# Computing on a Partially Eponymous Ring

## Abstract

We study the *partially eponymous* model of distributed computation, which simultaneously generalizes the *anonymous* and the *eponymous* models. In this model, processors have *identities*, which are neither necessarily all identical, nor necessarily unique; processors receive *inputs* and must reach *outputs* that respect a *relation*. We focus on the *partially eponymous ringR*, and we are interested in the computation of *circularly symmetric* relations on it.

∙ We distinguish between solvability and computability: in *solvability*, processors must *always* reach outputs that respect the relation; in *computability*, they must reach outputs that respect the relation whenever possible, and report impossibility otherwise.

– We provide an *efficient* characterization of solvability of an arbitrary (circularly symmetric) relation on an arbitrary set of rings. The characterization is topological and can be expressed as a number-theoretic property that can be checked efficiently.

– We present a *universal* distributed algorithm for computing any arbitrary (circularly symmetric) relation on any set of rings.

∙ Towards obtaining *message complexity* bounds, we derive a distributed algorithm for a natural generalization of *Leader Election*, in which a (non-zero) number of leaders are elected. We use this algorithm as a subroutine of our universal algorithm for collecting views; hence, we prove, as our main result, an upper bound on the message complexity of this particular instantiation of our universal algorithm to compute an *arbitrary* (circularly symmetric) relation on an *arbitrary* set of rings. The shown upper bound demonstrates a graceful degradation with the *Least Minimum Base*, a parameter indicating the degree of topological compatibility between the relation and the set of rings. We employ this universal upper bound to identify two interesting cases where an arbitrary relation can be computed with an efficient number of \(O(|R| \cdot \lg |R|)\) messages: The set of rings is *universal* (which allows the solvability of Leader Election), or *logarithmic* (where each identity appears at most \(\lg |R|\) times).

## Keywords

Input Vector Output Vector Local Algorithm Leader Election Message Complexity## Preview

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