# Setting Ports in an Anonymous Network: How to Reduce the Level of Symmetry?

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9988)

## Abstract

A fundamental question in the setting of anonymous graphs concerns the ability of nodes to spontaneously break symmetries, based on their local perception of the network. In contrast to previous work, which focuses on symmetry breaking under arbitrary port labelings, in this paper, we study the following design question: Given an anonymous graph G without port labels, how to assign labels to the ports of G, in interval form at each vertex, so that symmetry breaking can be achieved using a message-passing protocol requiring as few rounds of synchronous communication as possible?

More formally, for an integer $$l>0$$, the truncated view $$\mathcal {V}_l(v)$$ of a node v of a port-labeled graph is defined as a tree encoding labels encountered along all walks in the network which originate from node v and have length at most l, and we ask about an assignment of labels to the ports of G so that the views $${\mathcal {V}_{l}}(v)$$ are distinct for all nodes $$v\in V$$, with the goal being to minimize l.

We present such efficient port labelings for any graph G, and we exhibit examples of graphs showing that the derived bounds are asymptotically optimal in general. More precisely, our results imply the following statements.

1. 1.

For any graph G with n nodes and diameter D, a uniformly random port labeling achieves $$l = O(\min (D,\log n))$$, w.h.p.

2. 2.

For any graph G with n nodes and diameter D, it is possible to construct in polynomial time a labeling that satisfies $$l = O(\min (D,\log n))$$.

3. 3.

For any integers $$n\ge 2$$ and $$D \le \log _2 n-\log _2\log _2 n$$, there exists a graph G with n nodes and diameter D which satisfies $$l \ge \frac{1}{2} D - \frac{5}{2}$$.

## Keywords

Anonymous network Port-labeled network View Level of symmetry

## Notes

### Acknowlegdements

The authors would like to thank Philippe Duchon and David Ilcinkas for proposing the problem, and for some initial discussions.

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