## Project Details

### Description

Random Recursive Structures of Small Diameters

Abstract

The major aims of the AustrianTaiwanese

joint project are: (1) to get a deeper understanding

of the stochastic nature of random trees and more general structures which have a small

diameter, (2) development and advancement of general analytic tools, (3) stimulating

interdisciplinary research on an international level.

Tree structures play an important role in many areas like computer science, quantum

mechanics, biology and many more, and also in many subfields inside mathematics. They

serve for instance as data structure. If one faces random data and stores them in a tree, then

the resulting tree will be a random tree. In order to understand how this tree typically looks

like (for instance for designing efficient algorithms for information retrieval which exploit the

knowledge on the shape of the tree), it is necessary to consider large random trees and

analyze them systematically.

Trees may serve as a model of some real world phenomenon which we want to understand.

In order to analyze a model, one starts with simple models, like e.g. branching processes.

This model is very well understood, but its drawback is that there are many situations where it

is not realistic. There are other models like binary search trees which are wellsuited

for

certain frameworks in computer science. Many of those new models have the property that

the socalled

diameter is small in comparison to the old models.

In contrary to the branching process models, there exists no general framework for the new

models, but only many partial results for many different models of trees with small diameter,

Partial results indicate that such a framework can be described by means of Riccatilike

differential equations. Our aim is to explore this phenomenon and to enhance existing

methods for retrieving the desired information from these equations.

A further goal of the project is to deepen the understanding of the typical shape of various

classes of random trees. This is done by first studying the profile of random trees and then

several shape parameters simultaneously.

In applications one is often interested in the number of way to decompose a complex

structure in distinct parts. This is a highly nontrivial problem, but we hope that the tools we

develop during the project will enable us to obtain deep results on this question. In this

context, there are examples of tree models from biology, but also other interesting structures

from other fields of mathematics. Finally, a particular problem from information theory shall be

investigated: If we do not focus on the tree size but on some other parameter, then this gives

rise to a bias towards small diameter and probably trees with very different behaviour.

Abstract

The major aims of the AustrianTaiwanese

joint project are: (1) to get a deeper understanding

of the stochastic nature of random trees and more general structures which have a small

diameter, (2) development and advancement of general analytic tools, (3) stimulating

interdisciplinary research on an international level.

Tree structures play an important role in many areas like computer science, quantum

mechanics, biology and many more, and also in many subfields inside mathematics. They

serve for instance as data structure. If one faces random data and stores them in a tree, then

the resulting tree will be a random tree. In order to understand how this tree typically looks

like (for instance for designing efficient algorithms for information retrieval which exploit the

knowledge on the shape of the tree), it is necessary to consider large random trees and

analyze them systematically.

Trees may serve as a model of some real world phenomenon which we want to understand.

In order to analyze a model, one starts with simple models, like e.g. branching processes.

This model is very well understood, but its drawback is that there are many situations where it

is not realistic. There are other models like binary search trees which are wellsuited

for

certain frameworks in computer science. Many of those new models have the property that

the socalled

diameter is small in comparison to the old models.

In contrary to the branching process models, there exists no general framework for the new

models, but only many partial results for many different models of trees with small diameter,

Partial results indicate that such a framework can be described by means of Riccatilike

differential equations. Our aim is to explore this phenomenon and to enhance existing

methods for retrieving the desired information from these equations.

A further goal of the project is to deepen the understanding of the typical shape of various

classes of random trees. This is done by first studying the profile of random trees and then

several shape parameters simultaneously.

In applications one is often interested in the number of way to decompose a complex

structure in distinct parts. This is a highly nontrivial problem, but we hope that the tools we

develop during the project will enable us to obtain deep results on this question. In this

context, there are examples of tree models from biology, but also other interesting structures

from other fields of mathematics. Finally, a particular problem from information theory shall be

investigated: If we do not focus on the tree size but on some other parameter, then this gives

rise to a bias towards small diameter and probably trees with very different behaviour.

Status | Finished |
---|---|

Effective start/end date | 1/03/16 → 31/08/20 |