A Limit Theorem for the Six-length of Random Functional Graphs with a Fixed Degree Sequence

Kevin Leckey,Nick Wormald

doi:10.37236/7710

Abstract

We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v, f(v), \ldots, f^{i-1}(v)$ are all distinct. This has relevance to the study of algorithms for integer factorisation.

Highlights

We consider random directed graphs with all out-degrees equal to 1, which we call functional graphs or random mappings
The runtime depends on the six-length of a polynomial in Fp[x]. (Pollard’s first version used x2 − 1.) Under the assumption that a polynomial mod p ‘behaves like’ a random mapping, we are interested in the sixlength of random mappings
This section contains the proof of Theorem 1 for degree sequences that satisfy (A+), that is we prove the following statement: Proposition 5

Summary

Introduction

We consider random directed graphs with all out-degrees equal to 1, which we call functional graphs (see Section 2 for further notation) or random mappings. The electronic journal of combinatorics 26(4) (2019), #P4.35 functional graphs with given in-degree sequence, to give a baseline for comparison with random polynomial models. For a fixed set D, they considered a functional graph chosen uniformly at random among those with in-degrees in D. They studied various properties such as the in-degrees of vertices, tree size, tail length and six-length. In particular we give the limiting distribution of the six-length for functional graphs with given in-degree sequence, and asymptotics for the moments of the distribution, as well as the joint distribution of the tail- and six-lengths. Let sn(v) and tn(v) be sixand tail-length of a vertex v ∈ [n] in a random functional graph with degree sequence dn. These results support a conjecture by Brent and Pollard [3, Section 3] on the typical tailand cycle-length of polynomials mod p

Proofs for sequences with large coalescence

Limit theorem for the six-length

An extension to cases with small coalescence