A Mergeable Double-Ended Priority Queue

An implementation of a double-ended priority queue is discussed. This data structure referred to as min–max–pair heap can be built in linear time; the operations Delete-min, Delete-max and Insert take O(log n) time, while Find-min and Find-max run in O(1) time. In contrast to the min-max heaps, it is shown that two min–max–pair heaps can be merged in sublinear time. More precisely, two min–max–pair heaps of sizes n and k can be merged in time O(log (n/k) * log k)

A pointer-free data structure for merging heaps and min-max heaps

AbstractIn this paper a data structure for the representation of mergeable heaps and min-max heaps without using pointers is introduced. The supported operations are: Insert, DeleteMax, DeleteMin, FindMax, FindMin, Merge, NewHeap, DeleteHeap. The structure is analyzed in terms of amortized time complexity, resulting in a O(1) amortized time for each operation except for Insert, for which a O(lg n) bound holds

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

Efficient algorithms for minimax decisions under tree-structured incompleteness

When decisions must be based on incomplete (coarsened) observations and the coarsening mechanism is unknown, a minimax approach offers the best guarantees on the decision maker’s expected loss. Recent work has derived mathematical conditions characterizing minimax optimal decisions, but also found that computing such decisions is a difficult problem in general. This problem is equivalent to that of maximizing a certain conditional entropy expression. In this work, we present a highly efficient algorithm for the case where the coarsening mechanism can be represented by a tree, whose vertices are outcomes and whose edges are coarse observations

Automated Compilation Framework for Scratchpad-based Real-Time Systems

ScratchPad Memory (SPM) is highly adopted in real-time systems as it exhibits a predictable behaviour. SPM is software-managed by explicitly inserting instructions to move code and data transfers between the SPM and the main memory. However, it is a tedious job to decide how to manage the SPM and to manually modify the code to insert memory transfers. Hence, an automated compilation tool is essential to efficiently utilize the SPM. Another key problem with SPM is the latency suffered by the system due to memory transfers. Hiding this latency is important for high-performance systems. In this thesis, we address the problems of managing SPM and reducing the impact of memory latency. To realize the automation of our work, we develop a compilation framework based on the LLVM compiler to analyze and transform the program code. We exploit our framework to improve the performance of the execution of single and multi-tasks in real-time systems. For the single task execution, Worst-Case Execution Time (WCET) is of great importance to assure correct and safe behaviour of the system. So, we propose a WCET-driven allocation technique for data SPM that employs software prefetching to efficiently manage the SPM and to overlap the memory transfer and the task execution in a predictable way. On the other hand, multi-tasking requires the system to be schedulable such that all the tasks can meet their timing requirements. However, executing multiple tasks on a multi-processor platform suffers from the contention of the accesses to the shared main memory. To avoid the contention, several scheduling techniques adopted the 3-phase execution model which executes the task as a sequence of memory and computation phases. This provides the means to avoid the contention as well as to hide the memory latency by using a Direct Memory Access (DMA) engine. Executing memory transfers using the DMA allows overlapping the memory transfers with the computations on the processor. Using the 3-phase model in systems with limited sizes of local SPM may necessitate a segmentation of the task. Automating the segmentation process is necessary especially for systems with large task sets. Hence, we propose a set of efficient segmentation algorithms that follow the 3-phase execution model. The application of these algorithms shows a significant improvement in the system schedulability. For our segmentation algorithms to be more applicable, we extend the 3-phase model to allow programs with multiple paths represented as conditional Directed Acyclic Graphs (DAGs), unlike the previous works that targeted sequential programs. We also introduce a multi-steaming model to exploit the benefits of prefetching by overlapping the memory and computation phases of the same task, which was not allowed in the previous approaches. By combining the automated compilation with the proposed algorithms, we are able to achieve our goal to efficiently manage data SPM in real-time systems

Efficient Algorithms for Finding Maximum Matchings in Convex Bipartite Graphs and Related Problems

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / DAAB-07-72-C-0259 and DAAG-20-78-C-0016National Science Foundation / MCS76-1732

Implementation of operations in double-ended heaps

Existuje viacero spôsobov ako vytvoriť dvojkoncovú haldu z dvoch klasických háld. V tejto práci rozšírime dvojkoncovú haldu založenú na prepojení listov a vytvoríme novú schému nazvanú L-korešpondencia. Táto schéma rozšíri triedu možných klasických háld použiteľných pre vytvorenie dvojkoncovej haldy (napr. Fibonacci halda, Rank-pairing halda). Ďalej umožní operácie ``Zníž prioritu'' a ``Zvýš prioritu''. Tento prístup ukážeme na troch konkrétnych haldách a odhadneme časovú zložitosť pre všetky operácie. Ďalším výsledkom je, že pre tieto tri konkrétne haldy, očakávaný čas operácií ``Zníž prioritu'' a ``Zvýš prioritu'' je obmedzený konštantou.There are several approaches for creating double-ended heaps from the single-ended heaps. We build on one of them, the leaf correspondence heap, to create a generic double ended heap scheme called L-correspondence heap. This will broaden the class of eligible base single-ended heaps (e.g. by Fibonacci heap, Rank-pairing heap) and make the operations Decrease and Increase possible. We show this approach on specific examples for three different single-ended base heaps and give time complexity bounds for all operations. Another result is that for these three examples, the expected amortized time for Decrease and Increase operations in the L-correspondence heap is bounded by a constant.Department of Theoretical Computer Science and Mathematical LogicKatedra teoretické informatiky a matematické logikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Doctor of Philosophy

dissertationMatrices are essential data representations for many large-scale problems in data analytics; for example, in text analysis under the bag-of-words model, a large corpus of documents are often represented as a matrix. Many data analytic tasks rely on obtaining a summary (a.k.a sketch) of the data matrix. Using this summary in place of the original data matrix saves on space usage and run-time of machine learning algorithms. Therefore, sketching a matrix is often a necessary first step in data reduction, and sometimes has direct relationships to core techniques including PCA, LDA, and clustering. In this dissertation, we study the problem of matrix sketching over data streams. We first describe a deterministic matrix sketching algorithm called FrequentDirections. The algorithm is presented an arbitrary input matrix A∈ Rn&× d one row at a time. It performs O(dl) operations per row and maintains a sketch matrix B ∈ Rl× d such that for any k< l, ||ATA - BTB \|| 2 < ||A - Ak||F2 / (l-k) and ||A - πBk(A)||F2 ≤ (1 + k/l-k)||A-Ak||F2 . Here, Ak stands for the minimizer of ||A - Ak||F over all rank k matrices (similarly Bk), and πBk (A) is the rank k matrix resulting from projecting A on the row span of Bk. We show both of these bounds are the best possible for the space allowed, the sketch is mergeable, and hence trivially parallelizable. We propose several variants of FrequentDirections that improve its error-size tradeoff, and nearly matches the simple heuristic Iterative SVD method in practice. We then describe SparseFrequentDirections for sketching sparse matrices. It resembles the original algorithm in many ways including having the same optimal asymptotic guarantees with respect to the space-accuracy tradeoff in the streaming setting, but unlike FrequentDirections which runs in O(ndl) time, SparseFrequentDirections runs in Õ(nnz(A)l + nl2) time. We then extend our methods to distributed streaming model, where there are m distributed sites each observing a distinct stream of data, and which has a communication channel with a coordinator. The goal is to track an ε-approximation (for ε ∈ (0,1)) to the norm of the matrix along any direction. We present novel algorithms to address this problem. All our methods satisfy an additive error bound that for any unit vector x, | ||A x||2 - ||B x ||2 | ≤ |ε ||A||F2 holds