Abstract
The class of robot convergence tasks has been shown to capture fundamental aspects of fault-tolerant computability. A set of asynchronous robots that may fail by crashing, start from unknown places in some given space, and have to move towards positions close to each other. In this article, we study the case where the space is uni-dimensional, modeled as a graph G. In graph convergence, robots have to end up on one or two vertices of the same edge. We consider also a variant of robot convergence on graphs, edge covering, where additionally, it is required that not all robots end up on the same vertex. Remarkably, these two similar problems have very different computability properties, related to orthogonal fundamental issues of distributed computations: agreement and symmetry breaking. We characterize the graphs on which each of these problems is solvable, and give optimal time algorithms for the solvable cases. Although the results can be derived from known general topology theorems, the presentation serves as a self-contained introduction to the algebraic topology approach to distributed computing, and yields concrete algorithms and impossibility results.
Highlights
The family of robot convergence tasks plays a fundamental role in the theory of fault-tolerant distributed computing [21]
It is used to prove the wait-free computability theorem [25] that characterizes the tasks that are wait-free solvable in a read/write shared memory environment, and is intimately related to the simplicial approximate agreement theorem of topology
Impossibility results we present a series of impossibility results that fully characterize the solvability of graph convergence and edge covering
Summary
The family of robot convergence tasks plays a fundamental role in the theory of fault-tolerant distributed computing [21]. The space K could be the d-dimensional Euclidean space as in [28], and robots may be required to converge on regions spanned by the convex hull of their initial positions; if all start on the same point, they should remain there, and if all start in two points, they should converge to points close to each other along the straight line connecting the two points Another example is the loop agreement task [23], where there is a given loop in the space K, and three given distinguished points v1, v2, v3 on the loop.
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