Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, April 06, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Maxime Larcher

Efficient Generation of Size-Conditioned Galton-Watson Trees

A Galton-Watson tree is a random tree where each node has a random number of children, drawn i.i.d. according to some given probability distribution X. Sampling from uniformly from usual families of trees can sometimes be reduced to sampling a Galton-Watson tree (conditioned on the size) with an appropriate X. For instance, if X is uniform over {0,2} then the distribution of the Galton-Watson tree of size n is uniform over binary trees.

In 2012, Devroye showed it was possible to draw a Galton-Watson tree conditioned on having size exactly n in linear time, provided that E[X]=1 and the second moment of X is bounded. We extend his result and show that this is possible as long the a-th moment of X is bounded, for some a ∈ (1,2]. We also explore the case E[X] ≠ 1.

This is work in progress, joint with Johannes Lengler, Frank Mousset, Andreas Noever and Konstantinos Panagiotou.

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