Around Kolmogorov complexity: basic notions and results

Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author

Smooth heaps and a dual view of self-adjusting data structures

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures. Roughly speaking, we map an arbitrary heap algorithm within a natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature. Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal. Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. As corollaries of results known for Greedy, we obtain instance-specific upper bounds for the smooth heap, with applications in adaptive sorting. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a "power-of-two-choices" type of heuristic.Comment: Presented at STOC 2018, light revision, additional figure

Complex Query Operators on Modern Parallel Architectures

Identifying interesting objects from a large data collection is a fundamental problem for multi-criteria decision making applications.In Relational Database Management Systems (RDBMS), the most popular complex query operators used to solve this type of problem are the Top-K selection operator and the Skyline operator.Top-K selection is tasked with retrieving the k-highest ranking tuples from a given relation, as determined by a user-defined aggregation function.Skyline selection retrieves those tuples with attributes offering (pareto) optimal trade-offs in a given relation.Efficient Top-K query processing entails minimizing tuple evaluations by utilizing elaborate processing schemes combined with sophisticated data structures that enable early termination.Skyline query evaluation involves supporting processing strategies which are geared towards early termination and incomparable tuple pruning.The rapid increase in memory capacity and decreasing costs have been the main drivers behind the development of main-memory database systems.Although the act of migrating query processing in-memory has created many opportunities to improve the associated query latency, attaining such improvements has been very challenging due to the growing gap between processor and main memory speeds.Addressing this limitation has been made easier by the rapid proliferation of multi-core and many-core architectures.However, their utilization in real systems has been hindered by the lack of suitable parallel algorithms that focus on algorithmic efficiency.In this thesis, we study in depth the Top-K and Skyline selection operators, in the context of emerging parallel architectures.Our ultimate goal is to provide practical guidelines for developing work-efficient algorithms suitable for parallel main memory processing.We concentrate on multi-core (CPU), many-core (GPU), and processing-in-memory architectures (PIM), developing solutions optimized for high throughout and low latency.The first part of this thesis focuses on Top-K selection, presenting the specific details of early termination algorithms that we developed specifically for parallel architectures and various types of accelerators (i.e. GPU, PIM).The second part of this thesis, concentrates on Skyline selection and the development of a massively parallel load balanced algorithm for PIM architectures.Our work consolidates performance results across different parallel architectures using synthetic and real data on variable query parameters and distributions for both of the aforementioned problems.The experimental results demonstrate several orders of magnitude better throughput and query latency, thus validating the effectiveness of our proposed solutions for the Top-K and Skyline selection operators

Fast Parallel Algorithms for Basic Problems

Parallel processing is one of the most active research areas these days. We are interested in one aspect of parallel processing, i.e. the design and analysis of parallel algorithms. Here, we focus on non-numerical parallel algorithms for basic combinatorial problems, such as data structures, selection, searching, merging and sorting. The purposes of studying these types of problems are to obtain basic building blocks which will be useful in solving complex problems, and to develop fundamental algorithmic techniques. In this thesis, we study the following problems: priority queues, multiple search and multiple selection, and reconstruction of a binary tree from its traversals. The research on priority queue was motivated by its various applications. The purpose of studying multiple search and multiple selection is to explore the relationships between four of the most fundamental problems in algorithm design, that is, selection, searching, merging and sorting; while our parallel solutions can be used as subroutines in algorithms for other problems. The research on the last problem, reconstruction of a binary tree from its traversals, was stimulated by a challenge proposed in a recent paper by Berkman et al. ( Highly Parallelizable Problems, STOC 89) to design doubly logarithmic time optimal parallel algorithms because a remarkably small number of such parallel algorithms exist