Gossip and Random Walk Techniques for Network Coding

Ad hoc and sensor networks usually do not have a central entity for managing information spreading. Moreover, such wireless stations have limited energy and computational power. All this leads to a need for distributed and efficient algorithms for disseminating information across the network. Network coding in conjunction with random walks and gossip techniques proposes a local and distributed approach for compressing information and disseminating it across the network. In this work we study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading PUSH, PULL, and EXCHANGE. While traditionally algebraic gossip protocols used PUSH or PULL as their gossip algorithms, we prove that using the EXCHANGE algorithm can be unboundedly better. The stopping time of algebraic gossip is known to be linear for the complete graph; but the question of determining a tight upper bound for general graphs is still open. We take a major step in solving this question, and prove a linear tight upper bound for bounded degree graphs. Our proofs use a new method that relies on Jackson’s queuing theorem to analyze the stopping time of network coding, and this technique is likely become useful for future research. In addition, we compare algebraic gossip to an average computation task via EXCHANGE and show that there are graphs for which algebraic gossip can be unboundedly faster. In future work (Chapter 6) we will address many open questions that still remain. First, we are going to find tight upper and lower bounds for general graphs and find deterministic algorithm to compute these bounds as a function of a given graph. We would like to prove all bounds for the asynchronous time model and to give stopping time bounds with high probability along with the expected value results. We are also aiming to study gossip algorithms in general, and algebraic gossip (i.e., linear network coding over gossip algorithm) in particular, on dynamic graphs and SINR models ([1] and [2]). To the best of our knowledge, this, has not been addressed at all previously.