Linear Time Solution Research Articles

A Linear‐Time Solution for All‐SAT Problem Based on P System

A Linear‐Time Solution for All‐SAT Problem Based on P System

Queueing and glueing for optimal partitioning (functional pearl)

The queueing-glueing algorithm is the nickname we give to an algorithmic pattern that provides amortised linear time solutions to a number of optimal list partition problems that have a peculiar property: at various moments we know that two of three candidate solutions could be optimal. The algorithm works by keeping a queue of lists, glueing them from one end, while chopping from the other end, hence the name. We give a formal derivation of the algorithm, and demonstrate it with several non-trivial examples.

Top-k Term-Proximity in Succinct Space

Let $$\mathcal {D} = \{\mathsf {T}_1,\mathsf {T}_2, \ldots ,\mathsf {T}_D\}$$D={T1,T2,ź,TD} be a collection of D string documents of n characters in total, that are drawn from an alphabet set $$\varSigma =[\sigma ]$$Σ=[ź]. The top-k document retrieval problem is to preprocess $$\mathcal{D}$$D into a data structure that, given a query $$(P[1\ldots p],k)$$(P[1źp],k), can return the k documents of $$\mathcal{D}$$D most relevant to the pattern P. The relevance is captured using a predefined ranking function, which depends on the set of occurrences of P in $$\mathsf {T}_d$$Td. For example, it can be the term frequency (i.e., the number of occurrences of P in $$\mathsf {T}_d$$Td), or it can be the term proximity (i.e., the distance between the closest pair of occurrences of P in $$\mathsf {T}_d$$Td), or a pattern-independent importance score of $$\mathsf {T}_d$$Td such as PageRank. Linear space and optimal query time solutions already exist for the general top-k document retrieval problem. Compressed and compact space solutions are also known, but only for a few ranking functions such as term frequency and importance. However, space efficient data structures for term proximity based retrieval have been evasive. In this paper we present the first sub-linear space data structure for this relevance function, which uses only o(n) bits on top of any compressed suffix array of $$\mathcal{D}$$D and solves queries in $$O((p+k) {{\mathrm{polylog}}}\,\,n)$$O((p+k)polylogn) time. We also show that scores that consist of a weighted combination of term proximity, term frequency, and document importance, can be handled using twice the space required to represent the text collection.

Solving optimization problems by using networks of evolutionary processors with quantitative filtering

Searching for new efficient algorithms to solve complex optimization problems in big data scenarios is a priority, especially when the search space increases exponentially with the problem size, making impossible to find a solution through a mere blind search. Networks of Evolutionary Processors (NEP) is a formal framework formed of highly parallel and distributed computing models inspired and abstracted from biological evolution that is able to solve hard problems in an efficient way. However, NEP is not expressive enough to model quantitative aspects present in many problems. In this paper we propose NEPO, a new model based on the NEP evolutionary processors. NEPO deals with a class of data that is able to solve hard optimization problems and defines a novel selection process based on a quantitative filtering strategy. We present a linear time solution to a well known NP-complete optimization problem (the 0/1 Knapsack problem) in order to demonstrate NEPO advantages. This result suggests that NEPO's quantitative filtering is more suitable to tackle practical solutions to optimization problems in order to deploy them on highly scalable distributed computational platforms.

Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems

AbstractWe consider the fundamental problem of solving quadratic systems of equations in , and is unknown. We propose a novel method, which starts with an initial guess computed by means of a spectral method and proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach [13]. There are several key distinguishing features, most notably a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e., in time proportional to reading the data {ai} and {yi} as soon as the ratio m/n between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have and prove that our algorithms achieve a statistical accuracy, which is nearly unimprovable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size—hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least squares problem of the same size. © 2016 Wiley Periodicals, Inc.

RASCAL: A Randomized Approach for Coevolutionary Analysis.

A popular method for coevolutionary inference is cophylogenetic reconstruction where the branch length of the phylogenies have been previously derived. This approach, unlike the more generalized reconstruction techniques that are NP-Hard, can reconcile the shared evolutionary history of a pair of phylogenetic trees in polynomial time. This approach, while proven to be highly successful, requires a high polynomial running time. This is quickly becoming a limiting factor of this approach due to the continual increase in size of coevolutionary data sets. One existing method that combats this issue proposes a trade-off of accuracy for an asymptotic time complexity reduction. This technique in almost 70% of cases converges on Pareto optimal solutions in linear time. We build on this prior work by proposing an alternate linear time algorithm (RASCAL) that offers a significant accuracy increase, with RASCAL converging on Pareto optimal solutions in 85% of cases and unlike prior methods can ensure, with high probability, that all optimal solutions can be recovered, provided sufficient replicates are performed.

