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

.... log n) time, by counting the number of inversions in the permutation , using a straightforward tree insertion procedure [183] Moreover, we can implement this inversion counting procedure by a parallel sorting algorithm that takes O(log n) parallel steps and uses O(n) processors (e.g. the one in [29]) Hence, we can count the number of inversions in O(n log n) time sequentially or in O(log n) parallel time using O(n) processors. Plugging these algorithms into the parametric searching paradigm, we obtain an O(n log n) time algorithm for the slope selection problem. 2.3 Improvements and ....

....only by an additive logarithmic term, which leads to the improvement stated above. An ideal setup for Cole s improvement is when the parallel algorithm is described as a circuit (or network) each of whose gates has a constant fan out. Since sorting can be implemented efficiently by such a network [29], Cole s technique is applicable to problems whose decision procedure is based on sorting. Cole s idea therefore improves the running time of the slope selection algorithm to n) Note that the only step of the algorithm that depends on is the sorting that produces . The subsequent ....

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

....subjects. By the early eighties Ramsey type theorems scattered around in di#erent fields were put together to form Ramsey Theory. About the same time, capitalizing on the maturity of the subject theoretical computer science started to profit from it, initiated perhaps by Ajtai Komlos Szemeredi s [3, 4, 5] and Yao s [252] influential papers. Since then Ramsey theory has been applied in many di#erent ways in theoretical computer science and these have not been put together so far. Most of these applications are using existing theorems, but there are also papers, mostly by Alon [6, 7, 8] where new ....

....where 0 a b n, then we call the graph G (n, a, b) expanding. A superconcentrator is an expander graph with the additional properties of being acyclic and from any r inputs to any r outputs there are r vertex disjoint paths. Expander graphs were used by Ajtai, Komlos and Szemeredi (1983)[5] to establish an O(log n) upper bound for the complexity of parallel sorting networks. Superconcentrators or expanding graphs with small number of edges are also important in the construction of graphs with special connectivity properties (see Chung [67] in the study of lower bounds (see Valiant ....

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

....with Cole s weighted median strategy [10] Our sweep algorithm can be parallelized to run in O(log n) steps on O(n ) processors. Thus, the total time is O(n n) in general, and O(n log n) in the plane. The parametric search technique requires the construction of an AKS sorting network [3] with O(n ) inputs, one for each processor used by the parallel implementation of the fixed parameter algorithm. This network can be built in time and space O(n log n) 3] Lemma 5.3. If a point p is in the minimum circumradius k point set, then the set is contained in the O(k) rectilinear ....

....and O(n log n) in the plane. The parametric search technique requires the construction of an AKS sorting network [3] with O(n ) inputs, one for each processor used by the parallel implementation of the fixed parameter algorithm. This network can be built in time and space O(n log n)[3]. Lemma 5.3. If a point p is in the minimum circumradius k point set, then the set is contained in the O(k) rectilinear nearest neighbors of p. Proof: Let R be the optimal circumradius. The minimum circumradius set is contained in an axis aligned hypercube of width 4R,centered at p. This ....

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

....and high authentication probability independent of the graph size. An expander graph has the property that every subset of the vertices has many neighbors. Expander graphs enjoy wide use in computer science; a very incomplete list of applications includes network constructions [FFP88] sorting [AKS83, Pip87] complexity theory [Val76] cryptography [GIL 90] and pseudorandomness [AKS87] We consider two type of expanders: bipartite expanders and ordinary expander graphs. We use bipartite expanders in our construction of authentication graphs and ordinary expander graphs in our construction ....

M. Ajtai, J. Komlos, and E. Szemeredi. Sorting in c log n Parallel Steps. Combinatorica, 3:1-19, 1983.

....sorting network and odd even merge sorting network [4, 5] have cost O(p log p) and depth O(log p) where p is the network I O size. The time performance of a sorting network is the number of parallel steps performed, which is usually the depth of the network. Ajtai, Koml os and Szemer edi [1] proposed a sorting network, commonly called the AKS sorting network, of I O size p, depth O(log p) and cost O(p log p) Later, Leighton [13] and Paterson [21] developed comparator based sorting networks of I O size p, cost O(p) and depth O(log p) that sort p elements in O(log p) time. The AKS ....

