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Register a new userWait-free approximate agreement on graphs
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Added by
Joel Rybicki
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Approximate agreement is one of the few variants of consensus that can be solved in a wait-free manner in asynchronous systems where processes communicate by reading and writing to shared memory. In this work, we consider a natural generalisation of approximate agreement on arbitrary undirected connected graphs. Each process is given a vertex of the graph as input and, if non-faulty, must output a vertex such that - all the outputs are within distance 1 of one another, and - each output value lies on a shortest path between two input values. From prior work, it is known that there is no wait-free algorithm among $n ge 3$ processes for this problem on any cycle of length $c ge 4$, by reduction from 2-set agreement (Casta~neda et al., 2018). In this work, we investigate the solvability and complexity of this task on general graphs. We give a new, direct proof of the impossibility of approximate agreement on cycles of length $c ge 4$, via a generalisation of Sperners Lemma to convex polygons. We also extend the reduction from 2-set agreement to a larger class of graphs, showing that approximate agreement on on these graphs is unsolvable. Furthermore, we show that combinatorial arguments, used by both existing proofs, are necessary, by showing that the impossibility of a wait-free algorithm in the nonuniform iterated snapshot model cannot be proved via an extension-based proof. On the positive side, we present a wait-free algorithm for a class of graphs that properly contains the class of chordal graphs.
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Consider a distributed system with $n$ processors out of which $f$ can be Byzantine faulty. In the approximate agreement task, each processor $i$ receives an input value $x_i$ and has to decide on an output value $y_i$ such that - the output values are in the convex hull of the non-faulty processors input values, - the output values are within distance $d$ of each other. Classically, the values are assumed to be from an $m$-dimensional Euclidean space, where $m ge 1$. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph $G$ and the goal is to output vertices that are within distance $d$ of each other in $G$, but still remain in the graph-induced convex hull of the input values. For $d=0$, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any $d ge 1$, we show that the task is solvable in asynchronous systems when $G$ is chordal and $n > (omega+1)f$, where $omega$ is the clique number of~$G$. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.
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