Efficient solutions to hard computational problems by P systems with symport/antiport rules and membrane division

P systems are computing models inspired by some basic features of biological membranes. In this work, membrane division, which provides a way to obtain an exponential workspace in linear time, is introduced into (cell-like) P systems with communication (symport/antiport) rules, where objects are never modified but they just change their places. The computational efficiency of this kind of P systems is studied. Specifically, we present a (uniform) linear time solution to the NP-complete problem, Subset Sum by using division rules for elementary membranes and communication rules of length at most 3. We further prove that such P system allowing division rules for non-elementary membranes can efficiently solve the PSPACE-complete problem, QSAT in a uniform way.

Calculating a linear-time solution to the densest-segment problem

AbstractThe problem of finding a densest segment of a list is similar to the well-known maximum segment sum problem, but its solution is surprisingly challenging. We give a general specification of such problems, and formally develop a linear-time online solution, using a sliding-window style algorithm. The development highlights some elegant properties of densities, involving partitions that are decreasing and all right-skew.

Error diffusion on meshes

We consider the problem of quantization for surface graphs. In particular, we generalize the process of error diffusion to meshes and then compare different paths on the surface when used for error diffusion. We suggest paths for processing mesh elements that lead to better distributions of available neighbors for error diffusion. We demonstrate the potential benefit of error diffusion at several mesh processing applications: quantization of differential mesh coordinates, including an extension to animated geometry, and vertex subset selection for mesh simplification. These applications allow us to compare different paths objectively. We find that the linear time solution results in excellent overall performance, outperforming other traversals taken from the literature. We conclude that the proposed path can be taken as a starting point for any application of error diffusion on meshes.

A POLYNOMIAL-TIME ALGORITHM FOR COMPUTING THE RESILIENCE OF ARRANGEMENTS OF RAY SENSORS

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.

Reverse engineering of compact suffix trees and links: A novel algorithm

Invented in the 1970s, the Suffix Tree (ST) is a data structure that indexes all substrings of a text in linear space. Although more space demanding than other indexes, the ST remains likely an inspiring index because it represents substrings in a hierarchical tree structure. Along time, STs have acquired a central position in text algorithmics with myriad of algorithms and applications to for instance motif discovery, biological sequence comparison, or text compression. It is well known that different words can lead to the same suffix tree structure with different labels. Moreover, the properties of STs prevent all tree structures from being STs. Even the suffix links, which play a key role in efficient construction algorithms and many applications, are not sufficient to discriminate the suffix trees of distinct words. The question of recognising which trees can be STs has been raised and termed Reverse Engineering on STs. For the case where a tree is given with potential suffix links, a seminal work provides a linear time solution only for binary alphabets. Here, we also investigate the Reverse Engineering problem on ST with links and exhibit a novel approach and algorithm. Hopefully, this new suffix tree characterisation makes up a valuable step towards a better understanding of suffix tree combinatorics.

On the relationship between histogram indexing and block-mass indexing

Histogram indexing, also known as jumbled pattern indexing and permutation indexing is one of the important current open problems in pattern matching. It was introduced about 6 years ago and has seen active research since. Yet, to date there is no algorithm that can preprocess a text T in time o(|T|(2)/polylog|T|) and achieve histogram indexing, even over a binary alphabet, in time independent of the text length. The pattern matching version of this problem has a simple linear-time solution. Block-mass pattern matching problem is a recently introduced problem, motivated by issues in mass-spectrometry. It is also an example of a pattern matching problem that has an efficient, almost linear-time solution but whose indexing version is daunting. However, for fixed finite alphabets, there has been progress made. In this paper, a strong connection between the histogram indexing problem and the block-mass pattern indexing problem is shown. The reduction we show between the two problems is amazingly simple. Its value lies in recognizing the connection between these two apparently disparate problems, rather than the complexity of the reduction. In addition, we show that for both these problems, even over unbounded alphabets, there are algorithms that preprocess a text T in time o(|T|(2)/polylog|T|) and enable answering indexing queries in time polynomial in the query length. The contributions of this paper are twofold: (i) we introduce the idea of allowing a trade-off between the preprocessing time and query time of various indexing problems that have been stumbling blocks in the literature. (ii) We take the first step in introducing a class of indexing problems that, we believe, cannot be pre-processed in time o(|T|(2)/polylog|T|) and enable linear-time query processing.

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.

