Compressed Data Structures Research Articles

A parallel subdomain by subdomain implementation of the implicitly restarted Arnoldi/Lanczos method

This work presents a parallel implementation of the implicitly restarted Arnoldi/Lanczos method for the solution of eigenproblems approximated by the finite element method. The implicitly restarted Arnoldi/Lanczos uses a restart scheme in order to improve the convergence of the desired portion of the spectrum, addressing issues such as memory requirements and computational costs related to the generation and storage of the Krylov basis. The presented implementation is suitable for distributed memory architectures, especially PC clusters. In the parallel solution, a subdomain by subdomain approach was implemented and overlapping and non-overlapping mesh partitions were tested. Compressed data structures in the formats CSRC and CSRC/CSR were used to store the coefficient matrices. The parallelization of numerical linear algebra operations present in both Krylov and implicitly restarted methods are discussed. Numerical examples are shown, in order to point out the efficiency and applicability of the proposed method.

Enhancing Counting Bloom Filters Through Huffman-Coded Multilayer Structures

Bloom Filters are efficient randomized data structures for membership queries on a set with a certain known false positive probability. Counting Bloom Filters (CBFs) allow the same operation on dynamic sets that can be updated via insertions and deletions with larger memory requirements. This paper first presents a simple tight upper bound for counters overflow probability in CBFs, which is adopted in the design of more efficient CBFs. On the basis of such theoretical achievements, we introduce the idea of a hierarchical structure as well as the use of Huffman code to improve standard CBFs in terms of fast access and limited memory consumption (up to 50% of memory saving). The target could be the implementation of the compressed data structures in the small (but fast) local memory or on-chip SRAM of devices such as network processors. As an application of our algorithms, an anti-evasion system is finally proposed.

On compact representations of All-Pairs-Shortest-Path-Distance matrices

Let G be an unweighted and undirected graph of n nodes, and let D be the n × n matrix storing the All-Pairs-Shortest-Path Distances in G . Since D contains integers in [ n ] ∪ + ∞ , its plain storage takes n 2 log ( n + 1 ) bits. However, a simple counting argument shows that n 2 / 2 bits are necessary to store D . In this paper we investigate the question of finding a succinct representation of D that requires O ( n 2 ) bits of storage and still supports constant-time access to each of its entries. This is asymptotically optimal in the worst case, and far from the information-theoretic lower bound by a multiplicative factor log 2 3 ≃ 1.585 . As a result O ( 1 ) bits per pairs of nodes in G are enough to retain constant-time access to their shortest-path distance. We achieve this result by reducing the storage of D to the succinct storage of labeled trees and ternary sequences, for which we properly adapt and orchestrate the use of known compressed data structures. This approach can be easily and optimally extended to graphs whose edge weights are positive integers bounded by a constant value.

SIMD-scan

The availability of huge system memory, even on standard servers, generated a lot of interest in main memory database engines. In data warehouse systems, highly compressed column-oriented data structures are quite prominent. In order to scale with the data volume and the system load, many of these systems are highly distributed with a shared-nothing approach. The fundamental principle of all systems is a full table scan over one or multiple compressed columns. Recent research proposed different techniques to speedup table scans like intelligent compression or using an additional hardware such as graphic cards or FPGAs. In this paper, we show that utilizing the embedded Vector Processing Units (VPUs) found in standard superscalar processors can speed up the performance of mainmemory full table scan by factors. This is achieved without changing the hardware architecture and thereby without additional power consumption. Moreover, as on-chip VPUs directly access the system's RAM, no additional costly copy operations are needed for using the new SIMD-scan approach in standard main memory database engines. Therefore, we propose this scan approach to be used as the standard scan operator for compressed column-oriented main memory storage. We then discuss how well our solution scales with the number of processor cores; consequently, to what degree it can be applied in multi-threaded environments. To verify the feasibility of our approach, we implemented the proposed techniques on a modern Intel multi-core processor using Intel® Streaming SIMD Extensions (Intel® SSE). In addition, we integrated the new SIMD-scan approach into SAP® Netweaver® Business Warehouse Accelerator. We conclude with describing the performance benefits of using our approach for processing and scanning compressed data using VPUs in column-oriented main memory database systems.

Rank/select on dynamic compressed sequences and applications

Operations r a n k and s e l e c t over a sequence of symbols have many applications to the design of succinct and compressed data structures managing text collections, structured text, binary relations, trees, graphs, and so on. We are interested in the case where the collections can be updated via insertions and deletions of symbols. Two current solutions stand out as the best in the tradeoff of space versus time (when considering all the operations). One solution, by Mäkinen and Navarro, achieves compressed space (i.e., n H 0 + o ( n log σ ) bits) and O ( log n log σ ) worst-case time for all the operations, where n is the sequence length, σ is the alphabet size, and H 0 is the zero-order entropy of the sequence. The other solution, by Lee and Park, achieves O ( log n ( 1 + log σ log log n ) ) amortized time and uncompressed space, i.e. n log 2 σ + O ( n ) + o ( n log σ ) bits. In this paper we show that the best of both worlds can be achieved. We combine the solutions to obtain n H 0 + o ( n log σ ) bits of space and O ( log n ( 1 + log σ log log n ) ) worst-case time for all the operations. Apart from the best current solution to the problem, we obtain several byproducts of independent interest applicable to partial sums, text indexes, suffix arrays, the Burrows–Wheeler transform, and others.

