On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

Band Allocation for Cognitive Radios with Buffered Primary and Secondary Users

In this paper, we study band allocation of $\mathcal{M}_s$ buffered secondary users (SUs) to $\mathcal{M}_p$ orthogonal primary licensed bands, where each primary band is assigned to one primary user (PU). Each SU is assigned to one of the available primary bands with a certain probability designed to satisfy some specified quality of service (QoS) requirements for the SUs. In the proposed system, only one SU is assigned to a particular band. The optimization problem used to obtain the stability region's envelope (closure) is shown to be a linear program. We compare the stability region of the proposed system with that of a system where each SU chooses a band randomly with some assignment probability. We also compare with a fixed (deterministic) assignment system, where only one SU is assigned to one of the primary bands all the time. We prove the advantage of the proposed system over the other systems.Comment: Accepted in WCNC 201

Discovering Functional Communities in Dynamical Networks

Many networks are important because they are substrates for dynamical systems, and their pattern of functional connectivity can itself be dynamic -- they can functionally reorganize, even if their underlying anatomical structure remains fixed. However, the recent rapid progress in discovering the community structure of networks has overwhelmingly focused on that constant anatomical connectivity. In this paper, we lay out the problem of discovering_functional communities_, and describe an approach to doing so. This method combines recent work on measuring information sharing across stochastic networks with an existing and successful community-discovery algorithm for weighted networks. We illustrate it with an application to a large biophysical model of the transition from beta to gamma rhythms in the hippocampus.Comment: 18 pages, 4 figures, Springer "Lecture Notes in Computer Science" style. Forthcoming in the proceedings of the workshop "Statistical Network Analysis: Models, Issues and New Directions", at ICML 2006. Version 2: small clarifications, typo corrections, added referenc

A Survey and Analysis of Multi-Robot Coordination

International audienceIn the field of mobile robotics, the study of multi-robot systems (MRSs) has grown significantly in size and importance in recent years. Having made great progress in the development of the basic problems concerning single-robot control, many researchers shifted their focus to the study of multi-robot coordination. This paper presents a systematic survey and analysis of the existing literature on coordination, especially in multiple mobile robot systems (MMRSs). A series of related problems have been reviewed, which include a communication mechanism, a planning strategy and a decision-making structure. A brief conclusion and further research perspectives are given at the end of the paper