Average Binary Long-Lived Consensus: Quantifying the Stabilizing Role Played by Memory

Abstract

Consider a system composed of n sensors operating in synchronous rounds. In each round an input vector of sensor readings x is produced, where the i-th entry of x is a binary value produced by the i-th sensor. The sequence of input vectors is assumed to be smooth: exactly one entry of the vector changes from one round to the next one. The system implements a fault-tolerant averaging consensus function f. This function returns, in each round, a representative output value v of the sensor readings x. Assuming that at most t entries of the vector can be erroneous, f is required to return a value that appears at least t + 1 times in x. The instability of the system is the number of output changes over a random sequence of input vectors.

Our first result is to design optimal instability consensus systems with and without memory. Roughly, in the memoryless case, we show that an optimal system is D ₀, that outputs 1 unless it is forced by the fault-tolerance requirement to output 0 (on vectors with t or less 1’s). For the case of systems with memory, we show that an optimal system is D ₁, that initially outputs the most common value in the input vector, and then stays with this output unless forced by the fault-tolerance requirement to change (i.e., a single bit of memory suffices).

Our second result is to quantify the gain factor due to memory by computing c _n (t), the number of decision changes performed by D ₀ per each decision change performed by D ₁. If $t=\frac{n}{2}$ the system is always forced to decide the simple majority and, in that case, memory becomes useless. We show that the same type of phenomenon occurs when $\frac{n}{2}-t$ is constant. Nevertheless, as soon as $\frac{n}{2}-t \sim \sqrt{n}$, memory plays an important stabilizing role because the ratio c _n (t) grows like $\Theta(\sqrt{n})$. We also show that this is an upper bound: $c_n(t)=O(\sqrt{n})$ for every t.

Our results are average case versions of previous works where the sequence of input vectors was assumed to be, in addition to smooth, geodesic: the i-th entry of the input vector was allowed to change at most once over the sequence. It thus eliminates some anomalies that ocurred in the worst case, geodesic instability setting.