Ergodicity and Mixing Rate of One-Dimensional Cellular Automata

One-and two-dimensional cellular automata which are known to be fault-tolerant are very complex. On the other hand, only very simple cellular automata have actually been proven to lack fault-tolerance, i.e., to be mixing. The latter either have large noise probability ε or belong to the small family of two-state nearest-neighbor monotonic rules which includes local majority voting. For a certain simple automaton L called the soldiers rule, this problem has intrigued researchers for the last two decades since L is clearly more robust than local voting: in the absence of noise, L eliminates any finite island of perturbation from an initial configuration of all 0's or all 1's. The same holds for a 4-state monotonic variant of L, K, called two-line voting. We will prove that the probabilistic cellular automata Kε and Lε asymptotically lose all information about their initial state when subject to small, strongly biased noise. The mixing property trivially implies that the systems are ergodic. The finite-time information-retaining quality of a mixing system can be represented by its relaxation time Relax(⋅), which measures the time before the onset of significant information loss. This is known to grow as (1/ε)^c for noisy local voting. The impressive error-correction ability of L has prompted some researchers to conjecture that Relax(Lε) = 2^(c/ε). We prove the tight bound 2^(c1log^21/ε) ＜ Relax(Lε) ＜ 2^(c2log^21/ε) for a biased error model. The same holds for Kε. Moreover, the lower bound is independent of the bias assumption. The strong bias assumption makes it possible to apply sparsity/renormalization techniques, the main tools of our investigation, used earlier in the opposite context of proving fault-tolerance.

Angels: In Network Support for Minimum Distribution Time in P2P Overlays

This paper proposes the use of in-network caches (which we call Angels) to reduce the Minimum Distribution Time (MDT) of a file from a seeder – a node that possesses the file – to a set of leechers – nodes who are interested in downloading the file. An Angel is not a leecher in the sense that it is not interested in receiving the entire file, but rather it is interested in minimizing the MDT to all leechers, and as such uses its storage and up/down-link capacity to cache and forward parts of the file to other peers. We extend the analytical results by Kumar and Ross [1] to account for the presence of angels by deriving a new lower bound for the MDT. We show that this newly derived lower bound is tight by proposing a distribution strategy under assumptions of a fluid model. We present a GroupTree heuristic that addresses the impracticalities of the fluid model. We evaluate our designs through simulations that show that our Group-Tree heuristic outperforms other heuristics, that it scales well with the increase of the number of leechers, and that it closely approaches the optimal theoretical bounds.

Angels: In-Network Support for Minimum Distribution Time in P2P Overlays

This thesis proposes the use of in-network caches (which we call Angels) to reduce the Minimum Distribution Time (MDT) of a file from a seeder – a node that possesses the file – to a set of leechers – nodes who are interested in downloading the file. An Angel is not a leecher in the sense that it is not interested in receiving the entire file, but rather it is interested in minimizing the MDT to all leechers, and as such uses its storage and up/down-link capacity to cache and forward parts of the file to other peers. We extend the analytical results by Kumar and Ross (Kumar and Ross, 2006) to account for the presence of angels by deriving a new lower bound for the MDT. We show that this newly derived lower bound is tight by proposing a distribution strategy under assumptions of a fluid model. We present a GroupTree heuristic that addresses the impracticalities of the fluid model. We evaluate our designs through simulations that show that our GroupTree heuristic outperforms other heuristics, that it scales well with the increase of the number of leechers, and that it closely approaches the optimal theoretical bounds.

Real-Time Spatio-Temporal Query Processing in Mobile Ad-Hoc Sensor Networks

Personal communication devices are increasingly equipped with sensors that are able to collect and locally store information from their environs. The mobility of users carrying such devices, and hence the mobility of sensor readings in space and time, opens new horizons for interesting applications. In particular, we envision a system in which the collective sensing, storage and communication resources, and mobility of these devices could be leveraged to query the state of (possibly remote) neighborhoods. Such queries would have spatio-temporal constraints which must be met for the query answers to be useful. Using a simplified mobility model, we analytically quantify the benefits from cooperation (in terms of the system's ability to satisfy spatio-temporal constraints), which we show to go beyond simple space-time tradeoffs. In managing the limited storage resources of such cooperative systems, the goal should be to minimize the number of unsatisfiable spatio-temporal constraints. We show that Data Centric Storage (DCS), or "directed placement", is a viable approach for achieving this goal, but only when the underlying network is well connected. Alternatively, we propose, "amorphous placement", in which sensory samples are cached locally, and shuffling of cached samples is used to diffuse the sensory data throughout the whole network. We evaluate conditions under which directed versus amorphous placement strategies would be more efficient. These results lead us to propose a hybrid placement strategy, in which the spatio-temporal constraints associated with a sensory data type determine the most appropriate placement strategy for that data type. We perform an extensive simulation study to evaluate the performance of directed, amorphous, and hybrid placement protocols when applied to queries that are subject to timing constraints. Our results show that, directed placement is better for queries with moderately tight deadlines, whereas amorphous placement is better for queries with looser deadlines, and that under most operational conditions, the hybrid technique gives the best compromise.