A simple algorithm for the constrained sequence problems

One way to detect intruders on the Internet is to compare the similarity of two thumbprints. A thumbprint is a summary of a connection that characterizes the connection. The packet gap thumbprint consists of sequences of non-negative real number representing the time gaps between "send" packets. This paper formalized definitions of similarity between two non-negative real number sequences, by introducing /spl epsiv/-similarity, partial sum and longest /spl epsiv/-similar subsequence (LSS). Length of LSS is a measurement of similarity between two sequences. The longest /spl epsiv/-similar subsequence (LSS) problem is a generalization of the well known longest common subsequence (LCS) problem. The goal of this paper is to find an optimal solution to the LSS problem. We analyzed the property of partial sums and proposed to focus on the minimum matched partial sum which leads to an optimal solution to LSS while reduce the problem space. As the LSS problem has optimal structure, we proposed an algorithm based on dynamic programming technique. Time complexity of this algorithm is O(m/sup 2/n/sup 2/). By using a property of the partial sums, we reduced the time complexity to O(mn(m+n)).

In this paper, we study exact, exponential-time algorithms for a variant of the classic Longest Common Subsequence problem called the r-Repetition Longest Common Subsequence problem (or r-RLCS, for short): Given two sequences X and Y over an alphabet S, find a longest common subsequence of X and Y such that each symbol appears at most r times in the obtained subsequence. Without loss of generality, we will assume that \(|X| \le |Y|\) from here on. The special case of 1-RLCS, also known as the Repetition-Free Longest Common Subsequence problem (RFLCS), has been studied previously; e.g., in [1], Adi et al. presented an (exponential-time) integer linear programming-based exact algorithm for 1-RLCS. However, they did not analyze its time complexity, and to the best of our knowledge, there are no previous results on the running times of any exact algorithms for this problem. In this paper, we first propose a simple algorithm for 1-RLCS based on the strategy used in [1] and show explicitly that its running time is bounded by \(O(1.44225^{|X|}|X||Y|)\). Next, we provide a DP-based algorithm for r-RLCS and prove that its running time is \(O((r+1)^{|X|/(r+1)}|X||Y|)\) for any \(r \ge 1\). In particular, our new algorithm runs in \(O(1.41422^{|X|}|X||Y|)\) time for 1-RLCS, which is faster than the previous one.

The multiple sequence alignment (MSA) problem has become relevant to several areas in bioinformatics from finding sequences family, detecting structural homologies of protein/DNA sequences, determining functions of protein/DNA sequences to predict patients diseases by comparing DNAs of patients in disease discovery, etc. The MSA is a NP-hard problem. In this paper, two new methods based on a cultural algorithm, namely the method of musical composition, for the solution of the MSA problem are introduced. The performance of the first and second versions were evaluated and analyzed on 26 and 12 different benchmark alignments, respectively. Test instances were taken from BAliBASE 3.0. Alignment accuracies are computed using the QSCORE program, which is a quality scoring program that compares two multiple sequence alignments. Numerical results on the tackled instances indicate that the performance levels of the proposed versions of the MMC are promising. In particular, the experimental results show that the second version found the best alignment reported in the specialized literature in 25 $$\%$$% of the tested instances. Besides, for 50 $$\%$$% of the tested instances, the second version achieved the second best alignment. Finally, the significance of the numerical results were analyzed according to the Wilcoxon rank-sum test, which indicated that the second proposed version is statistically similar to some state-of-the-art techniques for the MSA problem.

The general algorithms which are followed to solve the Longest Common Subsequence (LCS) problems have both time complexity and space complexity of O(m * n). To reduce this complexity, two ways are proposed in this work. One is the use of Divide and Conquer approach, and another one is changing the data structure from two-dimensional array to queue. By using these approaches, an algorithm has been developed and implemented for the long length string in this paper. The time complexity after using these approaches becomes a function of logarithm O(n * log(n)) which has been shown with proper explanation and the space complexity after using queue data structure becomes linear O(n).

