Classifying rendezvous tasks of arbitrary dimension

The rendezvous is a type of distributed decision tasks including many well-known tasks such as set agreement, simplex agreement, and approximation agreement. An ndimensional rendezvous task, n ≥ 1, allows n + 2 distinct input values, and each execution produces at most n + 2 distinct output values. A rendezvous task is said to implement another if an instance of its solution, followed by a protocol based on shared read/write registers, solves the other. The notion of implementation induces a classification of rendezvous tasks of every dimension: two tasks belong to the same class if they implement each other. Previous work on classifying rendezvous tasks only focused on 1dimensional ones. This paper solves an open problemby presenting the classification of nice rendezvous of arbitrary dimension. An n-dimensional rendezvous task is said to be nice if the qth reduced homology group of its decision space is trivial for q 6= n, and free for q = n. Well-known examples are set agreement, simplex agreement, and approximation agreement. Each ndimensional rendezvous task is assigned an algebraic signature, which consists of the nth homology group of the decision space, as well as a distinguished element in the group. It is shown that an n-dimensional nice rendezvous task implements another if and only if there is a homomorphism from its signature to that of the other. Hence the computational power of a nice rendezvous task is completely characterized by its signature. In each dimension, there are infinitely many classes of rendezvous tasks, and exactly countable classes of nice ones. A representative is explicitly constructed for each class of nice rendezvous tasks. © 2009 Elsevier B.V. All rights reserved.