Polynomial-time word problems

We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble straight-line programs, which are widely studied in the theory of compressed data structures. In an effort to be self-contained we give a detailed exposition of the necessary results from computer science.

Compressed data structures: Dictionaries and data-aware measures

In this paper, we propose measures for compressed data structures, in which space usage is measured in a data-aware manner. In particular, we consider the fundamental dictionary problem on set data, where the task is to construct a data structure for representing a set S of n items out of a universe U = { 0 , … , u − 1 } and supporting various queries on S . We use a well-known data-aware measure for set data called gap to bound the space of our data structures. We describe a novel dictionary structure that requires gap + O ( n log ( u / n ) / log n ) + O ( n log log ( u / n ) ) bits. Under the RAM model, our dictionary supports membership, rank, and predecessor queries in nearly optimal time, matching the time bound of Andersson and Thorup’s predecessor structure [A. Andersson, M. Thorup, Tight(er) worst-case bounds on dynamic searching and priority queues, in: ACM Symposium on Theory of Computing, STOC, 2000], while simultaneously improving upon their space usage. We support select queries even faster in O ( log log n ) time. Our dictionary structure uses exactly gap bits in the leading term (i.e., the constant factor is 1) and answers queries in near-optimal time. When seen from the worst-case perspective, we present the first O ( n log ( u / n ) ) -bit dictionary structure that supports these queries in near-optimal time under the RAM model. We also build a dictionary which requires the same space and supports membership, select, and partial rank queries even more quickly in O ( log log n ) time. We go on to show that for many (real-world) datasets, data-aware methods lead to a worthwhile compression over combinatorial methods. To the best of our knowledge, these are the first results that achieve data-aware space usage and retain near-optimal time.

Parallel implementation of the finite element method using compressed data structures

This paper presents a parallel implementation of the finite element method designed for coarse-grain distributed memory architectures. The MPI standard is used for message passing and tests are run on a PC cluster and on an SGI Altix 350. Compressed data structures are employed to store the coefficient matrix and obtain iterative solutions, based on Krylov methods, in a subdomain-by-subdomain approach. Two mesh partitioning schemes are compared: non-overlapping and overlapping. The pros and cons of these partitioning methods are discussed. Numerical examples of symmetric and non-symmetric problems in two and three dimensions are presented.

Indexing compressed text

We design two compressed data structures for the full-text indexing problem that support efficient substring searches using roughly the space required for storing the text in compressed form.Our first compressed data structure retrieves the occ occurrences of a pattern P [1, p ] within a text T [1, n ] in O ( p + occ log 1+ε n ) time for any chosen ε, 0<ε<1. This data structure uses at most 5 n H k ( T ) + o ( n ) bits of storage, where H k ( T ) is the k th order empirical entropy of T . The space usage is Θ( n ) bits in the worst case and o ( n ) bits for compressible texts. This data structure exploits the relationship between suffix arrays and the Burrows--Wheeler Transform, and can be regarded as a compressed suffix array .Our second compressed data structure achieves O ( p + occ ) query time using O ( n H k ( T )log ε n ) + o ( n ) bits of storage for any chosen ε, 0<ε<1. Therefore, it provides optimal output-sensitive query time using o ( n log n ) bits in the worst case. This second data structure builds upon the first one and exploits the interplay between two compressors: the Burrows--Wheeler Transform and the LZ78 algorithm.

Resolution cannot polynomially simulate compressed-BFS

Many algorithms for Boolean satisfiability (SAT) work within the framework of resolution as a proof system, and thus on unsatisfiable instances they can be viewed as attempting to find proofs by resolution. However it has been known since the 1980s that every resolution proof of the pigeonhole principle (PHPnm), suitably encoded as a CNF instance, includes exponentially many steps [18]. Therefore SAT solvers based upon the DLL procedure [12] or the DP procedure [13] must take exponential time. Polynomial-sized proofs of the pigeonhole principle exist for different proof systems, but general-purpose SAT solvers often remain confined to resolution. This result is in correlation with empirical evidence. Previously, we introduced the Compressed-BFS algorithm to solve the SAT decision problem. In an earlier work [27], an implementation of a Compressed-BFS algorithm empirically solved $\overline{\mathrm{PHP}_{n}^{n+1}}$ instances in ?(n4) time. Here, we add to this claim, and show analytically that these instances are solvable in polynomial time by Compressed-BFS. Thus the class of tautologies efficiently provable by Compressed-BFS is different than that of any resolution-based procedure. We hope that the details of our complexity analysis shed some light on the proof system implied by Compressed-BFS. Our proof focuses on structural invariants within the compressed data structure that stores collections of sets of open clauses during the Compressed-BFS algorithm. We bound the size of this data structure, as well as the overall memory, by a polynomial. We then use this to show that the overall runtime is bounded by a polynomial.

