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

Tight Bounds for Set Disjointness in the Message Passing Model

In a multiparty message-passing model of communication, there are $k$ players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work \cite{phillips12,woodruff12,woodruff13}, strong lower bounds of the form $\Omega(n \cdot k)$ were obtained for several functions in the message-passing model; however, a lower bound on the classical Set Disjointness problem remained elusive. In this paper, we prove tight lower bounds of the form $\Omega(n \cdot k)$ for the Set Disjointness problem in the message passing model. Our bounds are obtained by developing information complexity tools in the message-passing model, and then proving an information complexity lower bound for Set Disjointness. As a corollary, we show a tight lower bound for the task allocation problem \cite{DruckerKuhnOshman} via a reduction from Set Disjointness

Opportunistic Information Dissemination in Mobile Ad-hoc Networks: adaptiveness vs. obliviousness and randomization vs. determinism

In this paper the problem of information dissemination in Mobile Ad-hoc Networks (MANET) is studied. The problem is to disseminate a piece of information, initially held by a distinguished source node, to all nodes in a set defined by some predicate. We use a model of MANETs that is well suited for dynamic networks and opportunistic communication. In this model nodes are placed in a plane, in which they can move with bounded speed, and communication between nodes occurs over a collision-prone single channel. In this setup informed and uninformed nodes can be disconnected for some time (bounded by a parameter alpha), but eventually some uninformed node must become neighbor of an informed node and remain so for some time (bounded by a parameter beta). In addition, nodes can start at different times, and they can crash and recover. Under the above framework, we show negative and positive results for different types of randomized protocols, and we put those results in perspective with respect to previous deterministic results

The Range of Topological Effects on Communication

We continue the study of communication cost of computing functions when inputs are distributed among $k$ processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges. Chattopadhyay, Radhakrishnan and Rudra (FOCS'14) recently initiated a study of the effect of topology of the network on the total communication cost using tools from $L_1$ embeddings. Their techniques provided tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. This work addresses two other kinds of natural functions. We show that for a large class of natural functions like Set-Disjointness the communication cost is essentially $n$ times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like $\text{ED} \circ \text{XOR}$ and $\text{XOR} \circ \text{ED}$, the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs. To obtain our results, we use some new tools in addition to ones used in Chattopadhyay et. al. These include (i) viewing the communication constraints via a linear program; (ii) using tools from the theory of tree embeddings to prove topology sensitive direct sum results that handle the case of composed functions and (iii) representing the communication constraints of certain problems as a family of collection of multiway cuts, where each multiway cut simulates the hardness of computing the function on the star topology

A Lower Bound Technique for Communication in BSP

Communication is a major factor determining the performance of algorithms on current computing systems; it is therefore valuable to provide tight lower bounds on the communication complexity of computations. This paper presents a lower bound technique for the communication complexity in the bulk-synchronous parallel (BSP) model of a given class of DAG computations. The derived bound is expressed in terms of the switching potential of a DAG, that is, the number of permutations that the DAG can realize when viewed as a switching network. The proposed technique yields tight lower bounds for the fast Fourier transform (FFT), and for any sorting and permutation network. A stronger bound is also derived for the periodic balanced sorting network, by applying this technique to suitable subnetworks. Finally, we demonstrate that the switching potential captures communication requirements even in computational models different from BSP, such as the I/O model and the LPRAM