Communication complexity

About: Communication complexity is a research topic. Over the lifetime, 3870 publications have been published within this topic receiving 105832 citations.

Improving the sample complexity using global data

Abstract: We study the sample complexity of proper and improper learning problems with respect to different q-loss functions. We improve the known estimates for classes which have relatively small covering numbers in empirical L/sub 2/ spaces (e.g. log-covering numbers which are polynomial with exponent p<2). We present several examples of relevant classes which have a "small" fat-shattering dimension, and hence fit our setup, the most important of which are kernel machines.

Amortized Communication Complexity

Abstract: In this work we study the direct-sum problem with respect to communication complexity: Consider a relation $f$ defined over $\{0,1\}^{n} \times \{0,1\}^{n}$. Can the communication complexity of simultaneously computing $f$ on $\cal l$ instances $(x_1,y_1),\ldots,(x_{\cal l},y_{\cal l})$ be smaller than the communication complexity of computing $f$ on the $\cal l$ instances, separately? Let the amortized communication complexity of $f$ be the communication complexity of simultaneously computing $f$ on $\cal l$ instances, divided by $\cal l$. We study the properties of the amortized communication complexity. We show that the amortized communication complexity of a relation can be smaller than its communication complexity. More precisely, we present a partial function whose (deterministic) communication complexity is $\Theta(\log n)$ and its amortized (deterministic) communication complexity is $O(1)$. Similarly, for randomized protocols, we present a function whose randomized communication complexity is $\Theta(\log n)$ and its amortized randomized communication complexity is $O(1)$. We also give a general lower bound on the amortized communication complexity of any function $f$ in terms of its communication complexity $C(f)$: for every function $f$ the amortized communication complexity of $f$ is $\Omega \left (\sqrt{C(f)} - \log n \right)$.

Information Equals Amortized Communication

Abstract: We show how to efficiently simulate the sending of a message to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver who has some information about the message. This is a generalization and strengthening of the Slepian Wolf theorem, which shows how to carry out such a simulation with low amortized communication in the case that the message is a deterministic function of an input. A caveat is that our simulation is interactive. As a consequence, we prove that the internal information cost(namely the information revealed to the parties) involved in computing any relation or function using a two party interactive protocol is exactly equal to the amortized communication complexity of computing independent copies of the same relation or function. We also show that the only way to prove a strong direct sum theorem for randomized communication complexity is by solving a particular variant of the pointer jumping problem that we define. Our work implies that a strong direct sum theorem for communication complexity holds if and only if efficient compression of communication protocols is possible.

A new distributed algorithm to find breadth first search trees

Abstract: A new distributed algorithm is presented for constructing breadth first search (BFS) trees. A BFS tree is a tree of shortest paths from a given root node to all other nodes of a network under the assumption of unit edge weights; such trees provide useful building blocks for a number of routing and control functions in communication networks. The order of communication complexity for the new algorithm is O(V^{1.6} + E) where V is the number of nodes and E the number of edges. For dense networks with E \geq V^{1.6} this order of complexity is optimum.