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22
A Strong Direct Product Theorem for Corruption and theMultiparty NOF Communication Complexity of Disjointness
, 2005
"... We prove that twoparty randomized communication complexity satisfies a strong direct productproperty, so long as the communication lower bound is proved by a "corruption" or "onesided discrepancy" method over a rectangular distribution. We use this to prove new n\Omega (1) lo ..."
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Cited by 12 (3 self)
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We prove that twoparty randomized communication complexity satisfies a strong direct productproperty, so long as the communication lower bound is proved by a "corruption" or "onesided discrepancy" method over a rectangular distribution. We use this to prove new n\Omega (1) lower bounds for numberontheforehead protocols in which the first player speaks once and then the other two players proceed arbitrarily. Using other techniques, we also establish an \Omega (n1/(k1)/(k 1)) lower bound for kplayerrandomized numberontheforehead protocols for the disjointness function in which all messages are broadcast simultaneously. A simple corollary of this is that general randomized numberontheforeheadprotocols require \Omega (log n/(k 1)) bits of communication to compute the disjointness function.
Multiparty Communication Complexity and Threshold Circuit Size of AC⁰
, 2008
"... We prove an nΩ(1) /2O(k) lower bound on the randomized kparty communication complexity of readonce depth 4 AC0 functions in the numberonforehead (NOF) model for up to Θ(log n) players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any AC0 fu ..."
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We prove an nΩ(1) /2O(k) lower bound on the randomized kparty communication complexity of readonce depth 4 AC0 functions in the numberonforehead (NOF) model for up to Θ(log n) players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any AC0 function for ω(log log n) players. For nonconstant k the bounds are larger than all previous lower bounds for any AC0 function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of AC0 by general MAJ ◦ SYMM ◦ AND circuits, showing that the wellknown quasipolynomial simulations of AC0 by such circuits are qualitatively optimal, even for readonce formulas of small constant depth. We also exhibit a readonce depth 5 formula in NP cc k − BPPcc k
Communication lower bounds via critical block sensitivity
 In Proc. 46th Annual ACM Symposium on Theory of Computing (STOC ’14
, 2014
"... Abstract. We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with ..."
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Abstract. We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with critical block sensitivity b, then every randomised twoparty protocol solving a certain twoparty lift of S requires Ω(b) bits of communication. Besides simplicity, our proof has the advantage of generalising to the multiparty setting. We combine these results with new critical block sensitivity lower bounds for Tseitin and Pebbling search problems to obtain the following applications. • Monotone circuit depth: We exhibit a monotone function on n variables whose monotone circuits require depth Ω(n / log n); previously, a bound of Ω( n) was known (Raz and Wigderson, JACM 1992). Moreover, we prove a tight Θ( n) monotone depth bound for a function in monotone P. This implies an averagecase hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013). • Proof complexity: We prove new rank lower bounds as well as obtain the first length–space lower bounds for semialgebraic proof systems, including Lovász– Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordström.ar
A Direct Sum Theorem for Corruption and the Multiparty NOF Communication Complexity of Set Disjointness
 IN PROCEEDINGS OF THE 20TH ANNUAL CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2005
"... We prove that corruption, one of the most powerful measures used to analyze 2party randomized communication complexity, satisfies a strong direct sum property under rectangular distributions. This direct sum bound holds even when the error is allowed to be exponentially close to 1. We use this to a ..."
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Cited by 5 (4 self)
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We prove that corruption, one of the most powerful measures used to analyze 2party randomized communication complexity, satisfies a strong direct sum property under rectangular distributions. This direct sum bound holds even when the error is allowed to be exponentially close to 1. We use this to analyze the complexity of the widelystudied set disjointness problem in the usual “numberontheforehead” (NOF) model of multiparty communication complexity.
NOFmultiparty information complexity bounds for pointer jumping
 In Proc. 31st International Symposium on Mathematical Foundations of Computer Science
, 2006
"... Abstract. We prove a lower bound on the communication complexity of pointer jumping for multiparty oneway protocols in the number on the forehead model that satisfy a certain information theoretical restriction: We consider protocols for which the ith player may only reveal information about the fi ..."
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Abstract. We prove a lower bound on the communication complexity of pointer jumping for multiparty oneway protocols in the number on the forehead model that satisfy a certain information theoretical restriction: We consider protocols for which the ith player may only reveal information about the first i + 1 inputs. To this end we extend the information complexity approach of Chakrabarti, Shi, Wirth, and Yao (2001) and BarYossef, Jayram, Kumar, and Sivakumar (2004) to our restricted version of the multiparty number on the forehead model. The best currently known multiparty protocol for pointer jumping by Damm, Jukna, and Sgall (1998) works in this model.
