Split Trees, Cuttings and Explosions

University dissertation from Uppsala : Department of Mathematics

Abstract: This thesis is based on four papers investigating properties of split trees and also introducing new methods for studying such trees. Split trees comprise a large class of random trees of logarithmic height and include e.g., binary search trees, m-ary search trees, quadtrees, median of (2k+1)-trees, simplex trees, tries and digital search trees. Split trees are constructed recursively, using “split vectors”, to distribute n “balls” to the vertices/nodes. The vertices of a split tree may contain different numbers of balls; in computer science applications these balls often represent “key numbers”.In the ?rst paper, it was tested whether a recently described method for determining the asymptotic distribution of the number of records (or cuts) in a deterministic complete binary tree could be extended to binary search trees. This method used a classical triangular array theorem to study the convergence of sums of triangular arrays to in?nitely divisible distributions. It was shown that with modi?cations, the same approach could be used to determine the asymptotic distribution of the number of records (or cuts) in binary search trees, i.e., in a well-characterized type of random split trees.In the second paper, renewal theory was introduced as a novel approach for studying split trees. It was shown that this theory is highly useful for investigating these types of trees. It was shown that the expected number of vertices (a random number) divided by the number of balls, n, converges to a constant as n tends to in?nity. Furthermore, it was demonstrated that the number of vertices is concentrated around its mean value. New results were also presented regarding depths of balls and vertices in split trees.In the third paper, it was tested whether the methods of proof to determine the asymptotic distribution of the number of records (or cuts) used in the binary search tree, could be extended to split trees in general. Using renewal theory it was demonstrated for the overall class of random split trees that the normalized number of records (or cuts) has asymptotically a weakly 1-stable distribution.In the fourth paper, branching Markov chains were introduced to investigate split trees with immigration, i.e., CTM protocols and their generalizations. It was shown that there is a natural relationship between the Markov chain and a multi-type (Galton-Watson) process that is well adapted to study stability in the corresponding tree. A stability condition was presented to de­scribe a phase transition deciding when the process is stable or unstable (i.e., the tree explodes). Further, the use of renewal theory also proved to be useful for studying split trees with immi­gration. Using this method it was demonstrated that when the tree is stable (i.e., ?nite), there is the same type of expression for the number of vertices as for normal split trees.