Optimal relay location and power allocation for low SNR broadcast relay channels

We consider the broadcast relay channel (BRC), where a single source transmits to multiple destinations with the help of a relay, in the limit of a large bandwidth. We address the problem of optimal relay positioning and power allocations at source and relay, to maximize the multicast rate from source to all destinations. To solve such a network planning problem, we develop a three-faceted approach based on an underlying information theoretic model, computational geometric aspects, and network optimization tools. Firstly, assuming superposition coding and frequency division between the source and the relay, the information theoretic framework yields a hypergraph model of the wideband BRC, which captures the dependency of achievable rate-tuples on the network topology. As the relay position varies, so does the set of hyperarcs constituting the hypergraph, rendering the combinatorial nature of optimization problem. We show that the convex hull C of all nodes in the 2-D plane can be divided into disjoint regions corresponding to distinct hyperarcs sets. These sets are obtained by superimposing all k-th order Voronoi tessellation of C. We propose an easy and efficient algorithm to compute all hyperarc sets, and prove they are polynomially bounded. Using the switched hypergraph approach, we model the original problem as a continuous yet non-convex network optimization program. Ultimately, availing on the techniques of geometric programming and $p$-norm surrogate approximation, we derive a good convex approximation. We provide a detailed characterization of the problem for collinearly located destinations, and then give a generalization for arbitrarily located destinations. Finally, we show strong gains for the optimal relay positioning compared to seemingly interesting positions.Comment: In Proceedings of INFOCOM 201

A graph rewriting programming language for graph drawing

This paper describes Grrr, a prototype visual graph drawing tool. Previously there were no visual languages for programming graph drawing algorithms despite the inherently visual nature of the process. The languages which gave a diagrammatic view of graphs were not computationally complete and so could not be used to implement complex graph drawing algorithms. Hence current graph drawing tools are all text based. Recent developments in graph rewriting systems have produced computationally complete languages which give a visual view of graphs both whilst programming and during execution. Grrr, based on the Spider system, is a general purpose graph rewriting programming language which has now been extended in order to demonstrate the feasibility of visual graph drawing

Information Recovery from Pairwise Measurements

A variety of information processing tasks in practice involve recovering $n$ objects from single-shot graph-based measurements, particularly those taken over the edges of some measurement graph $\mathcal{G}$. This paper concerns the situation where each object takes value over a group of $M$ different values, and where one is interested to recover all these values based on observations of certain pairwise relations over $\mathcal{G}$. The imperfection of measurements presents two major challenges for information recovery: 1) $\textit{inaccuracy}$: a (dominant) portion $1-p$ of measurements are corrupted; 2) $\textit{incompleteness}$: a significant fraction of pairs are unobservable, i.e. $\mathcal{G}$ can be highly sparse. Under a natural random outlier model, we characterize the $\textit{minimax recovery rate}$, that is, the critical threshold of non-corruption rate $p$ below which exact information recovery is infeasible. This accommodates a very general class of pairwise relations. For various homogeneous random graph models (e.g. Erdos Renyi random graphs, random geometric graphs, small world graphs), the minimax recovery rate depends almost exclusively on the edge sparsity of the measurement graph $\mathcal{G}$ irrespective of other graphical metrics. This fundamental limit decays with the group size $M$ at a square root rate before entering a connectivity-limited regime. Under the Erdos Renyi random graph, a tractable combinatorial algorithm is proposed to approach the limit for large $M$ ($M=n^{\Omega(1)}$), while order-optimal recovery is enabled by semidefinite programs in the small $M$ regime. The extended (and most updated) version of this work can be found at (http://arxiv.org/abs/1504.01369).Comment: This version is no longer updated -- please find the latest version at (arXiv:1504.01369