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 $n$) into a noise-resilient version while incurring a multiplicative overhead of only $O(\log n)$ 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($B$) model over a noisy beeping network. The simulation succeeds with high probability and incurs an asymptotic multiplicative overhead of $O(B\cdot \Delta \cdot \min(n,\Delta^2))$ in the round complexity, where $\Delta$ 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

Reliable Communication over Highly Connected Noisy Networks

We consider the task of multiparty computation performed over networks in the presence of random noise. Given an n-party protocol that takes R rounds assuming noiseless communication, the goal is to find a coding scheme that takes R’ rounds and computes the same function with high probability even when the communication is noisy, while maintaining a constant asymptotic rate, i.e., while keeping inf(n,R ->infinity) R/R’ positive. Rajagopalan and Schulman (STOC ‘94) were the first to consider this question, and provided a coding scheme with rate O(1/ log(d + 1)), where d is the maximal degree in the network. While that scheme provides a constant rate coding for many practical situations, in the worst case, e.g., when the network is a complete graph, the rate is O(1/ log n), which tends to 0 as n tends to infinity. We revisit this question and provide an efficient coding scheme with a constant rate for the interesting case of fully connected networks. We furthermore extend the result and show that if a (d-regular) network has mixing time m, then there exists an efficient coding scheme with rate O(1/m(3) log m). This implies a constant rate coding scheme for any n-party protocol over a d-regular network with a constant mixing time, and in particular for random graphs with n vertices and degrees n(Omega(1))

Noisy Interactive Quantum Communication

We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with n messages, designed for noiseless qudit channels (where d is arbitrary), our main result is a simulation method that fails with probability less than 2⁻ᶿ⁽ⁿᵋ⁾ and uses a qudit channel n(1 + Θ(√ε)) times, of which ε fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the √ε term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Surprisingly, this outperforms the best known overhead of 1 + O(√(ε log log 1/ε)) in the corresponding classical model, which is also conjectured to be optimal [Haeupler, FOCS’14]. Our work also improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS’14] for small ε