Hypergroup Deformations of Semigroups

We view the well-known example of the dual of a countable compact hypergroup, motivated by the orbit space of p-adic integers by Dunkl and Ramirez (1975), as hypergroup deformation of the max semigroup structure on the linearly ordered set $\mathbb{Z}_+$ of the non-negative integers along the diagonal. This works as motivation for us to study hypergroups or semi convolution spaces arising from "max" semigroups or general commutative semigroups via hypergroup deformation on idempotents.Comment: 28 pages, 1 Table, This version is a truncated version with fourth section deleted from version 3, which is being developed into a separate paper. The title and abstract have been changed accordingl

Optimal Splitters for Database Partitioning with Size Bounds

Partitioning is an important step in several database algorithms, including sorting, aggregation, and joins. Partitioning is also fundamental for dividing work into equal-sized (or balanced) parallel subtasks. In this paper, we aim to find, materialize and maintain a set of partitioning elements (splitters) for a data set. Unlike traditional partitioning elements, our splitters define both inequality and equality partitions, which allows us to bound the size of the inequality partitions. We provide an algorithm for determining an optimal set of splitters from a sorted data set and show that it has time complexity O(k lg_2 N), where k is the number of splitters requested and N is the size of the data set. We show how the algorithm can be extended to pairs of tables, so that joins can be partitioned into work units that have balanced cost. We demonstrate experimentally (a) that finding the optimal set of splitters can be done efficiently, and (b) that using the precomputed splitters can improve the time to sort a data set by up to 76%, with particular benefits in the presence of a few heavy hitters

A New Client-Server Architecture for Distributed Query Processing

This paper presents the idea of "tuple bit-vectors" for distributed query processing. Using tuple bit-vectors, a new two-way semijoin operator called 2SJ++ that enhances the semijoin with an essentially "free" backward reduction capability is proposed. We explore in detail the benefits and costs of 2SJ++ compared with other semijoin variants, and its effect on distributed query processing performance. We then focus on one particular distributed query processing algorithm, called the "one-shot" algorithm. We modify the one-shot algorithm by using 2SJ++ and demonstrate the improvements achieved in network transmission cost compared with the original one-shot technique. We use this improvement to demonstrate that equipped with the 2SJ++ technique, one can improve the performance of distributed query processing algorithms significantly without adding much complexity to the algorithms

Schema Polynomials and Applications

Conceptual complexity is emerging as a new bottleneck as database developers, application developers, and database administrators struggle to design and comprehend large, complex schemas. The simplicity and conciseness of a schema depends critically on the idioms available to express the schema. We propose a formal conceptual schema representation language that combines different design formalisms, and allows schema manipulation that exposes the strengths of each of these formalisms. We demonstrate how the schema factorization framework can be used to generate relational, object-oriented, and faceted physical schemas, allowing a wider exploration of physical schema alternatives than traditional methodologies. We illustrate the potential practical benefits of schema factorization by showing that simple heuristics can significantly reduce the size of a real-world schema description. We also propose the use of schema polynomials to model and derive alternative representations for complex relationships with constraints

Better Semijoins Using Tuple Bit-Vectors

This paper presents the idea of "tuple-bit-vectors" for distributed query processing. Using tuple bit-vectors, a new two-way semijoin operator called 2SJ++ that enhances the semijoin with an essentially "free" backward reduction capability is proposed. We explore in detail the benefits and costs of 2SJ++ compared with other semijoin variants, and its effect on distributed query processing performance. We then focus on one particular distributed query processing algorithm, called the "one-shot" algorithm. We modify the one-shot algorithm by using 2SJ++ and demonstrate the improvements achieved in network transmission cost compared with the original one-shot technique. We use this improvement to demonstrate that equipped with the 2SJ++ technique, one can improve the performance of distributed query processing algorithms significantly without adding much complexity to the algorithms

On the Cost of Transitive Closures in Relational Databases

We consider the question of taking transitive closures on top of pure relational systems (Sybase and Ingres in this case). We developed three kinds of transitive closure programs, one using a stored procedure to simulate a built-in transitive closure operator, one using the C language embedded with SQL statements to simulate the iterated execution of the transitive closure operation, and one using Floyd's matrix algorithm to compute the transitive closure of an input graph. By comparing and analyzing the respective performances of their different versions in terms of elapsed time spent on taking the transitive closure, we identify some of the bottlenecks that arise when defining the transitive closure operator on top of existing relational systems. The main purpose of the work is to estimate the costs of taking transitive closures on top of relational systems, isolate the different cost factors (such as logging, network transmission cost, etc.), and identify some necessary enhancements to existing relational systems in order to support transitive closure operation efficiently. We argue that relational databases should be augmented with efficient transitive closure operators if such queries are made frequently

Partitioned Blockmap Indexes for Multidimensional Data Access

Given recent increases in the size of main memory in modern machines, it is now common to to store large data sets in RAM for faster processing. Multidimensional access methods aim to provide efficient access to large data sets when queries apply predicates to some of the data dimensions. We examine multidimensional access methods in the context of an in-memory column store tuned for on-line analytical processing or scientific data analysis. We propose a multidimensional data structure that contains a novel combination of a grid array and several bitmaps. The base data is clustered in an order matching that of the index structure. The bitmaps contain one bit per block of data, motivating the term "blockmap." The proposed data structures are compact, typically taking less than one bit of space per row of data. Partition boundaries can be chosen in a way that reflects both the query workload and the data distribution, and boundaries are not required to evenly divide the data if there is a bias in the query distribution. We examine the theoretical performance of the data structure and experimentally measure its performance on three modern CPUs and one GPU processor. We demonstrate that efficient multidimensional access can be achieved with minimal space overhead

High Throughput Heavy Hitter Aggregation

Heavy hitters are data items that occur at high frequency in a data set. Heavy hitters are among the most important items for an organization to summarize and understand during analytical processing. In data sets with sufficient skew, the number of heavy hitters can be relatively small. We take advantage of this small footprint to compute aggregate functions for the heavy hitters in fast cache memory. We design cache-resident, shared-nothing structures that hold only the most frequent elements from the table. Our approach works in three phases. It first samples and picks heavy hitter candidates. It then builds a hash table and computes the exact aggregates of these candidates. Finally, if necessary, a validation step identifies the true heavy hitters from among the candidates based on the query specification. We identify trade-offs between the hash table capacity and performance. Capacity determines how many candidates can be aggregated. We optimize performance by the use of perfect hashing and SIMD instructions. SIMD instructions are utilized in novel ways to minimize cache accesses, be- yond simple vectorized operations. We use bucketized and cuckoo hash tables to increase capacity, to adapt to different datasets and query constraints. The performance of our method is an order of magnitude faster than in-memory aggregation over a complete set of items if those items cannot be cache resident. Even for item sets that are cache resident, our SIMD techniques enable significant performance improvements over previous work