p N mesh have been proposed [4, 6, 7, 11] In the model where there is initially one element per processor and the target order is snake like row major, Schnorr and Shamir [10] developed an algorithm that runs in time 3 p N o( p N ) which is asymptotically near optimal as a provable lower bound is 3 p N Gamma o( p N) 5, 10] However, their algorithm is only practical for very large N . More recently, Krizanc [3] presented the first deterministic sorting ....

.... 6, 7, 11] In the model where there is initially one element per processor and the target order is snake like row major, Schnorr and Shamir [10] developed an algorithm that runs in time 3 p N o( p N ) which is asymptotically near optimal as a provable lower bound is 3 p N Gamma o( p N) [5, 10]. However, their algorithm is only practical for very large N . More recently, Krizanc [3] presented the first deterministic sorting algorithm in a similar model that overcomes the 3 p N Gamma o( p N) bound given that input is drawn from integers in the A preliminary version of this paper ....

C. Schnorr and A. Shamir. An Optimal Sorting Algorithm for Mesh Connected Computers. Proc. 18th ACM Symp. on Theory of Computing, 1986, 255-263.

....intermediate destinations to their preliminary destinations with only k Delta n=8 o(k Delta n) steps. Both effects can be realized by scattering the packets: dividing the ring in subchains, and handing out the packets of the subchains in a fair way over the subchains. This idea goes back on [13]. For efficiently routing the packets to their preliminary destination we could use the deterministical algorithm of [5] However, the routing of the packets is very regular, and the packets can be routed simply along the shortest path to their destinations. These ideas give us the following ....

Schnorr, C.P., A. Shamir, `An Optimal Sorting Algorithm for Mesh Connected Computers,' Proc. 18th Symposium on Theory of Computing, pp. 255-263, ACM, 1986.

....designed for (blocked) snake like row major indexings. However, in many cases it is desirable to have the packets in the more natural row major or column major order. Furthermore, sorting in snake like order is unsuitable for scattering. In the one packet model considered by Schnorr and Shamir [18], the best known upper bound for row major sorting is higher than for sorting in snake like row major order. In our model a PU may hold a constant number of packets and packets may be copied. The results of this paper demonstrate that in this model, sorting in row major order is not substantially ....

....order. Kaufmann et al. 8] came with a deterministic version. These algorithms are considerably more involved then the algorithm of this paper, and have queue sizes around 20. The best uni axial row major sorting algorithm so far, appears to be a modification of the algorithm of Schnorr and Shamir [18]. It takes 4 Delta n o(n) steps. The first near optimal algorithm for k k sorting was discovered by Kaufmann and Sibeyn [7] Then in [11] by Kunde and slightly later also in [8] deterministic versions of this randomized algorithm were described. All these algorithms use blocked snake like ....

[Article contains additional citation context not shown here]

C. P. SCHNORR, A. SHAMIR, An Optimal Sorting Algorithm for Mesh Connected Computers, Proc. 18th Symposium on Theory of Computing, pp. 255--263, ACM, 1986.

....a given architecture under realistic situations. Let us first consider the disadvantages of using a metric other than isoefficiency. Many parallel algorithms for sorting are analyzed using the metric of runtime complexity when the number of processors equals the number of data elements (e.g. see [18, 23, 17]) However, this metric is not very useful for realistic situations when the number of data elements can far outnumber the number of processors. For example, one could easily expect to sort a million data elements, but it would be quite unreasonable to expect to have a million processor system to ....

....based on sequential quicksort for sorting data on a mesh multicomputer, and analyzes their scalability. We show that QSP2 matches the lower bound on the isoefficiency function for mesh multicomputers. The isoefficiency of QSP1 is also fairly close to optimal. Lang et al. 18] and Schnorr et al. [23] have developed parallel sorting algorithms for the mesh architecture that have either optimal [23] or close to optimal [18] run time complexity for the one element per processor case. Both QSP1 and QSP2 have worse performance than these algorithms for the one element per processor case. But QSP1 ....

[Article contains additional citation context not shown here]

C.P. Schnorr and A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Proceedings STOC, pages 255--263, 1986.

....2. In this context, several fairly simple randomized algorithms with running time 2 Delta n O(logn) and small constant queue size have been proposed [7, 16] Considerable attention has also been given to the problem of 1 1 sorting on two dimensional meshes. In particular, Schnorr and Shamir [17] gave a 3 Delta n o(n) step algorithm for sorting into snakelike row major order, and proved a nearly matching lower bound of 3 Delta n Gamma o(n) also independently discovered by Kunde [10] In their model of the mesh, a PU may only hold a single packet at any time. However, this lower ....

....i mod N . Notice that every PU is the destination of exactly k packets. The utility of the above sort and unshuffle operation for sorting on meshes was previously observed by Schnorr and Shamir, who used it in the design of their 3 Delta n o(n) sorting algorithm in the single packet model [17]. In the following, we will demonstrate that this operation can in many cases be employed as a substitute for randomization. Following a scheme originally proposed by Valiant and Brebner [20] many randomized algorithms for routing on meshes start by sending the packets to 4 Kaufmann et al. ....

Schnorr, C.P., A. Shamir, `An Optimal Sorting Algorithm for Mesh Connected Computers, ' Proc. 18th Symposium on Theory of Computing, pp. 255-263, ACM, 1986.

....of our clockwise transposition routing. The new algorithm appears to be much faster than the 8m sorting algorithm (due to Schimmler) used in [1] and its local control is very simple compared to the complicated recursive algorithms that achieve the 3m step lower bound on mesh sorting (cf. [16]) A physical realization of the mesh will contain many local faults (especially for devices that are wafer scale or larger, as discussed below) In the routingbased mesh, we can handle local defects by algorithmic means as follows. Each node shall contain 4 additional state bits, indicating ....

C.P. Schnorr, A. Shamir, An Optimal Sorting Algorithm for Mesh Connected Computers, Proceedings 16th ACM Symposium on Theory of Computing, 255-263, 1986

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C. P. Schnorr and A. Shamir, An optimal sorting algorithm for mesh connected computers. Proc. 18th Annual ACM Symposium on Theory of Computing. 1986, pp. 255--263.

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Schnorr, C. P. and Shamir, A.: An optimal sorting algorithm for mesh-connected computers, Proc. 18-th ACM Symp. on Theory of Computing, 255-263(1986).

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C. Schnorr and A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Proc. 1996.

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C.P. Schnorr and A. Shamir, An optimal sorting algorithm for mesh-connected computers, in: Proceedings 18-th ACM Symp. on Theory of Computing, (1986) 255-263.

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C.P. Schnorr and A. Shamir, An optimal sorting algorithm for mesh-connected computers, in: Proc. 18-th ACM Symp. on Theory of Computing, (1986) 255-263.

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Schnorr,C.P., and Shamir,A., "An Optimal Sorting Algorithm for Mesh Connected Computers, " Proc. ACM Symposium on Theory of Computing, 1986, pp.255-263.

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C. Schnorr and A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Symposium on the Theory of Computation, pages 255#263, 1986.

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