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 ....

[Article contains additional citation context not shown here]

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. ....

[Article contains additional citation context not shown here]

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 ....

[Article contains additional citation context not shown here]

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 ....

[Article contains additional citation context not shown here]

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.

....algorithms [1, 5, 3] were designed for snake like row major indexings. In many cases it is desirable to have the packets in the more natural row or column major order. Furthermore, sorting in snake like order is unsuited for scattering. In the one packet model considered by Schnorr and Shamir [7], the best known upper bound for row major sorting is higher than for sorting in snake like rowmajor 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 ....

....row major order. In [3] a deterministic version is presented. 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 [7]. It takes 4 Delta n o(n) steps. The first near optimal algorithm for k k sorting was discovered by Kaufmann and Sibeyn [2] Then in [5] by Kunde and in [3] deterministic versions of this randomized algorithm were described. All use blocked snake like row major indexings. We present the ....

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

....This implies that after each iteration of Step 3.a, at most one packet has to be routed to a destination in Step 3.b. Hence, the routing in Step 3.b is a partial 1 1 routing within the squares. For the routing and sorting operations in shortqroute we apply the algorithm of Schnorr and Shamir [19]. By this algorithm, the packets of an n Thetan are sorted in 3 Delta n O(n 3=4 ) steps in snake like order. This result does not require that the connections act as comparators when we accept queue size two. Partial permutation routing with short queues is slightly harder: Lemma 11 On an n ....

....in 4 Delta n O(n 3=4 ) steps and with working queue size two. Proof: We use a variant of the algorithm of Kunde [10] The mesh is divided in n=2 Theta n=2 submeshes. In every submesh the packets are sorted in snake like columnmajor order on their destination columns with the algorithm of [19]. This takes 3 Delta n=2 O(n 3=4 ) steps. In n=2 steps this is turned into a column major order. Subsequently the packets are routed row first in 2 Delta n steps to their destinations. 2 Theorem 8 shortqroute routes 1 k permutations in 5 Delta p k Delta n 6 Delta n O(k 5=8 Delta ....

[Article contains additional citation context not shown here]

Schnorr, C.P., A. Shamir, `An Optimal Sorting Algorithm for Mesh Connected Computers,' Proc. 18th Symposium on Theory of Computing, pp. 255263, ACM, 1986.

....of h h problems when each processor initially and finally contains exactly h packets. For two dimensional n Theta n meshes without diagonals 1 1 problems have been studied for more than twenty years. Several 1 1 sorting algorithms exist for buffer size 1. The fastest ones need 3n o(n) steps [12, 15, 17]. For buffer size 2 we showed how to solve the 1 1 sorting problem deterministically in 2:5n o(n) transport steps [6] Recently, Kaklamanis and Krizanc [1] presented a randomized algorithm (needing constant buffer size) that sorts in only 2n o(n) steps with high probability. For 1 1 routing ....

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

....in the snake like ordering of the mesh s processors. However, as Scherson et al. note in [8] the performance of such an algorithm is also Omega Gamma N) in the worst case substantially inferior to the 3 p N o( p N) time of asymptotically optimal sorting algorithms for the mesh, such as [9]. In fact, in the worst case, a bubblesort is generally outperformed even by simple sub optimal algorithms, such as ShearSort [8] which has a running time of Theta( p N lg N ) One exception is the algorithm of [10, 3] which sketches a short periodic sorting algorithm that does achieve time ....

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

....indexings. 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 [12], the lower 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. From the results of this paper it follows that in this model, sorting in row major order is not substantially harder ....

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.

....computers with mesh and torus topologies have been very successful in offering high computational power together with high speed processor interconnections. Past research on parallel algorithms has resulted in a well established model for computing on lower dimensional meshes and tori [TK77] SS88] Kun91a] A large number of algorithms designed and analyzed within that model can be found in the literature. Unfortunately, most of these algorithms lack the simplicity of their counterparts designed for the hypercube models. Most mesh sorting algorithms divide the surface of the mesh into ....

....model. Thompson and Kung show that the complexity of sorting on an n Theta n mesh is O(n) and give a 6n o(n) algorithm [TK77] Unfortunately for practical machine sizes, o(n) hides significantlow order terms. Further we have looked into the optimal 3n O(n 3 4 ) algorithm bySchnorr andShamir [SS88] The algorithm is provably optimal in the sense that it matches a very general lower bound of the 1 1 sorting. This bound, proven in [SS88] applies to a general MIMD model. Because of their simplicity we also considered the sub optimal algorithms ShearSort and RevSort described by [SS88] The ....

