Section: New Results

Communication and Fault Tolerance in Distributed Networks

Linear Space Bootstrap Communication Schemes

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Eli Gafni, Sergio Rajsbaum.

We consider in [18] , a system of $n$ processes with ids not a priori known, that are drawn from a large space, potentially unbounded. How can these $n$ processes communicate to solve a task? We show that $n$ a priori allocated Multi-Writer Multi-Reader (MWMR) registers are both needed and sufficient to solve any read-write wait free solvable task. This contrasts with the existing possible solution borrowed from adaptive algorithms that require $\Theta \left(n^2\right)$ MWMR registers. To obtain these results, the paper shows how the processes can non blocking emulate a system of $n$ Single-Writer Multi-Reader (SWMR) registers on top of $n$ MWMR registers. It is impossible to do such an emulation with $n\hspace{0.17em}-\hspace{0.17em}1$ MWMR registers. Furthermore, we want to solve a sequence of tasks (potentially infinite) that are sequentially dependent (processes need the previous task's outputs in order to proceed to the next task). A non blocking emulation might starve a process forever. By doubling the space complexity, using $2n\hspace{0.17em}-\hspace{0.17em}1$ rather than just $n$ registers, the computation is wait free rather than non blocking.

Black Art: Obstruction-Free $k$-set Agreement with |MWMR registers| < |proccesses|

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Eli Gafni, Sergio Rajsbaum.

When $n$ processes communicate by writing to and reading from $k\hspace{0.17em}<\hspace{0.17em}n$ MWMR registers the “communication bandwidth” precludes emulation of SWMR system, even non-blocking.

Nevertheless, recently a positive result was shown that such a system either wait-free or obstruction-free can solve an interesting one-shot task. This paper demonstrates another such result. It shows that $\left(n\hspace{0.17em}-\hspace{0.17em}1\right)-$set agreement can be solved obstruction-free with merely 2 MWMR registers. Achieving $k-$set agreement with $n\hspace{0.17em}-\hspace{0.17em}k\hspace{0.17em}+\hspace{0.17em}1$ registers is a challenge. In [17] , we make the first step toward it by showing $k-$set agreement with $2\left(n\hspace{0.17em}-\hspace{0.17em}k\right)$ registers.

Adaptive Register Allocation with a Linear Number of Registers

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Eli Gafni, Leslie Lamport.

In [16] , we give an adaptive algorithm in which processes use multi-writer multi- reader registers to acquire exclusive write access to their own single-writer, multi-reader registers. It is the first such algorithm that uses a number of registers linear in the number of participating processes. Previous adaptive algorithms require at least $\Theta \left(n^{3/2}\right)$ registers

Uniform Consensus with Homonyms and Omission Failures

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Hung Tran-The.

In synchronous message passing models in which some processes may be homonyms, i.e. may share the same id, we consider the consensus problem. Many results have already been proved concerning Byzantine failures in models with homonyms, we complete in [19] , the picture with crash and omission failures.

Let $n$ be the number of processes, $t$ the number of processes that may be faulty ($t\hspace{0.17em}<\hspace{0.17em}n$) and $l$ $(1\le \hspace{0.17em}l\le \hspace{0.17em}n)$ the number of identifiers. We prove that for crash failures and send-omission failures, uniform consensus is solvable even if $l\hspace{0.17em}=\hspace{0.17em}1$, that is with fully anonymous processes for any number of faulty processes.

Concerning omission failures, when the processes are numerate, i.e. are able to count the number of copies of identical messages they received in each round, uniform consensus is solvable even for fully anonymous processes for $n\hspace{0.17em}>\hspace{0.17em}2t$. If processes are not numerate, uniform consensus is solvable if and only if $l\hspace{0.17em}>\hspace{0.17em}2t$.

All the proposed protocols are optimal both in the number of communication steps needed, and in the number of processes that can be faulty.

All these results show, (1) that identifiers are not useful for crash and send-omission failures or when processes are numerate, (2) for general omission or for Byzantine failures the number of different ids becomes significant.

Byzantine agreement with homonyms

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Rachid Guerraoui, Anne-Marie Kermarrec, Hung Tran-The.

So far, the distributed computing community has either as- sumed that all the processes of a distributed system have distinct identifiers or, more rarely, that the processes are anonymous and have no identifiers. These are two extremes of the same general model: namely, $n$ processes use $l$ dif- ferent authenticated identifiers, where $1\le l\le n$. In this paper [3] , we ask how many identifiers are actually needed to reach agreement in a distributed system with t Byzantine processes. We show that having $3t+1$ identifiers is necessary and suf- ficient for agreement in the synchronous case but, more sur- prisingly, the number of identifiers must be greater than $(n+3t)/2$ in the partially synchronous case. This demonstrates two differences from the classical model (which has l = n): there are situations where relaxing synchrony to partial syn- chrony renders agreement impossible; and, in the partially synchronous case, increasing the number of correct processes can actually make it harder to reach agreement. The im- possibility proofs use the fact that a Byzantine process can send multiple messages to the same recipient in a round. We show that removing this ability makes agreement easier: then, $t+1$ identifiers are sufficient for agreement, even in the partially synchronous model.

Byzantine agreement with homonyms in synchronous systems

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Hung Tran-The.

We consider in [4] , the Byzantine agreement problem in synchronous systems with homonyms. In this model different processes may have the same authenticated identifier. In such a system of $n$ processes sharing a set of $l$ identifiers, we define a distribution of the identifiers as an integer partition of $n$ into $l$ parts $n_1...,n_l$ giving for each identifier $i$ the number of processes having this identifier.

Assuming that the processes know the distribution of identifiers we give a necessary and sufficient condition on the integer partition of $n$ to solve the Byzantine agreement with at most $t$ Byzantine processes. Moreover we prove that there exists a distribution of $l$ identifiers enabling to solve Byzantine agreement with at most $t$ Byzantine processes if and only if $n>3t$, $l>t$ and $l\frac{(n-r)t}{n-t-min(t,r)}$ where $r=nmodl$.

This bound is to be compared with the $l>3t$ bound proved in Delporte-Gallet et al. (2011) when the processes do not know the distribution of identifiers.

Convergence of the D-iteration algorithm: convergence rate and asynchronous distributed scheme

Participants : Dohy Hong, Fabien Mathieu, Gérard Burnside.

In this paper [25] , we define the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme, including for distributed computation framework. The approach can be understood as a decomposition of the matrix-vector product operation in elementary operations at the vector entry level.