A simple online competitive adaptation of Lempel-Ziv compression with efficient random access support

We present a simple adaptation of the Lempel Ziv 78' (LZ78) compression scheme ({\em IEEE Transactions on Information Theory, 1978}) that supports efficient random access to the input string. Namely, given query access to the compressed string, it is possible to efficiently recover any symbol of the input string. The compression algorithm is given as input a parameter \eps >0, and with very high probability increases the length of the compressed string by at most a factor of (1+\eps). The access time is O(\log n + 1/\eps^2) in expectation, and O(\log n/\eps^2) with high probability. The scheme relies on sparse transitive-closure spanners. Any (consecutive) substring of the input string can be retrieved at an additional additive cost in the running time of the length of the substring. We also formally establish the necessity of modifying LZ78 so as to allow efficient random access. Specifically, we construct a family of strings for which $\Omega(n/\log n)$ queries to the LZ78-compressed string are required in order to recover a single symbol in the input string. The main benefit of the proposed scheme is that it preserves the online nature and simplicity of LZ78, and that for {\em every} input string, the length of the compressed string is only a small factor larger than that obtained by running LZ78

NEEXP is Contained in MIP*

We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP ⊆ MIP*, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion of NEEXP = NTIME[2^(2poly(n))] ⊆ MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed

간결한 자료구조를 활용한 반구조화된 문서 형식들의 공간 효율적 표현법

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 전기·컴퓨터공학부, 2021. 2. Srinivasa Rao Satti.Numerous big data are generated from a plethora of sources. Most of the data stored as files contain a non-fixed type of schema, so that the files are suitable to be maintained as semi-structured document formats. A number of those formats, such as XML (eXtensible Markup Language), JSON (JavaScript Object Notation), and YAML (YAML Ain't Markup Language) are suggested to sustain hierarchy in the original corpora of data. Several data models structuring the gathered data - including RDF (Resource Description Framework) - depend on the semi-structured document formats to be serialized and transferred for future processing. Since the semi-structured document formats focus on readability and verbosity, redundant space is required to organize and maintain the document. Even though general-purpose compression schemes are widely used to compact the documents, applying those algorithms hinder future handling of the corpora, owing to loss of internal structures. The area of succinct data structures is widely investigated and researched in theory, to provide answers to the queries while the encoded data occupy space close to the information-theoretic lower bound. Bit vectors and trees are the notable succinct data structures. Nevertheless, there were few attempts to apply the idea of succinct data structures to represent the semi-structured documents in space-efficient manner. In this dissertation we propose a unified, space-efficient representation of various semi-structured document formats. The core functionality of this representation is its compactness and query-ability derived from enriched functions of succinct data structures. Incorporation of (a) bit indexed arrays, (b) succinct ordinal trees, and (c) compression techniques engineers the compact representation. We implement this representation in practice, and show by experiments that construction of this representation decreases the disk usage by up to 60% while occupying 90% less RAM. We also allow processing a document in partial manner, to allow processing of larger corpus of big data even in the constrained environment. In parallel to establishing the aforementioned compact semi-structured document representation, we provide and reinforce some of the existing compression schemes in this dissertation. We first suggest an idea to encode an array of integers that is not necessarily sorted. This compaction scheme improves upon the existing universal code systems, by assistance of succinct bit vector structure. We show that our suggested algorithm reduces space usage by up to 44% while consuming 15% less time than the original code system, while the algorithm additionally supports random access of elements upon the encoded array. We also reinforce the SBH bitmap index compression algorithm. The main strength of this scheme is the use of intermediate super-bucket during operations, giving better performance on querying through a combination of compressed bitmap indexes. Inspired from splits done during the intermediate process of the SBH algorithm, we give an improved compression mechanism supporting parallelism that could be utilized in both CPUs and GPUs. We show by experiments that this CPU parallel processing optimization diminishes compression and decompression times by up to 38% in a 4-core machine without modifying the bitmap compressed form. For GPUs, the new algorithm gives 48% faster query processing time in the experiments, compared to the previous existing bitmap index compression schemes.셀 수 없는 빅 데이터가 다양한 원본로부터 생성되고 있다. 이들 데이터의 대부분은 고정되지 않은 종류의 스키마를 포함한 파일 형태로 저장되는데, 이로 인하여 반구조화된 문서 형식을 이용하여 파일을 유지하는 것이 적합하다. XML, JSON 및 YAML과 같은 종류의 반구조화된 문서 형식이 데이터에 내재하는 구조를 유지하기 위하여 제안되었다. 수집된 데이터를 구조화하는 RDF와 같은 여러 데이터 모델들은 사후 처리를 위한 저장 및 전송을 위하여 반구조화된 문서 형식에 의존한다. 반구조화된 문서 형식은 가독성과 다변성에 집중하기 때문에, 문서를 구조화하고 유지하기 위하여 추가적인 공간을 필요로 한다. 문서를 압축시키기 위하여 일반적인 압축 기법들이 널리 사용되고 있으나, 이들 기법들을 적용하게 되면 문서의 내부 구조의 손실로 인하여 데이터의 사후 처리가 어렵게 된다. 데이터를 정보이론적 하한에 가까운 공간만을 사용하여 저장을 가능하게 하면서 질의에 대한 응답을 제공하는 간결한 자료구조는 이론적으로 널리 연구되고 있는 분야이다. 비트열과 트리가 널리 알려진 간결한 자료구조들이다. 그러나 반구조화된 문서들을 저장하는 데 간결한 자료구조의 아이디어를 적용한 연구는 거의 진행되지 않았다. 본 학위논문을 통해 우리는 다양한 종류의 반구조화된 문서 형식을 통일되게 표현하는 공간 효율적 표현법을 제시한다. 이 기법의 주요한 기능은 간결한 자료구조가 강점으로 가지는 특성에 기반한 간결성과 질의 가능성이다. 비트열로 인덱싱된 배열, 간결한 순서 있는 트리 및 다양한 압축 기법을 통합하여 해당 표현법을 고안하였다. 이 기법은 실재적으로 구현되었고, 실험을 통하여 이 기법을 적용한 반구조화된 문서들은 최대 60% 적은 디스크 공간과 90% 적은 메모리 공간을 통해 표현될 수 있다는 것을 보인다. 더불어 본 학위논문에서 반구조화된 문서들은 분할적으로 표현이 가능함을 보이고, 이를 통하여 제한된 환경에서도 빅 데이터를 표현한 문서들을 처리할 수 있다는 것을 보인다. 앞서 언급한 공간 효율적 반구조화된 문서 표현법을 구축함과 동시에, 본 학위논문에서 이미 존재하는 압축 기법 중 일부를 추가적으로 개선한다. 첫째로, 본 학위논문에서는 정렬 여부에 관계없는 정수 배열을 부호화하는 아이디어를 제시한다. 이 기법은 이미 존재하는 범용 코드 시스템을 개선한 형태로, 간결한 비트열 자료구조를 이용한다. 제안된 알고리즘은 기존 범용 코드 시스템에 비해 최대 44\% 적은 공간을 사용할 뿐만 아니라 15\% 적은 부호화 시간을 필요로 하며, 기존 시스템에서 제공하지 않는 부호화된 배열에서의 임의 접근을 지원한다. 또한 본 학위논문에서는 비트맵 인덱스 압축에 사용되는 SBH 알고리즘을 개선시킨다. 해당 기법의 주된 강점은 부호화와 복호화 진행 시 중간 매개인 슈퍼버켓을 사용함으로써 여러 압축된 비트맵 인덱스에 대한 질의 성능을 개선시키는 것이다. 위 압축 알고리즘의 중간 과정에서 진행되는 분할에서 영감을 얻어, 본 학위논문에서 CPU 및 GPU에 적용 가능한 개선된 병렬화 압축 매커니즘을 제시한다. 실험을 통해 CPU 병렬 최적화가 이루어진 알고리즘은 압축된 형태의 변형 없이 4코어 컴퓨터에서 최대 38\%의 압축 및 해제 시간을 감소시킨다는 것을 보인다. GPU 병렬 최적화는 기존에 존재하는 GPU 비트맵 압축 기법에 비해 48\% 빠른 질의 처리 시간을 필요로 함을 확인한다.Chapter 1 Introduction 1 1.1 Contribution 3 1.2 Organization 5 Chapter 2 Background 6 2.1 Model of Computation 6 2.2 Succinct Data Structures 7 Chapter 3 Space-efficient Representation of Integer Arrays 9 3.1 Introduction 9 3.2 Preliminaries 10 3.2.1 Universal Code System 10 3.2.2 Bit Vector 13 3.3 Algorithm Description 13 3.3.1 Main Principle 14 3.3.2 Optimization in the Implementation 16 3.4 Experimental Results 16 Chapter 4 Space-efficient Parallel Compressed Bitmap Index Processing 19 4.1 Introduction 19 4.2 Related Work 23 4.2.1 Byte-aligned Bitmap Code (BBC) 24 4.2.2 Word-Aligned Hybrid (WAH) 27 4.2.3 WAH-derived Algorithms 28 4.2.4 GPU-based WAH Algorithms 31 4.2.5 Super Byte-aligned Hybrid (SBH) 33 4.3 Parallelizing SBH 38 4.3.1 CPU Parallelism 38 4.3.2 GPU Parallelism 39 4.4 Experimental Results 40 4.4.1 Plain Version 41 4.4.2 Parallelized Version 46 4.4.3 Summary 49 Chapter 5 Space-efficient Representation of Semi-structured Document Formats 50 5.1 Preliminaries 50 5.1.1 Semi-structured Document Formats 50 5.1.2 Resource Description Framework 57 5.1.3 Succinct Ordinal Tree Representations 60 5.1.4 String Compression Schemes 64 5.2 Representation 66 5.2.1 Bit String Indexed Array 67 5.2.2 Main Structure 68 5.2.3 Single Document as a Collection of Chunks 72 5.2.4 Supporting Queries 73 5.3 Experimental Results 75 5.3.1 Datasets 76 5.3.2 Construction Time 78 5.3.3 RAM Usage during Construction 80 5.3.4 Disk Usage and Serialization Time 83 5.3.5 Chunk Division 83 5.3.6 String Compression 88 5.3.7 Query Time 89 Chapter 6 Conclusion 94 Bibliography 96 요약 109 Acknowledgements 111Docto

