An Elegant Algorithm for the Construction of Suffix Arrays

The suffix array is a data structure that finds numerous applications in string processing problems for both linguistic texts and biological data. It has been introduced as a memory efficient alternative for suffix trees. The suffix array consists of the sorted suffixes of a string. There are several linear time suffix array construction algorithms (SACAs) known in the literature. However, one of the fastest algorithms in practice has a worst case run time of $O(n^2)$. The problem of designing practically and theoretically efficient techniques remains open. In this paper we present an elegant algorithm for suffix array construction which takes linear time with high probability; the probability is on the space of all possible inputs. Our algorithm is one of the simplest of the known SACAs and it opens up a new dimension of suffix array construction that has not been explored until now. Our algorithm is easily parallelizable. We offer parallel implementations on various parallel models of computing. We prove a lemma on the $\ell$-mers of a random string which might find independent applications. We also present another algorithm that utilizes the above algorithm. This algorithm is called RadixSA and has a worst case run time of $O(n\log{n})$. RadixSA introduces an idea that may find independent applications as a speedup technique for other SACAs. An empirical comparison of RadixSA with other algorithms on various datasets reveals that our algorithm is one of the fastest algorithms to date. The C++ source code is freely available at http://www.engr.uconn.edu/~man09004/radixSA.zi

Parallel Out-of-Core Sorting: The Third Way

Sorting very large datasets is a key subroutine in almost any application that is built on top of a large database. Two ways to sort out-of-core data dominate the literature: merging-based algorithms and partitioning-based algorithms. Within these two paradigms, all the programs that sort out-of-core data on a cluster rely on assumptions about the input distribution. We propose a third way of out-of-core sorting: oblivious algorithms. In all, we have developed six programs that sort out-of-core data on a cluster. The first three programs, based completely on Leighton\u27s columnsort algorithm, have a restriction on the maximum problem size that they can sort. The other three programs relax this restriction; two are based on our original algorithmic extensions to columnsort. We present experimental results to show that our algorithms perform well. To the best of our knowledge, the programs presented in this thesis are the first to sort out-of-core data on a cluster without making any simplifying assumptions about the distribution of the data to be sorted

Fast Out-of-Core Sorting on Parallel Disk Systems

This paper discusses our implementation of Rajasekaran\u27s (l,m)-mergesort algorithm (LMM) for sorting on parallel disks. LMM is asymptotically optimal for large problems and has the additional advantage of a low constant in its I/O complexity. Our implementation is written in C using the ViC* I/O API for parallel disk systems. We compare the performance of LMM to that of the C library function qsort on a DEC Alpha server. qsort makes a good benchmark because it is fast and performs comparatively well under demand paging. Since qsort fails when the swap disk fills up, we can only compare these algorithms on a limited range of inputs. Still, on most out-of-core problems, our implementation of LMM runs between 1.5 and 1.9 times faster than qsort, with the gap widening with increasing problem size

Algorithmic ramifications of prefetching in memory hierarchy

External Memory models, most notable being the I-O Model [3], capture the effects of memory hierarchy and aid in algorithm design. More than a decade of architectural advancements have led to new features not captured in the I-O model - most notably the prefetching capability. We propose a relatively simple Prefetch model that incorporates data prefetching in the traditional I-O models and show how to design algorithms that can attain close to peak memory bandwidth. Unlike (the inverse of) memory latency, the memory bandwidth is much closer to the processing speed, thereby, intelligent use of prefetching can considerably mitigate the I-O bottleneck. For some fundamental problems, our algorithms attain running times approaching that of the idealized Random Access Machines under reasonable assumptions. Our work also explains the significantly superior performance of the I-O efficient algorithms in systems that support prefetching compared to ones that do not

Spin-the-bottle Sort and Annealing Sort: Oblivious Sorting via Round-robin Random Comparisons

We study sorting algorithms based on randomized round-robin comparisons. Specifically, we study Spin-the-bottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a \emph{temperature} parameter. Both algorithms are simple, randomized, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spin-the-bottle sort to require $\Omega(n^2\log n)$ expected time in order to succeed, and that in $O(n^2\log n)$ time this algorithm succeeds with high probability for any input. We also show there is an implementation of Annealing sort that runs in $O(n\log n)$ time and succeeds with very high probability.Comment: Full version of a paper appearing in ANALCO 2011, in conjunction with SODA 201

Efficient Bundle Sorting

AMS subject classification. 68W01 DOI. 10.1137/S0097539704446554Many data sets to be sorted consist of a limited number of distinct keys. Sorting such data sets can be thought of as bundling together identical keys and having the bundles placed in order; we therefore denote this as bundle sorting. We describe an efficient algorithm for bundle sorting in external memory, which requires at most c(N/B) logM/B k disk accesses, where N is the number of keys, M is the size of internal memory, k is the number of distinct keys, B is the transfer block size, and 2 < c < 4. For moderately sized k, this bound circumvents the Θ((N/B) logM/B(N/B)) I/O lower bound known for general sorting. We show that our algorithm is optimal by proving a matching lower bound for bundle sorting. The improved running time of bundle sorting over general sorting can be significant in practice, as demonstrated by experimentation. An important feature of the new algorithm is that it is executed “in-place,” requiring no additional disk space

Efficient Bundle Sorting

This is the published version. Copyright © 2006 Society for Industrial and Applied MathematicsMany data sets to be sorted consist of a limited number of distinct keys. Sorting such data sets can be thought of as bundling together identical keys and having the bundles placed in order; we therefore denote this as bundle sorting. We describe an efficient algorithm for bundle sorting in external memory, which requires at most c(N/B) logM/Bk disk accesses, where N is the number of keys, M is the size of internal memory, k is the number of distinct keys, B is the transfer block size, and 2 < c < 4. For moderately sized k, this bound circumvents the Theta((N/B) logM/B (N/B)) I/O lower bound known for general sorting. We show that our algorithm is optimal by proving a matching lower bound for bundle sorting. The improved running time of bundle sorting over general sorting can be significant in practice, as demonstrated by experimentation. An important feature of the new algorithm is that it is executed "in-place," requiring no additional disk space

Evaluating holistic aggregators efficiently for very large datasets

In data warehousing applications, numerous OLAP queries involve the processing of holistic aggregators such as computing the “top n,” median, quantiles, etc. In this paper, we present a novel approach called dynamic bucketing to efficiently evaluate these aggregators. We partition data into equiwidth buckets and further partition dense buckets into sub-buckets as needed by allocating and reclaiming memory space. The bucketing process dynamically adapts to the input order and distribution of input datasets. The histograms of the buckets and subbuckets are stored in our new data structure called structure trees. A recent selection algorithm based on regular sampling is generalized and its analysis extended. We have also compared our new algorithms with this generalized algorithm and several other recent algorithms. Experimental results show that our new algorithms significantly outperform prior ones not only in the runtime but also in accuracy