[Article contains additional citation context not shown here]

C. P. Schnorr and A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Proceedings of the 20thAnnual ACM SymposiumonTheory of Computing, pages 255--263, 1988.

....packet among the packets from a submesh can be determined up to N=M positions. There are N=M submeshes, so inac(N; M ) N=M ) 2 : 2) The inaccuracy should equal the number of splitters, and thus we should take MD1 = N 2=3 : This idea is the basis of [11, 12] and was already present in [13, 18]. In a deterministic algorithm, the routing in Step 5 of BASIC SORT can be achieved by performing two suitable unshuffles. This yields Theorem 2 [19] For deterministic k k sorting, for all k 4 Delta d, T D1 (k; d; n) k Delta n=2 O(k 1 Gamma 1 3 Deltad Delta n 2=3 ) 4 Subsplitter ....

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

....times every node can hold only one packet. Under this assumption one can derive interesting lower bounds using the so called joker zone arguments . Lemma 3. 10 Sorting on an MIMD mesh in the one packet model requires at least 3n Gamma O(n 1=2 ) steps for sorting in (snake like) row major order [156, 278]. Deeper methods are required for finding lower bounds for all indexing schemes. Lemma 3.11 There is no indexing scheme, with respect to which sorting in the one packet model can be performed in less than 2:27 n steps [97] Thompson and Kung [309] presented the first O(n) sorting algorithm. ....

....are required for finding lower bounds for all indexing schemes. Lemma 3.11 There is no indexing scheme, with respect to which sorting in the one packet model can be performed in less than 2:27 n steps [97] Thompson and Kung [309] presented the first O(n) sorting algorithm. Schnorr and Shamir [278] designed the first optimal algorithm. Theorem 3.12 Sorting on an MIMD mesh in one packet model with respect to the snakelike row major indexing can be performed in 3n O(n 3=4 ) steps [196, 278, 309] On a m Theta n mesh the algorithm takes m 2n o(n m) steps. Multi Packet Model. If we ....

[Article contains additional citation context not shown here]

Schnorr, C., and Shamir, A. An optimal sorting algorithm for mesh connected computers. Proc. 18th ACM Symp. Theory Comput. 1986, pp. 255--263.

....greedy algorithm, which gives optimal routing time. The problem that arises is that if the recursion has to stop early, that then long queues may build up during the greedy routing. Alternatively, any routing algorithm can be applied that requires O(m) steps, for example the sorting algorithm of [11]. This results in T = 2 Delta n O(1) but Q can be bounded. An important factor that determines the maximal queue size is the time T s for performing the scattering in s Theta s squares. For the algorithm to be correct it is required that T s m. If T s ff Delta s, for some constant ff, then ....

....order, which is fast for all sizes of the mesh. In addition this algorithm 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 [13] points out the algorithm RevSort of [11] as an exception, but this algorithm requires more than 2 1 = 2 Delta log log s 3 Delta s steps for sorting in snake like order, and is not uni axial. Our algorithms borrow some ideas from the recursive algorithm of Lang et al. 6] The following results are also interesting for their own sake; ....

[Article contains additional citation context not shown here]

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.

....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 cycle in subchains, and handing out the packets of the subchains in a fair way over the subchains. This idea goes back on [11]. 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.

....sorting algorithms: sorting is reduced to routing once the destination of the packets are (approximately) known. A sorting problem requires at least as many steps as the corresponding routing problems. Many algorithms have been proposed in the literature for 1 1 sorting under various mesh models [30, 11, 28, 18, 7]. Park and Balasubramanian [22] proved that 2 2 sorting can be completed in (3 o(1) Delta n steps on two dimensional meshes. Only recently the first non trivial algorithms for multi packet sorting have been developed. Kunde [13] showed that k k sorting can be done in (1 o(1) Delta k Delta ....

....there are about N 2=3 splitters, and the estimates of the ranks are accurate up to O(log 1=2 N Delta N 2=3 ) 2.8 Routing and Sorting on Submeshes In our sorting algorithm we need routing and sorting routines within submeshes. There are several algorithms, for example a modification of [28], that achieve the following Lemma 3 For any k 0, any k k routing or sorting problem within arrays of size L Theta L, can be solved on in O(k Delta L) steps with maximal queue size k O(1) 3 One Dimensional Routing We derive important results concerning routing on one dimensional arrays. ....

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.

