New Combinatorial Properties and Algorithms for AVL Trees

In this thesis, new properties of AVL trees and a new partitioning of binary search trees named core partitioning scheme are discussed, this scheme is applied to three binary search trees namely AVL trees, weight-balanced trees, and plain binary search trees. We introduce the core partitioning scheme, which maintains a balanced search tree as a dynamic collection of complete balanced binary trees called cores. Using this technique we achieve the same theoretical efficiency of modern cache-oblivious data structures by using classic data structures such as weight-balanced trees or height balanced trees (e.g. AVL trees). We preserve the original topology and algorithms of the given balanced search tree using a simple post-processing with guaranteed performance to completely rebuild the changed cores (possibly all of them) after each update. Using our core partitioning scheme, we simultaneously achieve good memory allocation, space-efficient representation, and cache-obliviousness. We also apply this scheme to arbitrary binary search trees which can be unbalanced and we produce a new data structure, called Cache-Oblivious General Balanced Tree (COG-tree). Using our scheme, searching a key requires O(log_B n) block transfers and O(log n) comparisons in the external-memory and in the cache-oblivious model. These complexities are theoretically efficient. Interestingly, the core partition for weight-balanced trees and COG-tree can be maintained with amortized O(log_B n) block transfers per update, whereas maintaining the core partition for AVL trees requires more than a poly-logarithmic amortized cost. Studying the properties of these trees also lead us to some other new properties of AVL trees and trees with bounded degree, namely, we present and study gaps in AVL trees and we prove Tarjan et al.'s conjecture on the number of rotations in a sequence of deletions and insertions

Random generation of RNA secondary structures according to native distributions

Nebel M, Scheid A, Weinberg F. Random generation of RNA secondary structures according to native distributions. Algorithms for Molecular Biology. 2011;6(1): 24

Advances and Novel Approaches in Discrete Optimization

Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

An almost-linear time algorithm for uniform random spanning tree generation

We give an $m^{1+o(1)}\beta^{o(1)}$-time algorithm for generating a uniformly random spanning tree in an undirected, weighted graph with max-to-min weight ratio $\beta$. We also give an $m^{1+o(1)}\epsilon^{-o(1)}$-time algorithm for generating a random spanning tree with total variation distance $\epsilon$ from the true uniform distribution. Our second algorithm's runtime does not depend on the edge weights. Our $m^{1+o(1)}\beta^{o(1)}$-time algorithm is the first almost-linear time algorithm for the problem --- even on unweighted graphs --- and is the first subquadratic time algorithm for sparse weighted graphs. Our algorithms improve on the random walk-based approach given in Kelner-M\k{a}dry and M\k{a}dry-Straszak-Tarnawski. We introduce a new way of using Laplacian solvers to shortcut a random walk. In order to fully exploit this shortcutting technique, we prove a number of new facts about electrical flows in graphs. These facts seek to better understand sets of vertices that are well-separated in the effective resistance metric in connection with Schur complements, concentration phenomena for electrical flows after conditioning on partial samples of a random spanning tree, and more

Notes on Randomized Algorithms

Lecture notes for the Yale Computer Science course CPSC 469/569 Randomized Algorithms. Suitable for use as a supplementary text for an introductory graduate or advanced undergraduate course on randomized algorithms. Discusses tools from probability theory, including random variables and expectations, union bound arguments, concentration bounds, applications of martingales and Markov chains, and the Lov\'asz Local Lemma. Algorithmic topics include analysis of classic randomized algorithms such as Quicksort and Hoare's FIND, randomized tree data structures, hashing, Markov chain Monte Carlo sampling, randomized approximate counting, derandomization, quantum computing, and some examples of randomized distributed algorithms