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Academic Commons Search Resultsen-usTwo Lower Bounds In Asynchronous Distributed Computation
https://academiccommons.columbia.edu/catalog/ac:142126
Duris, Pavol; Galil, Zvihttp://hdl.handle.net/10022/AC:P:11874Fri, 02 Dec 2011 14:09:40 +0000We introduce new techniques for deriving lower bounds on the message complexity in asynchronous distributed computation. These techniques combine the choice of specific patterns of communication delays and crossing sequence arguments with consideration of the speed of propagation of messages, together with careful counting of messages in different parts of the network. They enable us to prove the following results, settling two open problems: An Ω(n log* n) lower bound for the number of messages sent by an asynchronous algorithm for computing any nonconstant function on a bidirectional ring of n anonymous processors. An Ω(n log n) lower bound for the average number of messages sent by any maximum finding algorithm on a ring of n processors, in case n is known.Computer sciencezg1Computer ScienceTechnical reportsLower Bounds on Communication Complexity
https://academiccommons.columbia.edu/catalog/ac:140742
Duris, Pavol; Galil, Zvi; Schnitger, Georghttp://hdl.handle.net/10022/AC:P:11575Wed, 26 Oct 2011 10:48:20 +0000We prove the following four results on communication complexity: 1) For every k ≥ 2, the language Lk of encodings of directed graphs of out degree one that contain a path of length k+1 from the first vertex to the last vertex and can be recognized by exchanging O(k log n) bits using a simple k-round protocol requires exchanging Ω(n1/2/k4log3n) bits if any (k−1)- round protocol is used. 2) For every k ≥ 1 and for infinitely many n ≥ 1, there exists a collection of sets Lnk @@@@ {0,1}2n that can be recognized by exchanging O(k log n) bits using a k-round protocol, and any (k−1)-round protocol recognizing Lnk requires exchanging Ω(n/k) bits. 3) Given a set L @@@@ {0,1}2n, there is a set L@@@@{0,1}8n such that any (k-round) protocol recognizing L@@@@ can be transformed to a (k-round) fixed partition protocol recognizing L with the same communication complexity, and vice versa. 4) For every integer function f, 1 ≤ f(n) ≤ n, there are languages recognized by a one round deterministic protocol exchanging f(n) bits, but not by any nondeterministic protocol exchanging f(n)−1 bits. The first two results show in an incomparable way an exponential gap between (k−1)-round and k-round protocols, settling a conjecture by Papadimitriou and Sipser. The third result shows that as long as we are interested in existence proofs, a fixed partition of the input is not a restriction. The fourth result extends a result by Papadimitriou and Sipser who showed that for every integer function f, 1 ≤ f(n) ≤ n, there is a language accepted by a deterministic protocol exchanging f(n) bits but not by any deterministic protocol exchanging f(n) − 1 bits.Computer sciencezg1Computer ScienceTechnical reportsTwo Nonlinear Bounds for On-Line Computations
https://academiccommons.columbia.edu/catalog/ac:140694
Duris, Pavol; Galil, Zvi; Paul, Wolfgang; Reischuk, Ruedigerhttp://hdl.handle.net/10022/AC:P:11548Tue, 25 Oct 2011 13:54:45 +0000We prove the following lower bounds for on line computation. 1) Simulating two tape nondeterministic machines by one tape machine requires n(n log n) time. 2) Simulating k tape (deterministic) machines by machines with k pushdown stores requires n(n log 1/(k+l) n) time.Computer sciencezg1Computer ScienceTechnical reports