In this paper we propose an algorithm for the constrained two-dimensional cutting stock problem (TDC) in which a single stock sheet has to be cut into a set of small pieces, while maximizing the value of the pieces cut. The TDC problem is NP-hard in the strong sense and finds many practical applications in the cutting and packing area. The proposed algorithm is a hybrid approach in which a depth-first search using hill-climbing strategies and dynamic programming techniques are combined. The algorithm starts with an initial (feasible) lower bound computed by solving a series of single bounded knapsack problems. In order to enhance the first-level lower bound, we introduce an incremental procedure which is used within a top-down branch-and-bound procedure. We also propose some hill-climbing strategies in order to produce a good trade-off between the computational time and the solution quality. Extensive computational testing on problem instances from the literature shows the effectiveness of the proposed approach. The obtained results are compared to the results published by Alvarez-Valdes et al. (2002).

We propose a rate-distortion optimal algorithm for sinusoidal modeling of audio and speech. The algorithm determines, for a pre-specified target bit-rate, the optimal (variable-length) time segmentation, the optimal distribution of sinusoidal components over the segments and the optimal (scalar) quantizers for quantizing the sinusoid parameters. The optimization is done by jointly optimizing the segment lengths, number of sinusoids and quantizers using high-resolution quantization theory and dynamic programming techniques, which makes it possible to solve the algorithm in polynomial time. A particular advantage of the proposed method is that, given a target bit-rate, it solves the problem of finding the optimal balance between total number of sinusoids and number of bits per sinusoid.

In this paper, we consider a generalized longest common subsequence problem with multiple substring exclusive constraints. For the two input sequences X and Y of lengths n and m, and a set of d constraints \(P=\{P_1,\cdots ,P_d\}\) of total length r, the problem is to find a common subsequence Z of X and Y excluding each of constraint string in P as a substring and the length of Z is maximized. A very simple dynamic programming algorithm to this problem is presented in this paper. The correctness of the new algorithm is demonstrated. The time and space complexities of the new algorithm are both O(nmr).

An examination is made of the dynamic programming technique implemented on a distributed-memory multiprocessor system. The matrix chain product algorithm is investigated as an example problem. A method for parallelizing the dynamic programming technique for solving the matrix chain product problem is presented and load-balancing considerations are examined. The sequential dynamic programming technique is a fine grain algorithm and considered by many researchers to be too fine grain to execute effectively on a hypercube. The parallel algorithm yielded modest speedups for fixed-size problems. Scaled problems promise even better speedups. Results show that respectable performance for the dynamic programming algorithmic technique can be achieved on a hypercube. >

We introduce a novel method for learning voice leading using neural networks. Unlike earlier approaches for learning chord sequences where chords are predicted from a local context, this method uses a multi-layer neural network for learning chord assessment from music examples. The network employs a special constraint-based topology for transforming the relative comparison of chord pairs into an absolute assessment function. Using this chord assessment function, globally optimized chord sequences can be found in linear time using the dynamic programming technique. The ChordNet implementation of the approach learns stylistic aspects of voice leading directly from a set of training examples. In combination with the HARMONET system, it presents a powerful framework for solving practical harmonization tasks.

BackgroundThe inference of homologies among DNA sequences, that is, positions in multiple genomes that share a common evolutionary origin, is a crucial, yet difficult task facing biologists. Its computational counterpart is known as the multiple sequence alignment problem. There are various criteria and methods available to perform multiple sequence alignments, and among these, the minimization of the overall cost of the alignment on a phylogenetic tree is known in combinatorial optimization as the Tree Alignment Problem. This problem typically occurs as a subproblem of the Generalized Tree Alignment Problem, which looks for the tree with the lowest alignment cost among all possible trees. This is equivalent to the Maximum Parsimony problem when the input sequences are not aligned, that is, when phylogeny and alignments are simultaneously inferred.ResultsFor large data sets, a popular heuristic is Direct Optimization (DO). DO provides a good tradeoff between speed, scalability, and competitive scores, and is implemented in the computer program POY. All other (competitive) algorithms have greater time complexities compared to DO. Here, we introduce and present experiments a new algorithm Affine-DO to accommodate the indel (alignment gap) models commonly used in phylogenetic analysis of molecular sequence data. Affine-DO has the same time complexity as DO, but is correctly suited for the affine gap edit distance. We demonstrate its performance with more than 330,000 experimental tests. These experiments show that the solutions of Affine-DO are close to the lower bound inferred from a linear programming solution. Moreover, iterating over a solution produced using Affine-DO shows little improvement.ConclusionsOur results show that Affine-DO is likely producing near-optimal solutions, with approximations within 10% for sequences with small divergence, and within 30% for random sequences, for which Affine-DO produced the worst solutions. The Affine-DO algorithm has the necessary scalability and optimality to be a significant improvement in the real-world phylogenetic analysis of sequence data.