....et al. 18] 1 1 sorting 1:2n 2n Kaklamanis and Krizanc [3] Kaufmann et al. 6] 4 4 sorting 1:6n 4n Kunde [9] 8 8 sorting 1:86n 4n Kunde [10] Kaufmann et al. 6] 12 12 sorting 2n 6n Kunde [10] Kaufmann et al. 6] can store only one packet at each time. The fastest ones need 3n o(n) steps [19, 22, 27]. For buffer size 2 the 1 1 sorting problem can be solved deterministically in 2:5n o(n) transport steps [9] Kaklamanis and Krizanc [3] presented a randomized algorithm (with constant buffer size) that sorts in only 2n o(n) steps with high probability. Using derandomization techniques, this ....

.... [28] considered them as early as in 1981 and others maybe even earlier , data concentration was introduced a short time ago [9] The first use of concentration was to solve the 1 1 sorting problem in 2:5n o(n) steps, while the previous best known bound without using concentration was 3n o(n) [22]. Today an optimal 2n o(n) steps algorithm is known [3, 6] We solved 12 12 sorting in optimal time. So h h sorting with h 12 is a candidate for speed up via concentration. Let us start with the fastest algorithm for grids in this paper, an algorithm for the 1 1 routing problem. Theorem ....

[Article contains additional citation context not shown here]

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, 1986.

....to travel along each sequence and color the first n p k ordinary packets black (step 4) We will see later how step 4 takes place. 3.2 Time Analysis We first analyze algorithm Color. The sorting step can be performed in 3n o(n) routing steps using the sorting algorithm of Schnorr and Shamir [7]. Let us now consider the time (4,1) 4,1) 4,1) 4,1) 2,5) 2,5) 2,5) 2,5) 2,5) 2,5) 2,3) 2,3) 1,2) 1,2) 1,1) 4,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,2) 1,2) 1,2) 1,2) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,1) 1,2) 1,2) 2,3) 2,3) 2,5) 2,5) 2,5) ....

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

....a method to obtain deflection worm routing algorithms based on store and forward packet routing algorithms. However, the packet routing algorithms used in their method were restricted to use queues of at most four packets per processor. By employing the sorting algorithm of Schnorr and Shamir [3] they obtained an O(k 2:5 n) step deflection worm routing algorithm for routing permutations. They also presented an O(k 1:5 n) step offline algorithm. Newman and Schuster also observed that better results for routing permutations could be obtained if fast algorithms for 1 Gamma h routing ....

C. P. Schnorr and A. Shamir, "An optimal sorting algorithm for mesh connected computers", in Proceedings of the 18th Ann. ACM Symposium on Theory of Computing (Berkeley, CA). ACM, 1986, pp. 255--263, ACM Press.

....without sub buses where only nearest neighbor communication is possible. In odd even transposition sort, processors swap values with their adjacent processors until order is achieved. It is commonly used as a subcomponent of optimal sorting algorithms for a two dimensional array of processors [15, 16, 17]. In a similar way, our one dimensional sorts could be used as subcomponents of these same algorithms for application on a two dimensional sub bus array. There are a number of architectures related to the sub bus mesh array, namely, the mesh array with fixed buses and the many flavors of ....

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

....no hidden costs. Most of these practical SIMD sorting algorithms are based on recursive merging; Kunde has conjectured [16] that the lower bound on this approach is 4:5n interchanges, or 9n routing steps, bounding the approach to a factor of 2.25 from optimal. MIMD Sorting Schnorr and Schamir [32] have developed a 3n o(n) MIMD sorting algorithm, together with a matching lower bound. MIMD Routing Since optimal online MIMD routing algorithms exist, offline algorithms are not considered. In the MIMD model, PEs are assumed to be very powerful: usually at least one priority queue operating ....

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

....k o(k) flits) queues on k Gamma k routing and in k 2 n 3 2 n O( p kn log n) steps with O(k) flits) queues on cut through routing, whenever k 8. As far as k Gamma k sorting on the mesh is concerned, several optimal algorithms can be found in the literature for 1 Gamma1 sorting [30, 8, 29, 16, 9, 14] (all of which have a run time of 3n o(n) Recently Kaklamanis, Krizanc, Narayanan, and Tsantilas [5] showed that 1 Gamma 1 sorting can be accomplished within 2:5n o(n) steps and constant queue length using a randomized algorithm. Later, Kunde [11] matched this time bound with a deterministic ....

....being e O(k) as long as k 4r. 5 k Gamma k Sorting on the Mesh The problem we consider in this Section is to sort a mesh in which there are k packets to start with at each node. Many optimal algorithms have been proposed in the literature for 1 Gamma 1 sorting under various mesh models [30, 8, 29, 16]. Park and Balasubramanian [19] proved that 2 Gamma 2 sorting can be completed in 3n o(n) steps. Later Kunde [11] showed that k Gamma k sorting can be done in kn o(kn) steps, keeping the queue length as k. In this Section we show that k Gamma k sorting can be accomplished within k 2 n ....

