Magister Curationis / Binary search trees (BSTs) are important data structures which are widely used in various guises. Splay trees are a specific kind of binary search tree, one without explicit balancing. Skip lists use more space than BSTs and are related to them in terms of much of their run-time behavior. Even though binary search trees have been used for about half a century, there are still many open questions regarding their run-time performance and algorith mic complexity. In many instances, their worst-case, average-case, and best-case behaviors are unknown and need further research. Our analysis provides a basis for selecting more suitable data structures and algorithms for specific processes and applications. We contrast the empirical behavior of splay trees and skip lists with their theoretical behavior. In particular we explore when splay trees outperform skip lists and vice versa. The performance of a splay tree depends on the history of accesses to its el ements. On the other hand, the performance of a skip list depends on an indepen dent randomization of the height of links that lead to specific elements. Therefore, probabilistic methods are used to analyze the operation of splay trees and skip lists. Our main results are that splay trees are faster for sorted insertion, where AVL trees are faster for random insertion. For searching, skip lists are faster than single class top-down splay trees, but two-class and multi-class top-down splay trees can behave better than skip lists.

Longest prefix matching is a well-studied problem in the context of IP-packet forwarding, an area of computational specialization where high performance with low memory impact is of the essence. Numerous specialized data structures exist for longest prefix matching in this setting, among them is Waldvogels binary search on lengths, WBSL for short. This data structure has previously been augmented with a Bloom filter to improve its performance, resulting in a new data structure: WBSL-BF. This study concerns the potential performance improvement of adding length-awareness to WBSL-BF. Performance of the augmented data structure is evaluated by recording the number of hash table accesses performed to carry out a series of queries, along with false positive rates for the Bloom filters. A test program is created to run multiple series of queries with different test configurations. The data structure is evaluated using two datasets containing prefixes from routing information base snapshots from real route collectors. The two datasets contain roughly 80000 and 900000 prefixes respectively. The results indicate that on the given datasets, augmenting the Bloom filter with length-awareness can reduce the number of hash table accesses when performing longest prefix matching by up to 44% in a best case scenario without increasing memory usage. However, the performance gain is limited in other scenarios, since the optimally configured standard Bloom filter leaves little to improve upon when applied to Waldvogels binary search on lengths.

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 first 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 infinitely divisible distributions. It was shown that with modifications, 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 infinity. 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., finite), there is the same type of expression for the number of vertices as for normal split trees.

In this thesis we compared some variants of binary search trees that approach dynamic optimality: Tango trees, Multisplay trees, and Splay trees. We empirically tested the behavior of these three types of trees, as well as Red-Black trees. We measured the amount of visited nodes per operation and running time on real hardware. We proved that Tango trees and Multisplay trees are in most cases less efficient than Splay trees and Red-Black trees. Cache-related effects played a surprisingly large part in the behavior of Red-Black tree and Splay tree. 1

A simple search problem is studied in which a binary n-tuple is to be found in a list, by sequential bit comparisons with cost. The problem can be solved (for small n) using dynamic programming. We show how the “bottom up” part of the algorithm can be organized by means of Formal Concept Analysis.

No description available.

This master's thesis explains basics of the longest prefix match (LPM) problem. It analyzes and describes chosen LPM algorithms considering their speed, memory requirements and an ability to implement them in hardware. On the basis of former findings it proposes a new algorithm Generic Hash Tree Bitmap. It is much faster than many other approaches, while its memory requirements are even lower. An implementation of the proposed algorithm has become a part of the Netbench library.

The main objective of this work is to design and implement an application that can be used as an aid for the education of programming essentials. Particularly, the attention focuses on the domain of dynamic data structures. The target application will be implemented with the use of web technologies so that it can be run in an ordinary WWW browser. First of all, a brief introduction recapitulates the data structures to be covered. Then the work summarizes the usable technologies available within the web browsers with the focus on the particular technology (which is DHTML) that will become the target platform. The most significant part of this work then discusses the design of the final application. This rather theoretical part is then followed by the description of the practical implementation. A short user manual is also included.

Cette thèse comporte deux parties indépendantes. Dans la première partie, nous nous intéressons à l’analyse asymptotique de quelques statistiques sur les arbres binaires de recherche (ABR). Dans la deuxième partie, nous nous intéressons à l’étude du problème du collectionneur de coupons impatient. Dans la première partie, en suivant le modèle introduit par Aguech, Lasmar et Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], on définit la profondeur pondérée d’un nœud dans un arbre binaire enraciné étiqueté comme la somme de toutes les clés sur le chemin qui relie ce nœud à la racine. Nous analysons alors dans ABR, les profondeurs pondérées des nœuds avec des clés données, le dernier nœud inséré, les nœuds ordonnés selon le processus de recherche en profondeur, la profondeur pondérée des trajets, l’indice de Wiener pondéré et les profondeurs pondérées des nœuds avec au plus un enfant. Dans la deuxième partie, nous étudions la forme asymptotique de la courbe de la complétion de la collection conditionnée à T_n≤ (1+Λ), Λ>0, où T_n≃n ln⁡n désigne le temps nécessaire pour compléter la collection. Puis, en tant qu’application, nous étudions les automates déterministes et accessibles et nous fournissons une nouvelle dérivation d’une formule due à Korsunov [Kor78, Kor86] / This Phd thesis is divided into two independent parts. In the first part, we provide an asymptotic analysis of some statistics on the binary search tree. In the second part, we study the coupon collector problem with a constraint. In the first part, following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyze the following statistics : the weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search procees, the weighted path length, the weighted Wiener index and the weighted depths of nodes with at most one child in a random binary search tree. In the second part, we study the asymptotic shape of the completion curve of the collection conditioned to T_n≤ (1+Λ), Λ>0, where T_n≃n ln⁡n is the time needed to complete accessible automata, we provide a new derivation of a formula due to Korsunov [Kor78, Kor86]