This research is aimed at the probabilistic analysis of 'divide-and-conquer' algorithms, which include fundamental examples such as quicksort, partial match queries on k-d trees, and multiple selection.
In order to gain a detailed understanding of such algorithms, a fine stochastic analysis is needed which may include higher order moments, concentration around the mean and in the best base, a limiting distribution.
The contraction method, introduced by Rosler in 1991, was the first method capable of unlocking the exact limiting distribution of the quicksort algorithm. Since then, Ruschendorf, and in particular, Neininger have expanded and successfully used this method to achieve limit laws for tens of challenging problems.
The objectives of this research is to broaden the scope of tools to include this new powerful method, and then to investigate new problems which show particular promise in their ability to apply the contraction method. We would like to investigate properties such as the height of certain tree models, in particular, the d-ary pyramids introduced by Mahmoud.
In the area of multivariate search trees, there are still many holes in what is known regarding the partial match query. The aim is to achieve a limiting distribution for the cost o f partial match on the ordinary k-d tree. Multivariate treaps are generalized search trees augmented with heap-ordered time stamps.
This research will also explore the complexity of partial match on this tree using the contraction method. In fact, partial match complexity on random relaxed k-d trees and relaxed quadtrees is as yet totally unexplored.
The research will carefully combine other mathematical tools, such as concentration inequalities, which can nail down certain aspects of the algorithm, branching processes which can often be used to model certain growth behaviours of an algorithm or random structure, and properties of random continuous processes such as Brownian motions and excursions.
Field of science
- /natural sciences/mathematics/applied mathematics/statistics and probability
- /natural sciences/computer and information sciences/software/computer programming
Call for proposal
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