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

....routing algorithms. Their method was general enough to work for any routing patterns, not only permutations. However, the packet routing algorithms used in their method were restricted to use queues of at most four packets per processor. By employing the sorting algorithm of Schnorr and Shamir [13] they obtained an O(k 2:5 n) step deflection worm routing algorithm for routing permutations. They also presented an O(k 1:5 n) step offline algorithm. Newman and Schuster also observed that better results for routing permutations could be obtained if fast algorithms for 1 Gamma h routing ....

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

....410] Locality based algorithms have been developed for all kinds of interconnection network structures. For an introduction to this area, see e.g. 34, 123, 185, 224] The following is a short list of some of the more common network structures for which such algorithms have been produced: arrays [224, 292, 335, 360], trees and Sneptrees [165, 242, 264] pyramids [5, 284, 351] the mesh of trees structure [178, 224] fat trees [232, 233] hypercubes [70, 80, 93, 152, 153, 162, 182, 224, 266, 312, 404] the cube connected cycles architecture [224, 307] butterflies [224] and various shuffle exchange and de ....

....have optimal Theta(p) VLSI volume. The main disadvantage of arrays, for computations requiring a large amount of non local communication, is their high diameter. Although, again, they are very appropriate for implementing systolic algorithms, and others which require only local communication [224, 335, 360]. We now briefly discuss some of the properties of the remaining graphs in the table. The shuffle exchange, cube connected cycles, butterfly and de Bruijn graphs have very similar properties. We will refer only to the butterfly. Various fat tree designs have been proposed. They are all ....

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

....A further problem that comes with very large values of k is that the worst case of dlog k 1e passes becomes a more significant burden. To overcome this problem alternative grid sorting algorithms to shear sort may be used which work well for much larger values of k. For example, reverse sort[14] has a worst 7 The determination of the two conditions can be accomplished in linear time by first calculating the cumulative largest and cumulative smallest elements for each slice. case of dlog log ke for large k. For smaller k, however, it has no advantage over shear sort. It is difficult ....

C. P. Schnorr, An optimal Sorting Algorithm For Mesh Connected Computers, Proc. ACM Synmposium on the Theory of Computation, 1986

....[3] 9] approaches. The trivial greedy algorithm routes the packets to the correct column and then to the correct row in 2n Gamma 2 steps. The size of the queues, however, can be as bad as O(n) The nontrivial solutions given to this problem are based on parallel sorting algorithms [7] [8] Kunde [4] 5] was the first to use parallel sorting to obtain a deterministic algorithm that completes the routing in 2n O(n=f(n) steps with queues of size O(f(n) Later on, Leighton, Makedon and Tollis [6] derived a deterministic algorithm that completes the routing in 2n Gamma 2 steps ....

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

....implemented by using a standard deterministic algorithm for parallel prefix on the mesh. Then the embedding in step 4, which requires routing of the tree nodes, is implemented by using a standard deterministic algorithm for permutation routing on the mesh (e.g. the sorting algorithms of [10] or [9]) Note also that each of these algorithms only needs buffers of size one or two per edge. Notice that in step 4 the only tree nodes that are routed are the ones that are actually going to get embedded in the current phase. It could be easier and faster (within constant factors) to move all active ....

....of side 2l , except the top one (i.e. the one of which M is a submesh) contains between 4l 2 and 8l 2 alive nodes while the top one contains between 2l 2 and 4l 2 . Thus step 3 consists of sorting O(l 2 ) nodes on meshes of side 2l which can be done deterministically in time O(l) [9, 10]. Finally steps 2; 4 and 5 only involve movement of alive nodes by one standard submesh of side l or 2l up or down along the indexing P . This can be done in time O(l) by shifting in lockstep by one standard submesh and, then, rearranging by sorting inside each submesh. The above lemmas prove the ....

C.P. Schnorr and Adi Shamir. An optimal sorting algorithm for mesh-connected computers. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, 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 snake like 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 ....

....block 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 random intermediate ....

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

No context found.

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.

No context found.

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).

No context found.

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

No context found.

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.

No context found.

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.

No context found.

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.

No context found.

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

No context found.

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

No context found.

C. Schnorr and A. Shamir, An Optimal Sorting Algorithm for Mesh Connected Computers, Symposium on the Theory of Computation, pp. 255-263, 1986.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC