A Queueing Analysis of Hashing With Lazy Deletion

Hashing with lazy deletion is a simple method for maintaining a dynamic dictionary: items are inserted and sought as usual in a separate-chaining hash table; however, items that no longer need to be in the data structure remain until a later insertion operation stumbles on them and removes them from the table. Because hashing with lazy deletion does not delete items as soon as possible, it keeps more items in the dictionary than methods that use more careful deletion strategies. On the other hand, its space overhead is much smaller than those more careful methods, so if the number of extra items is not too large, hashing with lazy deletion can be a practical algorithm when space is scarce. In this paper, we analyze the expected amount of excess space used by hashing with lazy deletion

The Maximum Size of Dynamic Data Structures

This paper develops two probabilistic methods that allow the analysis of the maximum data structure size encountered during a sequence of insertions and deletions in data structures such as priority queues, dictionaries, linear lists, and symbol tables, and in sweepline structures for geometry and Very-Large-Scale-Integration (VLSI) applications. The notion of the "maximum" is basic to issues of resource preallocation. The methods here are applied to combinatorial models of file histories and probabilistic models, as well as to a non-Markovian process (algorithm) for processing sweepline information in an efficient way, called "hashing with lazy deletion" (HwLD). Expressions are derived for the expected maximum data structure size that are asymptotically exact, that is, correct up to lower-order terms; in several cases of interest the expected value of the maximum size is asymptotically equal to the maximum expected size. This solves several open problems, including longstanding questions in queueing theory. Both of these approaches are robust and rely upon novel applications of techniques from the analysis of algorithms. At a high level, the first method isolates the primary contribution to the maximum and bounds the lesser effects. In the second technique the continuous-time probabilistic model is related to its discrete analog--the maximum slot occupancy in hashing

Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with then changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n^{1/2}, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n^{1/3}. We also treat some variations, including priority queues. The proofs use methods originally developed for random graphs.Comment: 31 PAGE

Random trees in queueing systems with deadlines

AbstractWe survey our research on scheduling aperiodic tasks in real-time systems in order to illustrate the benefits of modelling queueing systems by means of random trees. Relying on a discrete-time single-server queueing system, we investigated deadline meeting properties of several scheduling algorithms employed for servicing probabilistically arriving tasks, characterized by arbitrary arrival and execution time distributions and a constant service time deadline T. Taking a non-queueing theory approach (i.e., without stable-stable assumptions) we found that the probability distribution of the random time sT where such a system operates without violating any task's deadline is approximately exponential with parameter λT = 1μT, with the expectation E[sT] = μT growing exponentially in T. The value μT depends on the particular scheduling algorithm, and its derivation is based on the combinatorial and asymptotic analysis of certain random trees. This paper demonstrates that random trees provide an efficient common framework to deal with different scheduling disciplines and gives an overview of the various combinatorial and asymptotic methods used in the appropriate analysis

Random hypergraphs for hashing-based data structures

This thesis concerns dictionaries and related data structures that rely on providing several random possibilities for storing each key. Imagine information on a set S of m = |S| keys should be stored in n memory locations, indexed by [n] = {1,…,n}. Each object x [ELEMENT OF] S is assigned a small set e(x) [SUBSET OF OR EQUAL TO] [n] of locations by a random hash function, independent of other objects. Information on x must then be stored in the locations from e(x) only. It is possible that too many objects compete for the same locations, in particular if the load c = m/n is high. Successfully storing all information may then be impossible. For most distributions of e(x), however, success or failure can be predicted very reliably, since the success probability is close to 1 for loads c less than a certain load threshold c^* and close to 0 for loads greater than this load threshold. We mainly consider two types of data structures: • A cuckoo hash table is a dictionary data structure where each key x [ELEMENT OF] S is stored together with an associated value f(x) in one of the memory locations with an index from e(x). The distribution of e(x) is controlled by the hashing scheme. We analyse three known hashing schemes, and determine their exact load thresholds. The schemes are unaligned blocks, double hashing and a scheme for dynamically growing key sets. • A retrieval data structure also stores a value f(x) for each x [ELEMENT OF] S. This time, the values stored in the memory locations from e(x) must satisfy a linear equation that characterises the value f(x). The resulting data structure is extremely compact, but unusual. It cannot answer questions of the form “is y [ELEMENT OF] S?”. Given a key y it returns a value z. If y [ELEMENT OF] S, then z = f(y) is guaranteed, otherwise z may be an arbitrary value. We consider two new hashing schemes, where the elements of e(x) are contained in one or two contiguous blocks. This yields good access times on a word RAM and high cache efficiency. An important question is whether these types of data structures can be constructed in linear time. The success probability of a natural linear time greedy algorithm exhibits, once again, threshold behaviour with respect to the load c. We identify a hashing scheme that leads to a particularly high threshold value in this regard. In the mathematical model, the memory locations [n] correspond to vertices, and the sets e(x) for x [ELEMENT OF] S correspond to hyperedges. Three properties of the resulting hypergraphs turn out to be important: peelability, solvability and orientability. Therefore, large parts of this thesis examine how hyperedge distribution and load affects the probabilities with which these properties hold and derive corresponding thresholds. Translated back into the world of data structures, we achieve low access times, high memory efficiency and low construction times. We complement and support the theoretical results by experiments.Diese Arbeit behandelt Wörterbücher und verwandte Datenstrukturen, die darauf aufbauen, mehrere zufällige Möglichkeiten zur Speicherung jedes Schlüssels vorzusehen. Man stelle sich vor, Information über eine Menge S von m = |S| Schlüsseln soll in n Speicherplätzen abgelegt werden, die durch [n] = {1,…,n} indiziert sind. Jeder Schlüssel x [ELEMENT OF] S bekommt eine kleine Menge e(x) [SUBSET OF OR EQUAL TO] [n] von Speicherplätzen durch eine zufällige Hashfunktion unabhängig von anderen Schlüsseln zugewiesen. Die Information über x darf nun ausschließlich in den Plätzen aus e(x) untergebracht werden. Es kann hierbei passieren, dass zu viele Schlüssel um dieselben Speicherplätze konkurrieren, insbesondere bei hoher Auslastung c = m/n. Eine erfolgreiche Speicherung der Gesamtinformation ist dann eventuell unmöglich. Für die meisten Verteilungen von e(x) lässt sich Erfolg oder Misserfolg allerdings sehr zuverlässig vorhersagen, da für Auslastung c unterhalb eines gewissen Auslastungsschwellwertes c* die Erfolgswahrscheinlichkeit nahezu 1 ist und für c jenseits dieses Auslastungsschwellwertes nahezu 0 ist. Hauptsächlich werden wir zwei Arten von Datenstrukturen betrachten: • Eine Kuckucks-Hashtabelle ist eine Wörterbuchdatenstruktur, bei der jeder Schlüssel x [ELEMENT OF] S zusammen mit einem assoziierten Wert f(x) in einem der Speicherplätze mit Index aus e(x) gespeichert wird. Die Verteilung von e(x) wird hierbei vom Hashing-Schema festgelegt. Wir analysieren drei bekannte Hashing-Schemata und bestimmen erstmals deren exakte Auslastungsschwellwerte im obigen Sinne. Die Schemata sind unausgerichtete Blöcke, Doppel-Hashing sowie ein Schema für dynamisch wachsenden Schlüsselmengen. • Auch eine Retrieval-Datenstruktur speichert einen Wert f(x) für alle x [ELEMENT OF] S. Diesmal sollen die Werte in den Speicherplätzen aus e(x) eine lineare Gleichung erfüllen, die den Wert f(x) charakterisiert. Die entstehende Datenstruktur ist extrem platzsparend, aber ungewöhnlich: Sie ist ungeeignet um Fragen der Form „ist y [ELEMENT OF] S?“ zu beantworten. Bei Anfrage eines Schlüssels y wird ein Ergebnis z zurückgegeben. Falls y [ELEMENT OF] S ist, so ist z = f(y) garantiert, andernfalls darf z ein beliebiger Wert sein. Wir betrachten zwei neue Hashing-Schemata, bei denen die Elemente von e(x) in einem oder in zwei zusammenhängenden Blöcken liegen. So werden gute Zugriffszeiten auf Word-RAMs und eine hohe Cache-Effizienz erzielt. Eine wichtige Frage ist, ob Datenstrukturen obiger Art in Linearzeit konstruiert werden können. Die Erfolgswahrscheinlichkeit eines naheliegenden Greedy-Algorithmus weist abermals ein Schwellwertverhalten in Bezug auf die Auslastung c auf. Wir identifizieren ein Hashing-Schema, das diesbezüglich einen besonders hohen Schwellwert mit sich bringt. In der mathematischen Modellierung werden die Speicherpositionen [n] als Knoten und die Mengen e(x) für x [ELEMENT OF] S als Hyperkanten aufgefasst. Drei Eigenschaften der entstehenden Hypergraphen stellen sich dann als zentral heraus: Schälbarkeit, Lösbarkeit und Orientierbarkeit. Weite Teile dieser Arbeit beschäftigen sich daher mit den Wahrscheinlichkeiten für das Vorliegen dieser Eigenschaften abhängig von Hashing Schema und Auslastung, sowie mit entsprechenden Schwellwerten. Eine Rückübersetzung der Ergebnisse liefert dann Datenstrukturen mit geringen Anfragezeiten, hoher Speichereffizienz und geringen Konstruktionszeiten. Die theoretischen Überlegungen werden dabei durch experimentelle Ergebnisse ergänzt und gestützt

Research on Improving Reliability, Energy Efficiency and Scalability in Distributed and Parallel File Systems

With the increasing popularity of cloud computing and Big data applications, current data centers are often required to manage petabytes or exabytes of data. To store this huge amount of data, thousands or tens of thousands storage nodes are required at a single site. This imposes three major challenges for storage system designers: (1) Reliability---node failure in these datacenters is a normal occurrence rather than a rare situation. This makes data reliability a great concern. (2) Energy efficiency---a data center can consume up to 100 times more energy than a standard office building. More than 10% of this energy consumption can be attributed to storage systems. Thus, reducing the energy consumption of the storage system is key to reducing the overall consumption of the data center. (3) Scalability---with the continuously increasing size of data, maintaining the scalability of the storage systems is essential. That is, the expansion of the storage system should be completed efficiently and without limitations on the total number of storage nodes or performance. This thesis proposes three ways to improve the above three key features for current large-scale storage systems. Firstly, we define the problem of reverse lookup , namely finding the list of objects (blocks) for a failed node. As the first step of failure recovery, this process is directly related to the recovery/reconstruction time. While existing solutions use metadata traversal or data distribution reversing methods for reverse lookup, which are either time consuming or expensive, a deterministic block placement can achieve fast and efficient reverse lookup. However, the deterministic placement solutions are designed for centralized, small-scale storage architectures such as RAID etc.. Due to their lacking of scalability, they cannot be directly applied in large-scale storage systems. In this paper, we propose Group-Shifted Declustering (G-SD), a deterministic data layout for multi-way replication. G-SD addresses the scalability issue of our previous Shifted Declustering layout and supports fast and efficient reverse lookup. Secondly, we define a problem: how to balance the performance, energy, and recovery in degradation mode for an energy efficient storage system? . While extensive researches have been proposed to tradeoff performance for energy efficiency under normal mode, the system enters degradation mode when node failure occurs, in which node reconstruction is initiated. This very process requires a number of disks to be spun up and requires a substantial amount of I/O bandwidth, which will not only compromise energy efficiency but also performance. Without considering the I/O bandwidth contention between recovery and performance, we find that the current energy proportional solutions cannot answer this question accurately. This thesis present PERP, a mathematical model to minimize the energy consumption for a storage systems with respect to performance and recovery. PERP answers this problem by providing the accurate number of nodes and the assigned recovery bandwidth at each time frame. Thirdly, current distributed file systems such as Google File System(GFS) and Hadoop Distributed File System (HDFS), employ a pseudo-random method for replica distribution and a centralized lookup table (block map) to record all replica locations. This lookup table requires a large amount of memory and consumes a considerable amount of CPU/network resources on the metadata server. With the booming size of Big Data , the metadata server becomes a scalability and performance bottleneck. While current approaches such as HDFS Federation attempt to horizontally extend scalability by allowing multiple metadata servers, we believe a more promising optimization option is to vertically scale up each metadata server. We propose Deister, a novel block management scheme that builds on top of a deterministic declustering distribution method Intersected Shifted Declustering (ISD). Thus both replica distribution and location lookup can be achieved without a centralized lookup table

Exact Results for the Distribution of the Partial Busy Period for a Multi-Server Queue

Exact explicit results are derived for the distribution of the partial busy period of the M/M/c multi-server queue for a general number of servers. A rudimentary spectral method leads to a representation that is amenable to efficient numerical computation across the entire ergodic region. An alternative algebraic approach yields a representation as a finite sum of Marcum Q-functions depending on the roots of certain polynomials that are explicitly determined for an arbitrary number of servers. Asymptotic forms are derived in the limit of a large number of servers under two scaling regimes, and also for the large-time limit. Connections are made with previous work. The present work is the first to offer tangible exact results for the distribution when the number of servers is greater than two