PatTrieSort - External String Sorting based on Patricia Tries

External merge sort belongs to the most efficient and widely used algorithms to sort big data: As much data as fits inside is sorted in main memory and afterwards swapped to external storage as so called initial run. After sorting all the data in this way block-wise, the initial runs are merged in a merging phase in order to retrieve the final sorted run containing the completely sorted original data. Patricia tries are one of the most space-efficient ways to store strings especially those with common prefixes. Hence, we propose to use patricia tries for initial run generation in an external merge sort variant, such that initial runs can become large compared to traditional external merge sort using the same main memory size. Furthermore, we store the initial runs as patricia tries instead of lists of sorted strings. As we will show in this paper, patricia tries can be efficiently merged having a superior performance in comparison to merging runs of sorted strings. We complete our discussion with a complexity analysis as well as a comprehensive performance evaluation, where our new approach outperforms traditional external merge sort by a factor of 4 for sorting over 4 billion strings of real world data

Scalable String and Suffix Sorting: Algorithms, Techniques, and Tools

This dissertation focuses on two fundamental sorting problems: string sorting and suffix sorting. The first part considers parallel string sorting on shared-memory multi-core machines, the second part external memory suffix sorting using the induced sorting principle, and the third part distributed external memory suffix sorting with a new distributed algorithmic big data framework named Thrill.Comment: 396 pages, dissertation, Karlsruher Instituts f\"ur Technologie (2018). arXiv admin note: text overlap with arXiv:1101.3448 by other author

Engineering Parallel String Sorting

We discuss how string sorting algorithms can be parallelized on modern multi-core shared memory machines. As a synthesis of the best sequential string sorting algorithms and successful parallel sorting algorithms for atomic objects, we first propose string sample sort. The algorithm makes effective use of the memory hierarchy, uses additional word level parallelism, and largely avoids branch mispredictions. Then we focus on NUMA architectures, and develop parallel multiway LCP-merge and -mergesort to reduce the number of random memory accesses to remote nodes. Additionally, we parallelize variants of multikey quicksort and radix sort that are also useful in certain situations. Comprehensive experiments on five current multi-core platforms are then reported and discussed. The experiments show that our implementations scale very well on real-world inputs and modern machines.Comment: 46 pages, extension of "Parallel String Sample Sort" arXiv:1305.115

Low Space External Memory Construction of the Succinct Permuted Longest Common Prefix Array

The longest common prefix (LCP) array is a versatile auxiliary data structure in indexed string matching. It can be used to speed up searching using the suffix array (SA) and provides an implicit representation of the topology of an underlying suffix tree. The LCP array of a string of length $n$ can be represented as an array of length $n$ words, or, in the presence of the SA, as a bit vector of $2n$ bits plus asymptotically negligible support data structures. External memory construction algorithms for the LCP array have been proposed, but those proposed so far have a space requirement of $O(n)$ words (i.e. $O(n \log n)$ bits) in external memory. This space requirement is in some practical cases prohibitively expensive. We present an external memory algorithm for constructing the $2n$ bit version of the LCP array which uses $O(n \log \sigma)$ bits of additional space in external memory when given a (compressed) BWT with alphabet size $\sigma$ and a sampled inverse suffix array at sampling rate $O(\log n)$. This is often a significant space gain in practice where $\sigma$ is usually much smaller than $n$ or even constant. We also consider the case of computing succinct LCP arrays for circular strings

External Batched Range Minimum Queries and LCP Construction

This work deals with I/O-optimal suffix array (SA) and longest common prefix (LCP) array construction in external memory. For this purpose, the I/O-optimale DC3 algorithm is enhanced by LCP construction and adapted accordingly to the external memory model. In this context we present a method to construct the required range minimum queries (RMQs) efficiently in external memory. The core of this work is a description and implementation of the resulting external DC3-LCP algorithm using the Stxxl - the C++ Standard Template Library for Extra Large Data Sets. Experimental results based on realistic, real-world instances rounds off this work

The Cost of Address Translation

Modern computers are not random access machines (RAMs). They have a memory hierarchy, multiple cores, and virtual memory. In this paper, we address the computational cost of address translation in virtual memory. Starting point for our work is the observation that the analysis of some simple algorithms (random scan of an array, binary search, heapsort) in either the RAM model or the EM model (external memory model) does not correctly predict growth rates of actual running times. We propose the VAT model (virtual address translation) to account for the cost of address translations and analyze the algorithms mentioned above and others in the model. The predictions agree with the measurements. We also analyze the VAT-cost of cache-oblivious algorithms.Comment: A extended abstract of this paper was published in the proceedings of ALENEX13, New Orleans, US

Lightweight LCP Construction for Very Large Collections of Strings

The longest common prefix array is a very advantageous data structure that, combined with the suffix array and the Burrows-Wheeler transform, allows to efficiently compute some combinatorial properties of a string useful in several applications, especially in biological contexts. Nowadays, the input data for many problems are big collections of strings, for instance the data coming from "next-generation" DNA sequencing (NGS) technologies. In this paper we present the first lightweight algorithm (called extLCP) for the simultaneous computation of the longest common prefix array and the Burrows-Wheeler transform of a very large collection of strings having any length. The computation is realized by performing disk data accesses only via sequential scans, and the total disk space usage never needs more than twice the output size, excluding the disk space required for the input. Moreover, extLCP allows to compute also the suffix array of the strings of the collection, without any other further data structure is needed. Finally, we test our algorithm on real data and compare our results with another tool capable to work in external memory on large collections of strings.Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ The final version of this manuscript is in press in Journal of Discrete Algorithm

Inducing Suffix and LCP Arrays in External Memory

We consider full text index construction in external memory (EM). Our first contribution is an inducing algorithm for suffix arrays in external memory, which utilizes an efficient EM priority queue and runs in sorting complexity. Practical tests show that this algorithm outperforms the previous best EM suffix sorter [Dementiev et al., JEA 2008] by a factor of about two in time and I/O-volume. Our second contribution is to augment the first algorithm to also construct the array of longest common prefixes (LCPs). This yields the first EM construction algorithm for LCP arrays. The overhead in time and I/O volume for this extended algorithm over plain suffix array construction is roughly two. Our algorithms scale far beyond problem sizes previously considered in the literature (text size of 80 GiB using only 4 GiB of RAM in our experiments).