Self-organization and evolution on large computer data structures

We study the long time evolution of a large data structure while inserting new items. It is implemented using a well known computer science approach based on 2-3 trees. We have seen self-organized critical behavior on this data structure. To tackle this problem we have introduced and studied experimentally three statistical magnitudes: the stress of a tree, the sequence of jump points and the distribution of subtrees inside a tree. The stress measures the amount of free space inside the 2-3 tree. When the stress increases some part of the tree is restructured in a way close to an avalanche. Experimentally we obtain a potential law for stress distribution. When the tree does not have more free space in any internal node, needs to grow up. When this happens, the height of the whole tree increases by one and we have a jump point. Experimentally these points have good expected behavior.A 2-3 tree is composed from a great number of other 2-3 trees called their subtrees. We have studied experimentally the distribution of the different subtrees inside the tree. Finally we analyze these results using simple theoretical models based on fringe analysis, Markov and branching processes. These models give us a quite good description of the long term process.Postprint (published version

Higher-Order Analysis of 2-3 Trees

We present a fourth-order fringe analysis for the expected behavior of 2-3 trees, which includes 97% of the elements in the tree. It is accomplished by exploiting the structure of the transition matrix. Our results improve a number of bounds, in particular the bounds on the expected number of nodes and the expected space utilization. We also study 2-3 trees built by using overflow techniques. 1 Introduction Fringe analysis was formally introduced by Yao in 1974 [Yao74, Yao78] as a method to analyze search trees that considers only the bottom part or fringe of the tree. From the behavior of the subtrees in the fringe, it is possible to obtain bounds on most complexity measures for the complete tree, as well as some exact results. Classical fringe analysis considers only insertions. The model assumes that the n! possible permutations of the n keys used as input are equally likely. A search tree built under this model is called a random tree. This is equivalent to saying that the n-th in..