Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound

We study the one-way number-on-the-forehead (NOF) communication complexity of the $k$-layer pointer jumping problem with $n$ vertices per layer. This classic problem, which has connections to many aspects of complexity theory, has seen a recent burst of research activity, seemingly preparing the ground for an $\Omega(n)$ lower bound, for constant $k$. Our first result is a surprising sublinear -- i.e., $o(n)$ -- upper bound for the problem that holds for $k \ge 3$, dashing hopes for such a lower bound. A closer look at the protocol achieving the upper bound shows that all but one of the players involved are collapsing, i.e., their messages depend only on the composition of the layers ahead of them. We consider protocols for the pointer jumping problem where all players are collapsing. Our second result shows that a strong $n - O(\log n)$ lower bound does hold in this case. Our third result is another upper bound showing that nontrivial protocols for (a non-Boolean version of) pointer jumping are possible even when all players are collapsing. Our lower bound result uses a novel proof technique, different from those of earlier lower bounds that had an information-theoretic flavor. We hope this is useful in further study of the problem

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

Dependent Random Graphs and Multiparty 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 $\frac{(1-o(1))\log n}{\log(1/p)}$, and the chromatic number will be at most $\frac{n \log(1/1-p)}{\log n}$. As an application and second main result, we give a new communication protocol for the k-player Multiparty Pointer Jumping (MPJ_k) problem in the number-on-the-forehead (NOF) model. Multiparty 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 MPJ_3 costs $O(\frac{n\log\log n}{\log n})$ communication, improving on a bound of Brody and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer jumping problem $\widehat{MPJ}_k$, achieving an upper bound which is o(n) for any $k >= 4$ players. This is the first o(n) bound for $\widehat{MPJ}_k$ and improves on a bound of Damm, Jukna, and Sgall [DJS98] which has stood for almost twenty years.Comment: 18 page

Some Communication Complexity Results and their Applications

Communication Complexity represents one of the premier techniques for proving lower bounds in theoretical computer science. Lower bounds on communication problems can be leveraged to prove lower bounds in several different areas. In this work, we study three different communication complexity problems. The lower bounds for these problems have applications in circuit complexity, wireless sensor networks, and streaming algorithms. First, we study the multiparty pointer jumping problem. We present the first nontrivial upper bound for this problem. We also provide a suite of strong lower bounds under several restricted classes of protocols. Next, we initiate the study of several non-monotone functions in the distributed functional monitoring setting and provide several lower bounds. In particular, we give a generic adversarial technique and show that when deletions are allowed, no nontrivial protocol is possible. Finally, we study the Gap-Hamming-Distance problem and give tight lower bounds for protocols that use a constant number of messages. As a result, we take a well-known lower bound for one-pass streaming algorithms for a host of problems and extend it so it applies to streaming algorithms that use a constant number of passes

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

The Function-Inversion Problem: Barriers and Opportunities

The task of function inversion is central to cryptanalysis: breaking block ciphers, forging signatures, and cracking password hashes are all special cases of the function-inversion problem. In 1980, Hellman showed that it is possible to invert a random function $f\colon [N] \to [N]$ in time $T = \widetilde{O}(N^{2/3})$ given only $S = \widetilde{O}(N^{2/3})$ bits of precomputed advice about $f$. Hellman’s algorithm is the basis for the popular “Rainbow Tables” technique (Oechslin, 2003), which achieves the same asymptotic cost and is widely used in practical cryptanalysis. Is Hellman’s method the best possible algorithm for inverting functions with preprocessed advice? The best known lower bound, due to Yao (1990), shows that $ST = \widetilde{\Omega}(N)$, which still admits the possibility of an $S = T = \widetilde{O}(N^{1/2})$ attack. There remains a long-standing and vexing gap between Hellman’s $N^{2/3}$ upper bound and Yao’s $N^{1/2}$ lower bound. Understanding the feasibility of an $S = T = N^{1/2}$ algorithm is cryptanalytically relevant since such an algorithm could perform a key-recovery attack on AES-128 in time $2^{64}$ using a precomputed table of size $2^{64}$. For the past 29 years, there has been no progress either in improving Hellman’s algorithm or in strengthening Yao’s lower bound. In this work, we connect function inversion to problems in other areas of theory to (1) explain why progress may be difficult and (2) explore possible ways forward. Our results are as follows: - We show that *any* improvement on Yao’s lower bound on function-inversion algorithms will imply new lower bounds on depth-two circuits with arbitrary gates. Further, we show that proving strong lower bounds on *non-adaptive* function-inversion algorithms would imply breakthrough circuit lower bounds on linear-size log-depth circuits. - We take first steps towards the study of the *injective* function-inversion problem, which has manifold cryptographic applications. In particular, we show that improved algorithms for breaking PRGs with preprocessing would give improved algorithms for inverting injective functions with preprocessing. - Finally, we show that function inversion is closely related to well-studied problems in communication complexity and data structures. Through these connections we immediately obtain the best known algorithms for problems in these domains

A New Approach to Multi-Party Peer-to-Peer Communication Complexity

We introduce new models and new information theoretic measures for the study of communication complexity in the natural peer-to-peer, multi-party, number-in-hand setting. We prove a number of properties of our new models and measures, and then, in order to exemplify their effectiveness, we use them to prove two lower bounds. The more elaborate one is a tight lower bound of Omega(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit function Disjointness, Disj_k^n. The other one is a tight lower bound of Omega(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit bitwise parity function, Par_k^n. Both lower bounds hold when n=Omega(k). The lower bound for Disj_k^n improves over the lower bound that can be inferred from the result of Braverman et al. (FOCS 2013), which was proved in the coordinator model and can yield a lower bound of Omega(kn/log k) in the peer-to-peer model. To the best of our knowledge, our lower bounds are the first tight (non-trivial) lower bounds on communication complexity in the natural peer-to-peer multi-party setting. In addition to the above results for communication complexity, we also prove, using the same tools, an Omega(n) lower bound on the number of random bits necessary for the (information theoretic) private computation of the function Disj_k^n