On total communication complexity of collapsing protocols for pointer jumping problem

This paper focuses on bounding the total communication complexity of collapsing protocols for multiparty pointer jumping problem ($MPJ_k^n$). Brody and Chakrabati in \cite{bc08} proved that in such setting one of the players must communicate at least $n - 0.5\log{n}$ bits. Liang in \cite{liang} has shown protocol matching this lower bound on maximum complexity. His protocol, however, was behaving worse than the trivial one in terms of total complexity (number of bits sent by all players). He conjectured that achieving total complexity better then the trivial one is impossible. In this paper we prove this conjecture. Namely, we show that for a collapsing protocol for $MPJ_k^n$, the total communication complexity is at least $n-2$ which closes the gap between lower and upper bound for total complexity of $MPJ_k^n$ in collapsing setting

Dependent Random Graphs And Multi-Party Pointer Jumping

We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs dependent random graphs . Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size ((1-o(1))log(n))/log(1/p), and the chromatic number will be at most (nlog(1/(1-p)))/log(n). We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping problem (MPJk) in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ3 costs O((n * log(log(n)))/log(n)) communication, improving on a bound from [BrodyChakrabarti08]. We extend our protocol to the non-Boolean pointer jumping problem, achieving an upper bound which is o(n) for any k \u3e= 4 players. This is the first o(n) protocol and improves on a bound of Damm, Jukna, and Sgall, which has stood for almost twenty years

NOF-multiparty information complexity bounds for pointer jumping

Abstract. We prove a lower bound on the communication complexity of pointer jumping for multiparty one-way 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 Bar-Yossef, 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

Information Complexity and Data Stream Algorithms for Basic Problems

Data stream algorithms obtain their input as a stream of data elements that have to be processed immediately as they arrive using only a very limited amount of memory. They solve a new class of algorithmic problems that emerged recently with the growing importance of computer networks and the ever-increasing size of the data sets that are processed algorithmically. In this thesis data stream algorithms for basic problems under extreme space restrictions are developed, namely counting and random sampling. Then we apply these algorithms to improve the space complexity of the celebrated data stream algorithm for the computation of frequency moments by Alon, Matias, and Szegedy for very long data streams. Lower bounds on the space complexity of data stream algorithms are usually proved by using communication complexity arguments. Information complexity is a related field that applies Shannon's information theory to obtain lower bounds on the communication complexity of functions. The development of information complexity is closely linked to the recent interest in data stream algorithms since important parts of this theory have been developed to prove a lower bound on the space complexity of data stream algorithms for the frequency moments. In this thesis we prove an optimal lower bound on the multi-party information complexity of the disjointness function, the underlying communication problem in the proof of the lower bound on the space complexity of data stream algorithms for the frequency moments. Additionally, we generalize and simplify known lower bounds on the one-way communication complexity of the index function by using information complexity and we present the first attempt to apply information complexity to multi-party one-way protocols in the number on the forehead model by Chandra, Furst, and Lipton