Efficient parallel algorithms and software for compressed octrees with applications to hierarchical methods

We describe the design and implementation of efficient parallel algorithms, and a software library for the parallel implementation of compressed octree data structures. Octrees are widely used in supporting hierarchical methods for scientific applications such as the N-body problem, molecular dynamics and smoothed particle hydrodynamics. The primary goal of our work is to identify and abstract the commonalities present in various hierarchical methods using octrees, design efficient parallel algorithms for them, and encapsulate them in a software library. We designed provably efficient parallel algorithms and implementation strategies that perform well irrespective of the spatial distribution of data in the computational domain. The library will enable rapid development of applications, allowing application developers to use efficient parallel algorithms developed for this purpose, without the necessity of having detailed knowledge of the algorithms or of implementing them. The software is developed in C using the Message Passing Interface (MPI). We report experimental results on an IBM xSeries parallel computer.

An architecture for high-performance scalable shared-memory multiprocessors exploiting on-chip integration

Recent technology improvements allow multiprocessor designers to put some key components inside the processor chip, such as the memory controller, the coherence hardware, and the network interface/router. In this paper, we exploit such integration scale, presenting a novel node architecture aimed at reducing the long L2 miss latencies and the memory overhead of using directories that characterize cc-NUMA machines and limit their scalability. Our proposal replaces the traditional directory with a novel three-level directory architecture, as well as it adds a small shared data cache to each of the nodes of a multiprocessor system. Due to their small size, the first-level directory and the shared data cache are integrated into the processor chip in every node, which enhances performance by saving accesses to the slower main memory. Scalability is guaranteed by having the second and third-level directories out of the processor chip and using compressed data structures. A taxonomy of the L2 misses, according to the actions performed by the directory to satisfy them, is also presented. Using execution-driven simulations, we show that significant latency reductions can be obtained by using the proposed node architecture, which translates into reductions of more than 30 percent in several cases in the application execution time.

The development of a standard component library for plastic injection mould design using an object-oriented approach

Very often, outsourced components (known as standard components) are used for reducing design effort and product cost in mould design. They are usually manufactured by specialised suppliers. 3D CAD parametric models of these components will significantly reduce mould design lead-time and cost, and enhance design flexibility. This paper presents the detailed object definition, design, and implementation of a Standard Component Library within a mould design software package, QuickMould. With many components from different suppliers implemented, it is believed that the object design has generic nature and can be expanded to include most mechanical components in a collaborative environment. The advantages of this implementation are the compressed data structure, the ease of use and simple customisation.

A Compressed Data Structure for Surface Representation

AbstractOne of the standard methods of simplifying the task of obtaining a shaded image of a solid object is to represent it by a polyhedron. Another method is to use sculptured surface modelling which represents surfaces by collections of surface patches. Using either method the surfaces can be approximated by facets, which are simple to shade according to photometry information.To obtain a smooth image in regions of high curvature, the surface would typically be required to have hundreds or thousands of facets. Because of the large number of facets involved, it is extremely important that geometrical and topological information is stored in an efficient manner. This information must include all that is required for an unambiguous representation of the solid(s) in question.The compressed data structure (CDS) is suitable for this purpose, and is capable of defining the majority of surfaces. The structure is intended to minimise the amount of data stored, with as much information as possible being implied. The CDS can be easily generated knowing the order of the vertices defining the surface.

Identification and validation of the statistics of the initial states of linear dynamic systems based on cross-sectional data

This note examines the problem of statistical inference of the initial states of a linear discrete dynamic system based on a set of cross-sectional data. Several compressed data structures are proposed to reduce the amount of the cross-sectional data obtained from multiple independent experiments. It is shown that these data structures are sufficient statistics for estimating the mean and the covariance of the initial states, given the entire raw data from multiple experiments. Thus, the identification and the validation of these parameters can be performed with reduced data structures without referring back to the entire raw data and the original dynamics. For the identification of these parameters, the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E-M</tex> procedure presented in [1] can be applied to this case. For the validation of these parameters having specified values, simple tests of "significance" type are proposed. The major advantage of these tests over the generalized likelihood ratio test is that their probability distributions are known and computable under both the null and the alternative hypotheses even for the finite sample case, i.e., the asymptotic assumption is not necessary.