Basic Dynamic Programming Algorithm Research Articles

Hitting a Prime in 2.43 Dice Rolls (On Average)

What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than . This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes.

Regular Language Constrained Sequence Alignment Revisited

Imposing constraints in the form of a finite automaton or a regular expression is an effective way to incorporate additional a priori knowledge into sequence alignment procedures. With this motivation, the Regular Expression Constrained Sequence Alignment Problem was introduced, which proposed an O(n²t⁴) time and O(n²t²) space algorithm for solving it, where n is the length of the input strings and t is the number of states in the input non-deterministic automaton. A faster O(n²t³) time algorithm for the same problem was subsequently proposed. In this article, we further speed up the algorithms for Regular Language Constrained Sequence Alignment by reducing their worst case time complexity bound to O(n²t³)/log t). This is done by establishing an optimal bound on the size of Straight-Line Programs solving the maxima computation subproblem of the basic dynamic programming algorithm. We also study another solution based on a Steiner Tree computation. While it does not improve the worst case, our simulations show that both approaches are efficient in practice, especially when the input automata are dense.

Faster Algorithms for Optimal Multiple Sequence Alignment Based on Pairwise Comparisons

Multiple Sequence Alignment (MSA) is one of the most fundamental problems in computational molecular biology. The running time of the best known scheme for finding an optimal alignment, based on dynamic programming, increases exponentially with the number of input sequences. Hence, many heuristics were suggested for the problem. We consider a version of the MSA problem where the goal is to find an optimal alignment in which matches are restricted to positions in predefined matching segments. We present several techniques for making the dynamic programming algorithm more efficient, while still finding an optimal solution under these restrictions. We prove that it suffices to find an optimal alignment of the predefined sequence segments, rather than single letters, thereby reducing the input size and thus improving the running time. We also identify "shortcuts" that expedite the dynamic programming scheme. Empirical study shows that, taken together, these observations lead to an improved running time over the basic dynamic programming algorithm by 4 to 12 orders of magnitude, while still obtaining an optimal solution. Under the additional assumption that matches between segments are transitive, we further improve the running time for finding the optimal solution by restricting the search space of the dynamic programming algorithm.

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.

An Optimized and Efficiently Parallelized Dynamic Programming for RNA Secondary Structure Prediction

RNA secondary structure prediction based on free energy rules remains a major computational method in computational biology. Its basic dynamic programming algorithm needs O(n) time to calculate the minimum free energy for RNA secondary structure, where n is the length of RNA sequence. There are two variants for handling this problem: either the internal loops are bounded by a maximal size k, giving a time complexity of O(n×k), or one uses the trick of Lyngso, which makes use of the rules of loop energies, to reduce time complexity to O(n) for suboptimal free energy without restriction. Only with additional O(n) space, a new algorithm is proposed to eliminate the redundant calculation in the energy of internal loops and reduce the time complexity to O(n) with unrestricted loop sizes for optimal free energy. While the optimized algorithm is time consuming, an efficient parallel algorithm with good load balancing in cluster systems is also proposed. The experimental results show that the parallel program achieves reasonable speedups.

A dynamic programming segmentation procedure for hydrological and environmental time series

We present a procedure for the segmentation of hydrological and environmental time series. The procedure is based on the minimization of Hubert’s segmentation cost or various generalizations of this cost. This is achieved through a dynamic programming algorithm, which is guaranteed to find the globally optimal segmentations with K=1, 2, ..., K max segments. Various enhancements can be used to speed up the basic dynamic programming algorithm, for example recursive computation of segment errors and “block segmentation”. The “true” value of K is selected through the use of the Bayesian information criterion. We evaluate the segmentation procedure with experiments which involve artificial as well as temperature and river discharge time series.

A versatile divide and conquer technique for optimal string alignment

Common string alignment algorithms such as the basic dynamic programming algorithm (DPA) and the time efficient Ukkonen algorithm use quadratic space to determine an alignment between two strings. In this paper we present a technique that can be applied to these algorithms to obtain an alignment using only linear space, while having little or no effect on the time complexity. This new technique has several advantages over previous methods for determining alignments in linear space, such as: simplicity, the ability to use essentially the same technique when using different cost functions, and the practical advantage of easily being able to trade available memory for running time.

Approximate string matching using withinword parallelism

AbstractGiven a text string, a pattern string, and an integer k, the problem of approximate string matching with k differences is to find all substrings of the text string whose edit distance from the pattern string is less than k. The edit distance between two strings is defined as the minimum number of differences, where a difference can be a substitution, insertion, or deletion of a single character. An implementation of the dynamic programming algorithm for this problem is given that packs several characters and mod‐4 integers into a computer word. Thus, it is a parallelization of the conventional implementation that runs on ordinary processors. Since a small alphabet means that characters have short binary codes, the degree of parallelism is greatest for small alphabets and for processors with long words. For an alphabet of size 8 or smaller and a 64 bit processor, a 21‐fold parallelism over the conventional algorithm can be obtained. Empirical comparisons to the basic dynamic programming algorithm, to a version of Ukkonen's algorithm, to the algorithm of Galil and Park, and to a limited implementation of the Wu‐Manber algorithm are given.

Trees, Stars, and Multiple Biological Sequence Alignment

One important problem in biological sequence comparison is how to simultaneously align several nucleic acid or protein sequences. A multiple alignment avoids possible inconsistencies among several pairwise alignments and can elucidate relationships not evident from pairwise comparisons. The basic dynamic programming algorithm for optimal multiple sequence alignment requires too much time to be practical for more than three sequences, the length of an average protein. Recently, Carrillo and Lipman (SIAMJ. Appl. Math., 48 (1988), pp. 1073–1082) have rendered feasible the optimal simultaneous alignment of as many as six sequences by showing that a consideration of minimal pairwise alignment costs can vastly decrease the number of cells a dynamic programming algorithm need consider. Their argument, however, requires the cost of a multiple alignment to be a weighted sum of the costs of its projected pairwise alignments.This paper presents an extension of Carrillo and Lipman's algorithm to the definition of mul...

A state transition model for distributed query processing

A state transition model for the optimization of query processing in a distributed database system is presented. The problem is parameterized by means of a state describing the amount of processing that has been performed at each site where the database is located. A state transition occurs each time a new join or semijoin is executed. Dynamic programming is used to compute recursively the costs of the states and the globally optimal solution, taking into account communication and local processing costs. The state transition model is general enough to account for the possibility of parallel processing among the various sites, as well as for redundancy in the database. The model also permits significant reductions of the necessary computations by taking advantage of simple additivity and site-uniformity properties of a cost model, and of clever strategies that improve on the basic dynamic programming algorithm.