Leveraging Structural Knowledge for Solving Election in Anonymous Networks with Shared Randomness

We study the classical Election problem in anonymous net- works, where solutions can rely on the use of random bits, which may be either shared or unshared among nodes. We provide a complete char- acterization of the conditions under which a randomized Election algo- rithm exists, for arbitrary structural knowledge. Our analysis considers both Las Vegas and Monte Carlo randomized algorithms, under the as- sumptions of shared and unshared randomness. In our setting, random sources are considered shared if the output bits are identical across spe- cific subsets of nodes. The algorithms and impossibility proofs are extensions of those of [5] for the deterministic setting. Our results are a complete generalization of those from [8]. Moreover, as applications, we consider many specific knowledge: no knowledge, a bound on the size, a bound on the number of nodes sharing a source, the size, or the full topology of the network. For each of them, we show how the general characterizations apply, showing they actually correspond to classes of structural knowledge. We also de- scribe also how randomized Election algorithms from the literature fits in this landscape. We therefore provide a comprehensive picture illustrating how knowledge influences the computability of the Election problem in arbitrary anonymous graphs with shared randomness.

💡 Research Summary

The paper investigates the classic leader election problem in anonymous networks under the presence of random bits, distinguishing between shared and unshared sources of randomness. Its central contribution is a complete characterization of when a randomized election algorithm exists, given arbitrary structural knowledge about the network. The authors model knowledge as a recursive family F of connected symmetric B‑labeled digraphs, where each node’s random source label b(v) identifies the set of nodes sharing the same source (a B‑class).

Two key graph‑theoretic notions are introduced: B‑coverings and B‑quasi‑coverings. A B‑covering is a homomorphism that preserves the B‑labels and is locally bijective; a graph is B‑minimal if every B‑covering is an isomorphism. A B‑quasi‑covering is a partial homomorphism that behaves like a covering on a subgraph, with a parameter called radius measuring how far the covering extends.

Theorem 2.1 (Las Vegas case) states that a Las Vegas election algorithm for F exists iff every labeled digraph in F is B‑minimal and there is a recursive function τ such that no B‑quasi‑covering of radius larger than τ(D) exists for any D ∈ F except the trivial identity covering. This condition guarantees that the algorithm can terminate with probability p > 0 while always producing a correct final configuration.

Theorem 2.2 (Monte Carlo case) relaxes the B‑minimal requirement: a Monte Carlo election algorithm exists for F iff there is a recursive τ such that for every D ∈ F no proper B‑quasi‑covering of radius larger than τ(D) exists. Here “proper” means the quasi‑covering is not a full covering. This allows some non‑minimal graphs to be handled at the cost of a bounded error probability.

The authors then apply these general results to several standard knowledge classes:

1. No knowledge (F = all graphs). Neither Las Vegas nor Monte Carlo algorithms can guarantee explicit termination because non‑minimal graphs and arbitrarily large quasi‑coverings always exist.

2. Upper bound on size (|G| ≤ S). By setting τ proportional to S, a Monte Carlo algorithm can be designed that terminates always but succeeds with probability depending on τ.

3. Exact size (|G| = S). With precise size information, τ can be set to 0, yielding a Las Vegas algorithm. Moreover, if at least one node has an unshared random source, the graph automatically becomes B‑minimal, reinforcing the result.

4. Size interval (½ T < |G| ≤ T). Similar to the exact‑size case, a bounded τ allows a Monte Carlo algorithm, while a Las Vegas algorithm requires the exact size.

5. Full topology (F = {G}). The existence of a B‑minimal labeling of G determines whether a Las Vegas algorithm is possible; otherwise only a Monte Carlo algorithm may exist.

The paper also relates its findings to prior work. The deterministic quasi‑covering characterization of