Allowing Each Node to Communicate Only Once in a Distributed System: Shared Whiteboard Models
"... In this paper we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed and streami ..."
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Cited by 4 (2 self)
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In this paper we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed and streaming way. When computing graphtheoretic properties, nodes become natural units for distributed computation. Links do not necessarily represent communication channels between the computing units and therefore do not restrict the communication flow. Our goal is to model and analyze the computational power of such distributed systems where one computing unit is assigned to each node. Communication takes place on a whiteboard where each node is allowed to write at most one message. Every node can read the contents of the whiteboard and, when activated, can write one small message based on its local knowledge. When the protocol terminates its output is computed from the final contents of the whiteboard. We describe four synchronization models for accessing the whiteboard. We show that message size and synchronization power constitute two orthogonal hierarchies for these systems. We exhibit problems that separate these models, i.e., that can be solved in one model but not in a weaker one, even with increased message size. These problems are related to maximal independent set and connectivity. We also exhibit problems that require a given message size independently of the synchronization model.
Separating deterministic from nondeterministic NOF multiparty communication complexity (Extended Abstract)
 IN ICALP
, 2007
"... We solve some fundamental problems in the numberonforehead (NOF) kparty communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a onesided error probability of 1/3 but which has linear communication complexit ..."
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Cited by 4 (2 self)
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We solve some fundamental problems in the numberonforehead (NOF) kparty communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a onesided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for k = n O(1) players, where n is the number of bits on each players ’ forehead. This separates the analogues of RP and P in the NOF communication model. We also show that there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential and we do not know of any explicit function which allows such separations. However, for the 3player case we exhibit an explicit function which has Ω(log log n) randomized complexity for private coins but only constant complexity for public coins. It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic kparty communication complexity does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the numberinhand model, is not complete in the NOF model under cylindrical reductions.
The NOF Multiparty Communication Complexity of Composed Functions
"... We study the kparty ‘number on the forehead ’ communication complexity of composed functions f ◦ g, where f: {0,1} n → {±1}, g: {0,1} k → {0,1} and for (x1,...,xk) ∈ ({0,1} n) k, f ◦g(x1,...,xk) = f (...,g(x1,i,...,xk,i),...). We show that there is an O(log 3 n) cost simultaneous protocol for SYM ..."
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We study the kparty ‘number on the forehead ’ communication complexity of composed functions f ◦ g, where f: {0,1} n → {±1}, g: {0,1} k → {0,1} and for (x1,...,xk) ∈ ({0,1} n) k, f ◦g(x1,...,xk) = f (...,g(x1,i,...,xk,i),...). We show that there is an O(log 3 n) cost simultaneous protocol for SYM ◦ g when k> 1 + logn, SYM is any symmetric function and g is any function. Previously, an efficient protocol was only known for SYM ◦ g when g is symmetric and “compressible”. We also get a nonsimultaneous protocol for SYM ◦ g of cost O(n/2 k · logn + k logn) for any k ≥ 2. In the setting of k ≤ 1 + logn, we study more closely functions of the form MAJORITY ◦g, MODm ◦g, and NOR ◦g, where the latter two are generalizations of the wellknown and studied functions Generalized Inner Product and Disjointness respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM Journal on Computing, 33:137–166, 2004) and determine the communication complexity of MAJORITY ◦ QCSBk, where QCSBk is the “quadratic character of the sum of the bits” function.
Separating Deterministic from Randomized Multiparty Communication Complexity
, 2008
"... We solve some fundamental problems in the numberonforehead (NOF) kplayer communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a onesided error probability of 1/3 but which has linear communication complex ..."
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Cited by 2 (0 self)
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We solve some fundamental problems in the numberonforehead (NOF) kplayer communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a onesided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for k = nO(1) players, where n is the number of bits on each players ’ forehead. This separates the analogues of RP and P in the NOF communication model. We also show that there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential and we do not know of any explicit function which allows such separations. However, for the 3player case we exhibit an explicit function which has Ω(log logn) randomized complexity for private coins but only constant complexity for public coins. It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic kplayer communication complexity does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the numberinhand model, is not complete in the NOF model under cylindrical reductions.