Ajtai, M., Koml os, J., and Szemer edi, E. Sorting in c log n parallel steps. Combinatorica 3, 1 (1983), 1-19.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from a given range, such as addresses in memory or destinations in an interconnection network. Aspnes, Herlihy, and Shavit [5] offered a new approach to solving such problems by introducing counting networks, a class of networks that can be used to count. Counting networks, like sorting networks [6, 4, 11], are constructed from simple two input, two output computing elements called balancers, connected to one another by wires. A process accesses the counting network by issuing tokens. A counting network counts any number of input tokens even if they arrive at different times. The tokens may be ....

....they prove the existence of a counting network of depth #######, that matches the lower bound ## ##. Klugerman [19] extends this result and presents a polynomial time method to construct an ##### ## depth counting network. All the above constructions use as a building block the AKS sorting network [4] whose depth expression ##### ## hides huge constants and subsequently all these networks are impractical. 2.3 Linearizability Herlihy, Shavit, and Waarts [16] defined the class of linearizable counting networks. These networks assure that the order of the values returned by the network reflect ....

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AJTAI,M.,KOML OS,J.,AND SZEMER EDI,E.An### ### ## sorting network. Combinatorica 3 (1983), 1--19.

....from a given range, such as addresses in memory or destinations in an interconnection network. Aspnes, Herlihy, and Shavit [5] offered a new approach to solving such problems by introducing counting networks, a class of networks that can be used to count. Counting networks, like sorting networks [6, 4, 11], are constructed from simple two input, two output computing elements called balancers, connected to one another by wires. A process accesses the counting network by issuing tokens. A counting network counts any number of input tokens even if they arrive at different times. The tokens may be ....

....prove the existence of a counting network of depth O(logw) that matches the lower bound und w) Klugerman [19] extends this result and presents a polynomial time method to construct an O(log w) depth counting network. All the above constructions use as a building block the AKS sorting network [4] whose depth expression O(log w) hides huge constants and subsequently all these networks are impractical. 2.3 Linearizability Herlihy, Shavit, and Waarts [16] defined the class of linearizable counting networks. These networks assure that the order of the values returned by the network reflect ....

[Article contains additional citation context not shown here]

AJTAI, M., KOML OS, J., AND SZEMER EDI, E. An O(n log n) sorting network. Combinatorica 3 (1983), 1--19.

....to fail, and later possibly revive and proceed (in an undetectable manner) would also sort under wait free assumptions. It is possible to convert any PRAM algorithm to work in this failure model. However such transformations are expensive. One might start with an O(log N) sorting algorithm [2, 7, 11] and apply a transformation technique which simulates a reliable PRAM on a faulty one. This idea was first introduced by Kanellakis and Shvartsman in [22] and later improved upon by Kedem et al. 23] Both of these results are for the failstop model. In the general asynchronous model the results ....

Ajtai, M., Koml' os, J., and Szemer' edi, E. An O(n log n) sorting network. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (Boston, Massachusetts, 25--27 Apr. 1983), pp. 1--9.

....Theorem 2. 7 Any Omega or Inverse Omega permutation can be off line routed on the MIMD G Theta H network in T (G) T (H) steps [330] Expander graphs became increasingly important in search for asymptotically optimal permutation routing and sorting networks [127] The AKS sorting network [3, 4] and the multibutterfly are the only two known bounded degree networks which can sort in O(log N) bit steps with deterministic algorithms [200, 248] Recent proof that multibutterflies can sort is based on a constant load, congestion and dilation embedding of an N node AKS network onto a 3n 2 ....

Ajtai, M., Koml'os, J., and Szemer'edi, E. Sorting in c log n parallel steps. Combinatorica. 3, 1983, pp. 1--19.

....intermediate destinations prevent worst case behavior in butterfly routing. Instead of using randomness on line, results involving expanders utilize the random like connections of the network to obtain good worst case performance from deterministic routing algorithms. The AKS sorting network [1] used expander graphs to demonstrate that sorting could be done in parallel in c log N steps, for a sufficiently large value of c. More practical work on using expanders for routing has centered on the multibutterfly network studied in [16] and [13] In a butterfly network, the each bit of the ....

Ajtai, M., Komlos, J., and Szemeredi, E. Sorting in c log n parallel steps. Combinatorica. 3, (1983), 1-19.

....AND PETER ROSSMANITH complementing circuit of Borodin et al. 1989) can be replaced by vulnerable unbounded OR gates, i.e. all of them have at most one input evaluating to one. Moreover, the so called THRESHOLD gates additionally needed there can be replaced by monotone NC 1 circuits (Ajtai et al. 1983; Borodin et al. 1989) thus unbounded OR gates are not necessary in this case. 6. Normal forms for AuxPDAs In this section, we will utilize the characterizations of AuxPDAs by semi unbounded fan in circuits to prove some normal form results. First, we deal with the restriction of push down ....

Ajtai, M., Koml'os, J., and Szemer'edi, E. (1983). Sorting in c log n parallel steps.

.... easy to see that the time processor product of the algorithms above in this case is bounded only by O(n log m=log log n) i.e. the integer sorting algorithms are more efficient than algorithms for general (comparison based) sorting, which can be done in O(log n) time using O(n log n) operations (Ajtai et al. 1983; Cole, 1988) by a factor of at most Theta(log log n) to be compared with the potential maximum gain of Theta(log n) This is true even of a more recent EREW PRAM algorithm by Rajasekaran and Sen (1992) that sorts n integers in the range 1 : n stably in O(log n log log n) time using O(n log ....

Ajtai, M., Koml' os, J., and Szemer' edi, E. (1983), An O(n log n) sorting network, in "Proceedings, 15th Annual ACM Symposium on Theory of Computing," pp. 1--9.

....2 n) time using O(n) processors. Since then there has been a considerable amount of work done for this important problem (e.g. see Bitton et al. 7] J aJ a [15] Karp and Ramachandran [16] and Reif [14] Nevertheless, it was not until 1983 that it was shown, by Ajtai, Koml os, and Szemer edi [2], that one can sort in O(log n) time with an O(n log n) sized sorting network (see also Paterson [23] In 1985 Leighton [20] extended this result to show that one can produce an O(n) node bounded degree network capable of sorting n numbers in O(log n) steps. One drawback of these algorithms, ....

....Our method for this problem also runs in O(log n) time using an optimal O(n) number of processors. We leave open the following questions: ffl Can one solve the set expression evaluation problem optimally as a circuit (say, by extending the sorting network of Ajtai, Koml os, and Szemer edi [2]) ffl What is the complexity of sorting on a CRCW PPM ....

Ajtai, M., Koml' os, J., and Szemer' edi, E. Sorting in c log n parallel steps. Combinatorica 3 (1983), 1--19.

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Ajtai, M., Koml os, J., and Szemer edi, E. Sorting in c log n parallel steps. Combinatorica 3, 1 (1983), 1-19.

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Ajtai, M., Koml'os, J., and Szemer'edi, E. (1983). Sorting in c log n parallel steps. Combinatorica, 3:1--19.

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AJTAI, M., KOML OS, J., AND SZEMER EDI, E. 1983. Sorting in c log n parallel steps. Combinatorica 3, 1, 1--19.

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