LeCo: Lightweight Compression via Learning Serial Correlations

Lightweight data compression is a key technique that allows column stores to exhibit superior performance for analytical queries. Despite a comprehensive study on dictionary-based encodings to approach Shannon's entropy, few prior works have systematically exploited the serial correlation in a column for compression. In this paper, we propose LeCo (i.e., Learned Compression), a framework that uses machine learning to remove the serial redundancy in a value sequence automatically to achieve an outstanding compression ratio and decompression performance simultaneously. LeCo presents a general approach to this end, making existing (ad-hoc) algorithms such as Frame-of-Reference (FOR), Delta Encoding, and Run-Length Encoding (RLE) special cases under our framework. Our microbenchmark with three synthetic and six real-world data sets shows that a prototype of LeCo achieves a Pareto improvement on both compression ratio and random access speed over the existing solutions. When integrating LeCo into widely-used applications, we observe up to 3.9x speed up in filter-scanning a Parquet file and a 16% increase in Rocksdb's throughput

A Space-Optimal Grammar Compression

A grammar compression is a context-free grammar (CFG) deriving a single string deterministically. For an input string of length N over an alphabet of size sigma, the smallest CFG is O(log N)-approximable in the offline setting and O(log N log^* N)-approximable in the online setting. In addition, an information-theoretic lower bound for representing a CFG in Chomsky normal form of n variables is log (n!/n^sigma) + n + o(n) bits. Although there is an online grammar compression algorithm that directly computes the succinct encoding of its output CFG with O(log N log^* N) approximation guarantee, the problem of optimizing its working space has remained open. We propose a fully-online algorithm that requires the fewest bits of working space asymptotically equal to the lower bound in O(N log log n) compression time. In addition we propose several techniques to boost grammar compression and show their efficiency by computational experiments

GraCT: A Grammar based Compressed representation of Trajectories

We present a compressed data structure to store free trajectories of moving objects (ships over the sea, for example) allowing spatio-temporal queries. Our method, GraCT, uses a $k^2$-tree to store the absolute positions of all objects at regular time intervals (snapshots), whereas the positions between snapshots are represented as logs of relative movements compressed with Re-Pair. Our experimental evaluation shows important savings in space and time with respect to a fair baseline.Comment: This research has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie Actions H2020-MSCA-RISE-2015 BIRDS GA No. 69094