Dynamic Programming Matrix Research Articles

Calculating PSSM probabilities with lazy dynamic programming

Position-specific scoring matrices are one way to represent approximate string patterns, which are commonly encountered in the field of bioinformatics. An important problem that arises with their application is calculating the statistical significance of matches. We review the currently most efficient algorithm for this task, and show how it can be implemented in Haskell, taking advantage of the built-in non-strictness of the language. The resulting program turns out to be an instance of dynamic programming, using lists rather the typical dynamic programming matrix.

Bit-Parallel Witnesses and Their Applications to Approximate String Matching

We present a new bit-parallel technique for approximate string matching. We build on two previous techniques. The first one, BPM (Myers, 1999), searches for a pattern of length m in a text of length n permitting k differences in $O(\lceil m/w \rceil n)$ time, where w is the width of the computer word. The second one, ABNDM (Navarro and Raffinot, 2000), extends a sublinear-time exact algorithm to approximate searching. ABNDM relies on another algorithm, BPA (Wu and Manber, 1992), which makes use of an $O(k \lceil m/w \rceil n)$ time algorithm for its internal workings. BPA is slow but flexible enough to support all operations required by ABNDM. We improve previous ABNDM analyses, showing that it is average-optimal in number of inspected characters, although the overall complexity is higher because of the $O(k \lceil m/w \rceil )$ work done per inspected character. We then show that the faster BPM can be adapted to support all the operations required by ABNDM. This involves extending it to compute edit distance, to search for any pattern suffix, and to detect in advance the impossibility of a later match. The solution to those challenges is based on the concept of a witness, which permits sampling some dynamic programming matrix values to bound, deduce or compute others fast. The resulting algorithm is average-optimal for m ≤ w, assuming the alphabet size is constant. In practice, it performs better than the original ABNDM and is the fastest algorithm for several combinations of m, k and alphabet sizes that are useful, for example, in natural language searching and computational biology. To show that the concept of witnesses can be used in further scenarios, we also improve a recent variant of BPM. The use of witnesses greatly improves the running time of this algorithm too.

A table-driven, full-sensitivity similarity search algorithm.

Searching a database for a local alignment to a query under a typical scoring scheme, such as PAM120 or BLOSUM62 with affine gap costs, is a computation that has resisted algorithmic improvement due to its basis in dynamic programming and the weak nature of the signals being searched for. In a query preprocessing step, a set of tables can be built that permit one to (a) eliminate a large fraction of the dynamic programming matrix from consideration and (b) to compute several steps of the remainder with a single table lookup. While this result is not an asymptotic improvement over the original Smith-Waterman algorithm, its complexity is characterized in terms of some sparse features of the matrix and it yields the fastest software implementation to date for such searches.

A Subquadratic Sequence Alignment Algorithm for Unrestricted Scoring Matrices

Given two strings of size $n$ over a constant alphabet, the classical algorithm for computing the similarity between two sequences [D. Sankoff and B. Kruskal, eds., {Time Warps, String Edits, and Macromolecules}; Addison--Wesley, Reading, MA, 1983; T. F. Smith and M. S. Waterman, { J. Molec. Biol., 147 (1981), pp. 195--197] uses a dynamic programming matrix and compares the two strings in O(n2) time. We address the challenge of computing the similarity of two strings in subquadratic time for metrics which use a scoring matrix of unrestricted weights. Our algorithm applies to both {local} and {global} similarity computations. The speed-up is achieved by dividing the dynamic programming matrix into variable sized blocks, as induced by Lempel--Ziv parsing of both strings, and utilizing the inherent periodic nature of both strings. This leads to an $O(n^2 / \log n)$, algorithm for an input of constant alphabet size. For most texts, the time complexity is actually $O(h n^2 / \log n)$, where $h \le 1$ is the entropy of the text. We also present an algorithm for comparing two {run-length} encoded strings of length m and n, compressed into m' and n' runs, respectively, in O(m'n + n'm) complexity. This result extends to all distance or similarity scoring schemes that use an additive gap penalty.

A fast bit-vector algorithm for approximate string matching based on dynamic programming

The approximate string matching problem is to find all locations at which a query of length m matches a substring of a text of length n with k -or-fewer differences. Simple and practical bit-vector algorithms have been designed for this problem, most notably the one used in agrep . These algorithms compute a bit representation of the current state-set of the k -difference automaton for the query, and asymptotically run in either O ( nm/w ) or O ( nm log σ/ w ) time where w is the word size of the machine (e.g., 32 or 64 in practice), and σ is the size of the pattern alphabet. Here we present an algorithm of comparable simplicity that requires only O ( nm/w) time by virtue of computing a bit representation of the relocatable dynamic programming matrix for the problem. Thus, the algorithm's performance is independent of k , and it is found to be more efficient than the previous results for many choices of k and small m . Moreover, because the algorithm is not dependent on k , it can be used to rapidly compute blocks of the dynamic programming matrix as in the 4-Russians algorithm of Wu et al.(1996). This gives rise to an O(kn/w) expected-time algorithm for the case where m may be arbitrarily large. In practice this new algorithm, that computes a region of the dynamic progr amming (d.p.) matrx w entries at a time using the basic algorithm as a subroutine is significantly faster than our previous 4-Russians algorithm, that computes the same region 4 or 5 entries at a time using table lookup. This performance improvement yields a code that is either superior or competitive with all existing algorithms except for some filtration algorithms that are superior when k/m is sufficiently small.

