61 research outputs found
Beeping a Deterministic Time-Optimal Leader Election
The beeping model is an extremely restrictive broadcast communication model that relies only on carrier sensing. In this model, we solve the leader election problem with an asymptotically optimal round complexity of O(D + log n), for a network of unknown size n and unknown diameter D (but with unique identifiers). Contrary to the best previously known algorithms in the same setting, the proposed one is deterministic. The techniques we introduce give a new insight as to how local constraints on the exchangeable messages can result in efficient algorithms, when dealing with the beeping model.
Using this deterministic leader election algorithm, we obtain a randomized leader election algorithm for anonymous networks with an asymptotically optimal round complexity of O(D + log n) w.h.p. In previous works this complexity was obtained in expectation only.
Moreover, using deterministic leader election, we obtain efficient algorithms for symmetry-breaking and communication procedures: O(log n) time MIS and 5-coloring for tree networks (which is time-optimal), as well as k-source multi-broadcast for general graphs in O(min(k,log n) * D + k log{(n M)/k}) rounds (for messages in {1,..., M}). This latter result improves on previous solutions when the number of sources k is sublogarithmic (k = o(log n))
The Computational Power of Beeps
In this paper, we study the quantity of computational resources (state
machine states and/or probabilistic transition precision) needed to solve
specific problems in a single hop network where nodes communicate using only
beeps. We begin by focusing on randomized leader election. We prove a lower
bound on the states required to solve this problem with a given error bound,
probability precision, and (when relevant) network size lower bound. We then
show the bound tight with a matching upper bound. Noting that our optimal upper
bound is slow, we describe two faster algorithms that trade some state
optimality to gain efficiency. We then turn our attention to more general
classes of problems by proving that once you have enough states to solve leader
election with a given error bound, you have (within constant factors) enough
states to simulate correctly, with this same error bound, a logspace TM with a
constant number of unary input tapes: allowing you to solve a large and
expressive set of problems. These results identify a key simplicity threshold
beyond which useful distributed computation is possible in the beeping model.Comment: Extended abstract to appear in the Proceedings of the International
Symposium on Distributed Computing (DISC 2015
On the Computational Power of Radio Channels
Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity.
Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature
On Bioelectric Algorithms
Cellular bioelectricity describes the biological phenomenon in which cells in living tissue generate and maintain patterns of voltage gradients across their membranes induced by differing concentrations of charged ions. A growing body of research suggests that bioelectric patterns represent an ancient system that plays a key role in guiding many important developmental processes including tissue regeneration, tumor suppression, and embryogenesis. This paper applies techniques from distributed algorithm theory to help better understand how cells work together to form these patterns. To do so, we present the cellular bioelectric model (CBM), a new computational model that captures the primary capabilities and constraints of bioelectric interactions between cells and their environment. We use this model to investigate several important topics from the relevant biology research literature. We begin with symmetry breaking, analyzing a simple cell definition that when combined in single hop or multihop topologies, efficiently solves leader election and the maximal independent set problem, respectively - indicating that these classical symmetry breaking tasks are well-matched to bioelectric mechanisms. We then turn our attention to the information processing ability of bioelectric cells, exploring upper and lower bounds for approximate solutions to threshold and majority detection, and then proving that these systems are in fact Turing complete - resolving an open question about the computational power of bioelectric interactions
Noisy Beeping Networks
We introduce noisy beeping networks, where nodes have limited communication
capabilities, namely, they can only emit energy or sense the channel for
energy. Furthermore, imperfections may cause devices to malfunction with some
fixed probability when sensing the channel, which amounts to deducing a noisy
received transmission. Such noisy networks have implications for
ultra-lightweight sensor networks and biological systems.
We show how to compute tasks in a noise-resilient manner over noisy beeping
networks of arbitrary structure. In particular, we transform any algorithm that
assumes a noiseless beeping network (of size ) into a noise-resilient
version while incurring a multiplicative overhead of only in its
round complexity, with high probability. We show that our coding is optimal for
some tasks, such as node-coloring of a clique.
We further show how to simulate a large family of algorithms designed for
distributed networks in the CONGEST() model over a noisy beeping network.
The simulation succeeds with high probability and incurs an asymptotic
multiplicative overhead of in the
round complexity, where is the maximal degree of the network. The
overhead is tight for certain graphs, e.g., a clique. Further, this simulation
implies a constant overhead coding for constant-degree networks
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