Algorithms for the Problems of Length-Constrained Heaviest Segments

We present algorithms for length-constrained maximum sum segment and maximum density segment problems, in particular, and the problem of finding length-constrained heaviest segments, in general, for a sequence of real numbers. Given a sequence of n real numbers and two real parameters L and U (L <= U), the maximum sum segment problem is to find a consecutive subsequence, called a segment, of length at least L and at most U such that the sum of the numbers in the subsequence is maximum. The maximum density segment problem is to find a segment of length at least L and at most U such that the density of the numbers in the subsequence is the maximum. For the first problem with non-uniform width there is an algorithm with time and space complexities in O(n). We present an algorithm with time complexity in O(n) and space complexity in O(U). For the second problem with non-uniform width there is a combinatorial solution with time complexity in O(n) and space complexity in O(U). We present a simple geometric algorithm with the same time and space complexities. We extend our algorithms to respectively solve the length-constrained k maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the length-constrained $k$ maximum density segments problem in O(n min{k, U-L}) time and O(U+k) space. We present extensions of our algorithms to find all the length-constrained segments having user specified sum and density in O(n+m) and O(nlog (U-L)+m) times respectively, where m is the number of output. Previously, there was no known algorithm with non-trivial result for these problems. We indicate the extensions of our algorithms to higher dimensions. All the algorithms can be extended in a straight forward way to solve the problems with non-uniform width and non-uniform weight.Comment: 21 pages, 12 figure

Encodings of Range Maximum-Sum Segment Queries and Applications

Given an array A containing arbitrary (positive and negative) numbers, we consider the problem of supporting range maximum-sum segment queries on A: i.e., given an arbitrary range [i,j], return the subrange [i' ,j' ] \subseteq [i,j] such that the sum of the numbers in A[i'..j'] is maximized. Chen and Chao [Disc. App. Math. 2007] presented a data structure for this problem that occupies {\Theta}(n) words, can be constructed in {\Theta}(n) time, and supports queries in {\Theta}(1) time. Our first result is that if only the indices [i',j'] are desired (rather than the maximum sum achieved in that subrange), then it is possible to reduce the space to {\Theta}(n) bits, regardless the numbers stored in A, while retaining the same construction and query time. We also improve the best known space lower bound for any data structure that supports range maximum-sum segment queries from n bits to 1.89113n - {\Theta}(lg n) bits, for sufficiently large values of n. Finally, we provide a new application of this data structure which simplifies a previously known linear time algorithm for finding k-covers: i.e., given an array A of n numbers and a number k, find k disjoint subranges [i_1 ,j_1 ],...,[i_k ,j_k ], such that the total sum of all the numbers in the subranges is maximized.Comment: 19 pages + 2 page appendix, 4 figures. A shortened version of this paper will appear in CPM 201

Measuring the accuracy of genome-size multiple alignments

A novel computational approach can assess the accuracy of genomic alignments, and reveals suspicious regions in the 17-vertebrate MULTIZ alignment available on the UCSC Genome Browser

Design and Evaluation of a BLAST Ungapped Extension Accelerator, Master\u27s Thesis

The amount of biosequence data being produced each year is growing exponentially. Extracting useful information from this massive amount of data is becoming an increasingly difficult task. This thesis focuses on accelerating the most widely-used software tool for analyzing genomic data, BLAST. This thesis presents Mercury BLAST, a novel method for accelerating searches through massive DNA databases. Mercury BLAST takes a streaming approach to the BLAST computation by offloading the performance-critical sections onto reconfigurable hardware. This hardware is then used in combination with the processor of the host system to deliver BLAST results in a fraction of the time of the general-purpose processor alone. Mercury BLAST makes use of new algorithms combined with reconfigurable hardware to accelerate BLAST-like similarity search. An evaluation of this method for use in real BLAST-like searches is presented along with a characterization of the quality of results associated with using these new algorithms in specialized hardware. The primary focus of this thesis is the design of the ungapped extension stage of Mercury BLAST. The architecture of the ungapped extension stage is described along with the context of this stage within the Mercury BLAST system. The design is compact and performs over 20× faster than that of the standard software ungapped extension, yielding close to 50× speedup over the complete software BLAST application. The quality of Mercury BLAST results is essentially equivalent to the standard BLAST results

Cgaln: fast and space-efficient whole-genome alignment

<p>Abstract</p> <p>Background</p> <p>Whole-genome sequence alignment is an essential process for extracting valuable information about the functions, evolution, and peculiarities of genomes under investigation. As available genomic sequence data accumulate rapidly, there is great demand for tools that can compare whole-genome sequences within practical amounts of time and space. However, most existing genomic alignment tools can treat sequences that are only a few Mb long at once, and no state-of-the-art alignment program can align large sequences such as mammalian genomes directly on a conventional standalone computer.</p> <p>Results</p> <p>We previously proposed the CGAT (Coarse-Grained AlignmenT) algorithm, which performs an alignment job in two steps: first at the block level and then at the nucleotide level. The former is "coarse-grained" alignment that can explore genomic rearrangements and reduce the sizes of the regions to be analyzed in the next step. The latter is detailed alignment within limited regions. In this paper, we present an update of the algorithm and the open-source program, Cgaln, that implements the algorithm. We compared the performance of Cgaln with those of other programs on whole genomic sequences of several bacteria and of some mammalian chromosome pairs. The results showed that Cgaln is several times faster and more memory-efficient than the best existing programs, while its sensitivity and accuracy are comparable to those of the best programs. Cgaln takes less than 13 hours to finish an alignment between the whole genomes of human and mouse in a single run on a conventional desktop computer with a single CPU and 2 GB memory.</p> <p>Conclusions</p> <p>Cgaln is not only fast and memory efficient but also effective in coping with genomic rearrangements. Our results show that Cgaln is very effective for comparison of large genomes, especially of intact chromosomal sequences. We believe that Cgaln provides novel viewpoint for reducing computational complexity and will contribute to various fields of genome science.</p

Data Structures for Efficient String Algorithms

This thesis deals with data structures that are mostly useful in the area of string matching and string mining. Our main result is an O(n)-time preprocessing scheme for an array of n numbers such that subsequent queries asking for the position of a minimum element in a specified interval can be answered in constant time (so-called RMQs for Range Minimum Queries). The space for this data structure is 2n+o(n) bits, which is shown to be asymptotically optimal in a general setting. This improves all previous results on this problem. The main techniques for deriving this result rely on combinatorial properties of arrays and so-called Cartesian Trees. For compressible input arrays we show that further space can be saved, while not affecting the time bounds. For the two-dimensional variant of the RMQ-problem we give a preprocessing scheme with quasi-optimal time bounds, but with an asymptotic increase in space consumption of a factor of log(n). It is well known that algorithms for answering RMQs in constant time are useful for many different algorithmic tasks (e.g., the computation of lowest common ancestors in trees); in the second part of this thesis we give several new applications of the RMQ-problem. We show that our preprocessing scheme for RMQ (and a variant thereof) leads to improvements in the space- and time-consumption of the Enhanced Suffix Array, a collection of arrays that can be used for many tasks in pattern matching. In particular, we will see that in conjunction with the suffix- and LCP-array 2n+o(n) bits of additional space (coming from our RMQ-scheme) are sufficient to find all occ occurrences of a (usually short) pattern of length m in a (usually long) text of length n in O(m*s+occ) time, where s denotes the size of the alphabet. This is certainly optimal if the size of the alphabet is constant; for non-constant alphabets we can improve this to O(m*log(s)+occ) locating time, replacing our original scheme with a data structure of size approximately 2.54n bits. Again by using RMQs, we then show how to solve frequency-related string mining tasks in optimal time. In a final chapter we propose a space- and time-optimal algorithm for computing suffix arrays on texts that are logically divided into words, if one is just interested in finding all word-aligned occurrences of a pattern. Apart from the theoretical improvements made in this thesis, most of our algorithms are also of practical value; we underline this fact by empirical tests and comparisons on real-word problem instances. In most cases our algorithms outperform previous approaches by all means