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Two Lower Bounds In Asynchronous Distributed Computation
https://academiccommons.columbia.edu/catalog/ac:142126
Duris, Pavol; Galil, Zvi
10.7916/D8RX9M58
Mon, 19 Jun 2017 14:39:03 +0000
We 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 Î©(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 Î©(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 science
zg1
Computer Science
Reports

Lower Bounds on Communication Complexity
https://academiccommons.columbia.edu/catalog/ac:140742
Duris, Pavol; Galil, Zvi; Schnitger, Georg
10.7916/D8RN3GT3
Mon, 19 Jun 2017 13:34:55 +0000
We 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 kround protocol requires exchanging Ω(n1/2/k4log3n) bits if any (k1) 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 kround protocol, and any (k1)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 (kround) protocol recognizing L can be transformed to a (kround) 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 (k1)round and kround 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 science
zg1
Computer Science
Reports

Two Nonlinear Bounds for OnLine Computations
https://academiccommons.columbia.edu/catalog/ac:140694
Duris, Pavol; Galil, Zvi; Paul, Wolfgang; Reischuk, Ruediger
10.7916/D87S7WRD
Mon, 19 Jun 2017 13:34:28 +0000
We prove the following lower bounds for online 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 science
zg1
Computer Science
Reports