D. Dolev, C. Dwork, N. Pippenger, and A. Wigderson, "Superconcentrators, Generalizers, and Generalized Connectors with Limited Depth." In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1983, pp. 42-51.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that such a circuit must have the structure of a special type of graph called a prefix graph. They then prove upper and lower bounds on the size of prefix graphs of bounded depth [6, 7] We demonstrate a connection between prefix graphs and another family of graphs called weak superconcentrators [11]. Using this idea, we present a simple proof that improves the lower bound of [7] and shows that the construction in [6] is optimal. Dolev,Dwork, Pippenger and Wigderson [11] showed a lower bound on weak superconcentrators of bounded depth. Our lower bound for oblivious chaining algorithms is ....

....depth [6, 7] We demonstrate a connection between prefix graphs and another family of graphs called weak superconcentrators [11] Using this idea, we present a simple proof that improves the lower bound of [7] and shows that the construction in [6] is optimal. Dolev,Dwork, Pippenger and Wigderson [11] showed a lower bound on weak superconcentrators of bounded depth. Our lower bound for oblivious chaining algorithms is obtained by interpreting such algorithms as graphs and using the techniques of [11] to analyze their properties. It is worth noting that there are chaining algorithms whose ....

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D. Dolev, C. Dwork, N. Pippenger and A. Wigderson, "Superconcentrators, Generalizers and Generalized Connectors with Limited Depth", Proc. of the 15th ACM STOC, 1983.

....algorithm for parenthesis matching with nesting level. Without the nesting level information, PARITY can be reduced to this problem; hence it requires Omega Gammaequ n=log log n) time [5] Our lower bound argument for chaining is based on the work of Dolev, Dwork, Pippenger and Wigderson [15], who used a clever and versatile averaging argument to show that a weak superconcentrator with a linear number of edges must have Omega Gamma ff(n) depth. Chaudhuri [11] adapted their method to obtain the lower bound in the oblivious case. Our proof is a further extension of this method. 1.2 ....

....bounds for chaining, prefix maxima, range maxima and parenthesis matching on small domains. The bounds are tight for the chaining problem and parenthesis matching, but we do not know about the other two problems. Our work extends the techniques developed in Dolev, Dwork, Pippenger and Wigderson [15] and Chaudhuri [11] The techniques used in this paper have since been sharpened and applied to several other problems. In Chaudhuri [13] they have been used to obtain Omega Gammabta log n) lower bounds for the problem of approximate compaction, which is the problem of relocating a distinguished ....

D. Dolev, C. Dwork, N. Pippenger and A. Wigderson, "Superconcentrators, Generalizers and Generalized Connectors with Limited Depth", Proc. of the 15th ACM STOC, 1983, 42--51.

.... for depth 2 is O(n 3=2 ) Mes] and for depth 2k 1 are of size O(n (k 3) k 2) Alo1] On the other hand, non explicit constructions were known of size O(n log 2 n) for depth 2 [Pip2] and O(n(k; n) for depth 2k, k 2, for an extremely slowly growing (k; n) e.g. 2; n) log n) [DDPW]. Here, we give an explicit construction for depth 2 of size O (n) This is our biggest improvement: a factor of O ( p n) We use this construction to give the first explicit construction of a linear sized superconcentrator with sublogarithmic depth (namely, depth (log n) 2=3 o(1) The ....

D. Dolev, C. Dwork, N. Pippenger, and A. Wigderson, "Superconcentrators, Generalizers, and Generalized Connectors with Limited Depth." In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1983, pp. 42-51.

....k there are k vertex disjoint paths from I 0 to O 0 in G. Let c d (n) denote the size of the of the smallest n superconcentrator of depth d. Determining the functions c d for various values of d, as well as finding explicit small and shallow superconcentrators has been a major object of study [Pi78, Pi82, DDPW]. We shall need the following two upper bounds on the size of superconcentrators of depth 2, the first being nonconstructive and the second explicit. Theorem 3 [Pi82] c 2 (n) O(n(log n) 2 ) Theorem 4 [WZ93] There is a polynomial time algorithm which for every n outputs an ....

D. Dolev, C. Dwork, N. Pippenger, and A. Wigderson, "Superconcentrators, Generalizers, and Generalized Connectors with Limited Depth," Proc. of the 15th STOC, pp. 42-51, 1983. 11

.... for depth 2 is O(n 3=2 ) Mes] and for depth 2k 1 are of size O(n (k 3) k 2) Alo1] On the other hand, non explicit constructions were known of size O(n log 2 n) for depth 2 [Pip2] and O(n(k; n) for depth 2k, k 2, for an extremely slowly growing (k; n) e.g. 2; n) log n) [DDPW]. Here, we give an explicit construction for depth 2 of size O (n) This is our biggest improvement: a factor of O ( p n) We use this construction of a depth 2 superconcentrator to give the first explicit construction of a linear sized superconcentrator with sublogarithmic depth (namely, ....

....non explicit constructions: Theorem 4.9 There exist linear sized n superconcentrators with depth log (k; n) for any constant k, where (k; n) is the inverse of Ackermann s function (so e.g. 2; n) log n) Proof: Use the non explicit analog of Lemma 4. 7 and the superconcentrators given in [DDPW]. 2 5 Concentrators and Non Blocking Networks In this section we show how similar ideas can be used to explicitly construct non blocking networks. Before we do this, we define wide sense nonblocking generalized connectors, following [FFP] A route in a network is a directed path from an input ....

D. Dolev, C. Dwork, N. Pippenger, and A. Wigderson, "Superconcentrators, Generalizers, and Generalized Connectors with Limited Depth," 15th STOC, 1983, pp. 42-51.

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