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

....Thinking Machines CM5 1. Introduction. Sorting of numeric or alphabetic data is required in many computing applications. Knuth [8] gives several examples in his classic work on serial sorting algorithms. Many papers have discussed the task of sorting on parallel computers. See, for example, [1, 2, 12]. Most of these papers have dealt with the problem from a theoretical point of view, neglecting many issues that are important in a practical implementation of a parallel sorting algorithm [4, 10, 14] This paper describes a practical parallel sorting algorithm which is suitable for efficient ....

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

....part by NSF grant CCR 9820806 and by a Guggenheim Fellowship. 2 Jin Yi Cai Gabber Galil proof also has the added advantage of being relatively elementary. We will follow the proofs of [10] closely. There is an extensive literature on expanders and their applications to the theory of computing [1][2] 4] 5] 6] 8] 9] 14] 15] It was realized that expansion properties are closely related to the second largest eigenvalues of the graph #(G) see [6] and for d regular graphs the gap between d and #(G) provides estimates for both upper and lower bound for the expansion constant. The best ....

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

....[8] Their depth can be improved by a constant by interrupting the recursive construction early and substituting the best known sorting network on a small fixed number of inputs. For extremely large n the asymptotically optimal O(log n) depth sorting network of Ajtai, Koml os and Szemer edi [1, 2] has superior depth. It was known that no nine input sorting network of depth five can exist, but the best that had been obtained was depth seven. These bounds remained unimproved for over fifteen years. We undertook to determine by exhaustive search whether a nine input sorting Research ....

....2 downto 1 do 8. begin 9. swap item[i] with item[n Gamma 1] 10. match(item,n Gamma 2) 11. end; 12. temp : item[n Gamma 1] 13. for i : n Gamma 1 downto 2 do 14. item[i] item[i Gamma 1] 15. item[1] temp 16. end 17. end; The perfect matchings of the n values in item[1] item[2]; item[n] are processed with a call to match(item,n) The correctness of the procedure can be verified by induction on even n. The hypothesis is certainly true for n = 2. Now suppose that the hypothesis is true for match(item,n Gamma 2) In line 6, all perfect matchings of n items with a ....

M. Ajtai, J. Koml'os, and E. Szemer'edi. Sorting in c log n parallel steps. Combinatorica, 3:1--48, 1983.

....and lower bounds. 1.5 Unbalanced expanders with near optimal expansion An expander graph has the property that every not toolarge subset of the vertices has many neighbors, relative to its degree. Expanders have had numerous applications in computer science: network constructions [6] sorting [1, 16], complexity theory [30] cryptography [8] and pseudorandomness [2] Many of these applications require bipartite graphs, where only subsets on one side are required to expand. Definition 1.5.1 (expander) A bipartite graph G = V1 ; V2 ; E) is (K; c) expanding if for every A V1 of cardinality ....

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

....point of C r i (p i ) The third step of Calc Depth sorts the intersection points of C r i (p i ) with the other circles of that radius. Here we use (a serial simulation of) a parallel sorting scheme that uses O(n) processors and O(log n) parallel steps, such as the scheme of Ajtai et al. [4]. Each parallel step of the sorting attempts to perform O(n) comparisons, each asking for the relative order along C r i (p i ) of two of its intersections with, say C r i (p j ) and C r i (p k ) It is easy to see that, as we vary the radius of these circles (in the range where all 4 ....

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

....n is prime, we have oe n = id implies oe is a long cycle or oe = id. The analog of Lemma 1.3.2 is also straightforward. Therefore we have a (positive) bias in the number of trivial components in random element g n 2 G. Now, a general theory (the Ajtai Koml os Szemer edi sorting network in [AKS]) shows how to construct a short monotone Boolean circuit which detects the bias. The problem is to construct a short word w from the circuit. Basically we need to find probabilistic simulation of AND and OR by group operations. Formally, let H be a group and g 2 H. Consider the predicate E(g) ....

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

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

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

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

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

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

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

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

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M. Ajtai, J. Kolmos and E. Szermeredi, \Sorting in c log n parallel steps", Combinatorica 3, 1983, 1-19.

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

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M. Ajtai, J. Koml'os, and E. Szemer'edi. Sorting in c log n parallel steps. Combinatorica, 3:1--19, 1983. 16

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

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

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M. Ajtai, J. Kolmos and E. Szermeredi, \Sorting in c log n parallel steps", Combinatorica 3, 1983, 1-19.

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

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