In this paper, we revisit the classic and well-studied longest common subsequence (LCS) problem and study some new variants, first introduced and studied by Rahman and Iliopoulos [Algorithms for Computing Variants of the Longest Common Subsequence Problem, ISAAC 2006]. Here we define a generalization of these variants, the longest parameterized common subsequence (LPCS) problem, and show how to solve it in O(n 2) and time. Furthermore, we show how to compute two variants of LCS, RELAG and RIFIG in time.KeywordsEdit DistanceLonge Common SubsequenceLonge Common SubsequenceEmpty QueueLonge Common Subsequence ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We use the dynamic programming and approximation techniques to prove the existence of an optimal synthesis for a class of linear discrete time systems with quadratic costs and a fixed target point, in which the optimal control law is given in the feedback form of time and state. Our result establishes an analogy in synthesizing fixed target point linear quadratic regulator problems between linear continuous time systems and this class of linear discrete time systems.

Mining web log datasets has been extensively studied using Frequent Pattern Mining (FPM) and its various other forms. Identifyingfrequent patterns in different sequences can help in analyzing the most common sub-sequences (e.g., the pages visitedtogether). However, this approach would not be able to identify general structures spanning over multiple sequences. In responseto understanding general structures, we introduce a new form of sequential pattern mining called super-sequence frequent patternmining (SS-FPM). In contrast to sub-sequences determined by FPM, SS-FPM determines the super-sequences that can containthe common parts from different sequences. This can be useful in understanding the general behavior/flow of users in web usagemining, classifying web pages and users, making predictions etc. In essence, finding frequent super-sequence patterns turnsout to be related to the well-known heaviest (longest) path problem in graphs, which is known to be NP-hard. Accordingly,we transform a given sequential dataset into a sequence graph and formulate the problem as k-hop heaviest path problem. Wethen propose an efficient heuristic called sequence matrix method using dynamic programming techniques. We compared ourmethod to the existing Heavypath method. The results show that our method is more efficient especially on large datasets.

In the context of non-coding RNA (ncRNA) multiple structural alignment, Davydov and Batzoglou introduced in [7] the problem of finding the largest nested linear graph that occurs in a set ${\mathcal{G}}$ of linear graphs, the so-called Max-NLS problem. This problem generalizes both the longest common subsequence problem and the maximum common homeomorphic subtree problem for rooted ordered trees. In the present paper, we give a fast algorithm for finding the largest nested linear subgraph of a linear graph and a polynomial-time algorithm for a fixed number (k) of linear graphs. Also, we strongly strengthen the result of [7] by proving that the problem is NP-complete even if ${\mathcal{G}}$ is composed of nested linear graphs of height at most 2, thereby precisely defining the borderline between tractable and intractable instances of the problem. Of particular importance, we improve the result of [7] by showing that the Max-NLS problem is approximable within ratio O(logmopt) in O(kn2) running time, where mopt is the size of an optimal solution. We also present ${{\mathcal O}}(1)$-approximation of Max-NLS problem running in ${{\mathcal O}}(kn)$ time for restricted linear graphs. In particular, for ncRNA derived linear graphs, an $\frac{1}{4}$-approximation is presented.

One of the most challenging tasks in bioinformatics is the resolution of Multiple Sequence Alignment (MSA) problem. It consists in comparing a set of protein or DNA sequences, in aim of predicting their structure and function. This paper introduces a new bio-inspired approach to solve such problem. This approach named BA-MSA is based on Bat Algorithm. Bat Algorithm (BA) is a recent evolutionary algorithm inspired from Bats behavior seeking their prey. The proposed approach includes new mechanism to generate initial population. It consists in generating a guide tree for each solution with progressive approach by varying some parameters. The generated guide tree will be enhanced by Hill-Climbing algorithm. In addition, to deal with the premature convergence of BA, a new restart technique is proposed to introduce more diversification when detecting premature convergence. Balibase 2.0 datasets are used for experiments. The comparison with well-known methods as MSA-GA MSA-GA (w\prealign), ClustalW, and SAGA and recent method (BBOMP) shows the effectiveness of the proposed approach.