Constrained sequence alignment.

This paper presents a dynamic programming algorithm for aligning two sequences when the alignment is constrained to lie between two arbitrary boundary lines in the dynamic programming matrix. For affine gap penalties, the algorithm requires only O(F) computations time and O(M+N) space, when F is the area of the feasible region and M and N are the sequence lengths. The result extends to concave gap penalities, with somewhat increased time and space bounds.

Dynamic programming alignment of sequences representing cyclic patterns

String alignment by dynamic programming is generalized to include cyclic shift and corresponding optimal alignment cost for strings representing cyclic patterns. A guided search algorithm uses bounds on alignment costs to find all optimal cyclic shifts. The bounds are derived from submatrices of an initial dynamic programming matrix. Algorithmic complexity is analyzed for major stages in the search. The applicability of the method is illustrated with satellite DNA sequences and circularly permuted protein sequences. >

Error detection in arrays via dependency graphs

This paper describes a methodology based on dependency graphs for doing concurrent run-time error detection in systolic arrays and wavefront processors. It combines the projection method of deriving systolic arrays from dependency graphs with the idea of input-triggered testing. We call the method ITRED, forInput-driven Time-Redundancy Error Detection. Tests are triggered by inserting special symbols in the input, and so the approach gives the user flexibility in trading off throughput for error coverage. Correctness of timing is proved at the dependency graph level. The method requires no extraPEs and little extra hardware. We propose several variations of the general approach and derive corresponding constraints on the modified dependency graphs that guarantee correctness. One variation performs run-time error correction using majority voting. Examples are given, including a dynamic programming algorithm, convolution, and matrix multiplication.

STRUCTURAL ASPECTS OF BEEF PRODUCTION ON PASTURES IN THE SUMMER RAINFALL GRAIN PRODUCING AREAS OF SOUTH AFRICA

ABSTRACT A dynamic linear programming matrix is assembled in order to model the South African summer grain industry. Supply, demand and production risk, including interactions with production substitutes, are considered. The model is tested by imposing all of the policies which were in operation at that time to see if it adequately simulates the prevailing situation. The model is used to evaluate the possible structural effects of subsidising farmers to change over land under cash crop production to pastures for beef, taking into account the inter-relationships in grain markets in Southern Africa. These effects are evaluated under different marketing policies with regard to changes in land use, prices, labour requirements and producer and consumer welfare. The main conclusion reached from this analysis is that by reforming or helping some industry or group of producers, its problems are merely transferred to other industries or producers owing to the inter-relationships in agriculture and the economy. It emphasises the need for an overall policy plan.

Interrelationships in maize markets in Southern Africa I: Structural effects of the farmer support programme

A dynamic linear programming matrix is assembled in order to model the Southern African summer grain industry. Supply, demand and production risk, including interactions with production substitutes, are considered. The model is used to evaluate the possible structural effects of the farmer support programme (FSP) taking into account the interrelationships in grain markets in Southern Africa. The effects are evaluated under different marketing policies with respect to changes in land use, prices and labour requirements. A macro view is taken. An effective FSP will lead to structural changes in land use, prices of products and labour requirements. The effect of the FSP on production, prices and labour will not only vary according to the success in changing over from subsistence maize production to commercial farming in the developing areas, but is also dependent on marketing policies in general. The interrelationships in agriculture and the economy emphasise the need for an overall policy plan.

Sequential inference of Markov networks by dynamic programming for structural pattern recognition

Inference of Markov networks from finite sets of sample strings is formulated using dynamic programming. Strings are installed in a network sequentially via string-to-network alignments computed with a dynamic programming matrix, the cost function of which is based on relative frequency estimates of transition probabilities to emphasize landmark substrings common to the sample set. A classification experiment indicates the potential of the method for pattern description and recognition.

Dynamic Programming Inference of Markov Networks from Finite Sets of Sample Strings

Inference of Markov networks from finite sets of sample strings is formulated using dynamic programming. Strings are installed in a network sequentially via optimal string-to-network alignments computed with a dynamic programming matrix, the cost function of which uses relative frequency estimates of transition probabilities to emphasize landmark substrings common to the sample set. Properties of an inferred network are described and the method is illustrated with artificial data and with data representing banded human chromosomes.