Communication complexity

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

Communication modeling of heterogeneous networks of workstations for performance characterization of collective operations

Abstract: Networks of workstations (NOW) have become an attractive alternative platform for high performance computing. Due to the commodity nature of workstations and interconnects and due to the multiplicity of vendors and platforms, the NOW environments are being gradually redefined as heterogeneous networks of workstations (HNOW). Having an accurate model for the communication in HNOW systems is crucial for design and evaluation of efficient communication layers for such systems. In this paper we present a model for point-to-point communication in HNOW systems and show how it can be used for characterizing the performance of different collective communication operations. In particular, we show how the performance of broadcast, scatter, and gather operations can be modeled and analyzed. We also verify the accuracy of our proposed model by using an experimental HNOW testbed. Furthermore, it is shown how this model can be used for comparing the performance of different collective communication algorithms. We also show how the effect of heterogeneity on the performance of collective communication operations can be predicted.

Lower bounds in communication complexity based on factorization norms

Abstract: We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. This approach gives us access toseveral powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for deriving lower bounds in communication complexity. As we show,our method subsumes most of the previously known general approaches to lower bounds on communication complexity. Moreover, we extend all (but one) of these lower bounds to the realm of quantum communication complexity with entanglement. Our results also shed some light on the question how much communication can be saved by using entanglement.It is known that entanglement can save one of every two qubits, and examples for which this is tight are also known. It follows from our results that this bound on the saving in communication is tight almost always.

Comparison-based time-space lower bounds for selection

Abstract: We establish the first nontrivial lower bounds on time-space trade-offs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(nlog logSn) expected time in the RAM model (or more generally in the comparison branching program model), if we have S bits of extra space besides the read-only input array. This bound is tight for all S > log n, and remains true even if the array is given in a random order. Our result thus answers a 16-year-old question of Munro and Raman l1996r, and also complements recent lower bounds that are restricted to sequential access, as in the multipass streaming model lChakrabarti et al. 2008br.We also prove that any comparison-based, deterministic, multipass streaming algorithm for finding the median requires Ω(nloga(n/s)+ nlogsn) worst-case time (in scanning plus comparisons), if we have s cells of space. This bound is also tight for all s >log2n. We get deterministic lower bounds for I/O-efficient algorithms as well.The proofs in this article are self-contained and do not rely on communication complexity techniques.

Separations in query complexity based on pointer functions

Abstract: In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/log(n)) while its zero-error randomized query complexity is O(√n). We further show that the quantum query complexity of the same function is O(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/log(n)) and bounded-error randomized query complexity R(g) = O(√n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is QE(g) = O(√n). These functions show that the relations D(f) = O(R1(f)2) and R0(f) = O(R(f)2) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R0, a 3/2-power separation between QE and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Goos, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

Query-to-Communication Lifting for BPP

Abstract: For any $n$-bit boolean function $f$, we show that the randomized communication complexity of the composed function $f\circ g^n$, where $g$ is an index gadget, is characterized by the randomized decision tree complexity of $f$. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.