Unrooted Tree Reconciliation: A Unified Approach

Tree comparison functions are widely used in phylogenetics for comparing evolutionary trees. Unrooted trees can be compared with rooted trees by identifying all rootings of the unrooted tree that minimize some provided comparison function between two rooted trees. The plateau property is satisfied by the provided function, if all optimal rootings form a subtree, or plateau, in the unrooted tree, from which the rootings along every path toward a leaf have monotonically increasing costs. This property is sufficient for the linear-time identification of all optimal rootings and rooting costs. However, the plateau property has only been proven for a few rooted comparison functions, requiring individual proofs for each function without benefitting from inherent structural features of such functions. Here, we introduce the consistency condition that is sufficient for a general function to satisfy the plateau property. For consistent functions, we introduce general linear-time solutions that identify optimal rootings and all rooting costs. Further, we identify novel relationships between consistent functions in terms of plateaus, especially the plateau of the well-studied duplication-loss function is part of a plateau of every other consistent function. We introduce a novel approach for identifying consistent cost functions by defining a formal language of Boolean costs. Formulas in this language can be interpreted as cost functions. Finally, we demonstrate the performance of our general linear-time solutions in practice using empirical and simulation studies.

On left and right seeds of a string

We consider the problem of finding the repetitive structure of a given string y of length n. A factor u of y is a cover of y, if every letter of y lies within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring of y. A left seed of y is a prefix of y, that is a cover of a superstring of y. Similarly, a right seed of y is a suffix of y, that is a cover of a superstring of y. An integer array LS is the minimal left-seed (resp. maximal left-seed) array of y, if LS[i] is the minimal (resp. maximal) length of left seeds of y[0..i]. The minimal right-seed (resp. maximal right-seed) arrayRS of y is defined in a similar fashion. In this article, we present linear-time algorithms for computing all left and right seeds of y, a linear-time algorithm for computing the minimal left-seed array of y, a linear-time solution for computing the maximal left-seed array of y, an O(nlogn)-time algorithm for computing the minimal right-seed array of y, and a linear-time solution for computing the maximal right-seed array of y. All algorithms use linear auxiliary space.

Low-Complexity Energy-Efficient Broadcasting in One-Dimensional Wireless Networks

In this paper, we investigate the transmission range assignment for N wireless nodes located on a line (a linear wireless network) for broadcasting data from one specific node to all the nodes in the network with minimum energy. Our goal is to find a solution that has low complexity and yet performs close to optimal. We propose an algorithm for finding the optimal assignment (which results in the minimum energy consumption) with complexity O(N2) . An approximation algorithm with complexity O(N) is also proposed. It is shown that, for networks with uniformly distributed nodes, the linear-time approximate solution obtained by this algorithm, on average, performs practically identical to the optimal assignment. Both the optimal and the suboptimal algorithms require full knowledge of the network topology and are thus centralized. We also propose a distributed algorithm of negligible complexity, i.e., with complexity O(1), which only requires knowledge of the adjacent neighbors at each wireless node. Our simulations demonstrate that the distributed solution, on average, performs almost as good as the optimal solution for networks with uniformly distributed nodes.

Heuristic Based Task Scheduling in Multiprocessor Systems with Genetic Algorithm by Choosing the Eligible Processor

In multiprocessor systems, one of the main factors of systems' performance is task scheduling. The well the task be distributed among the processors the well be the performance. Again finding the optimal solution of scheduling the tasks into the processors is NP-complete, that is, it will take a lot of time to find the optimal solution. Many evolutionary algorithms (e.g. Genetic Algorithm, Simulated annealing) are used to reach the near optimal solution in linear time. In this paper we propose a heuristic for genetic algorithm based task scheduling in multiprocessor systems by choosing the eligible processor on educated guess. From comparison it is found that this new heuristic based GA takes less computation time to reach the suboptimal solution.

Parameterized longest previous factor

Given a string W, the longest previous factor (LPF) problem is to determine the maximum length of a previously occurring factor for each suffix occurring in W. The LPF problem is defined for traditional strings exclusively from the constant alphabet Σ. A parameterized string (p-string) is a string composed of symbols from a constant alphabet Σ and a parameter alphabet Π. We formulate the LPF problem in terms of p-strings by defining the parameterized longest previous factor (pLPF) problem. Subsequently, we present an expected linear time solution to construct the parameterized longest previous factor (pLPF) array. Given our pLPF solution, we show how to construct the pLCP (parameterized longest common prefix) array with the same general algorithm. We exploit the properties of the pLPF data structure to also construct the standard LPF (longest previous factor) and LCP (longest common prefix) arrays all in linear time. Further, we provide insight into the practicality of our construction algorithms.

Maximum Segment Sum, Monadically (distilled tutorial)

The maximum segment sum problem is to compute, given a list of integers, the largest of the sums of the contiguous segments of that list. This problem specification maps directly onto a cubic-time algorithm; however, there is a very elegant linear-time solution too. The problem is a classic exercise in the mathematics of program construction, illustrating important principles such as calculational development, pointfree reasoning, algebraic structure, and datatype-genericity. Here, we take a sideways look at the datatype-generic version of the problem in terms of monadic functional programming, instead of the traditional relational approach; the presentation is tutorial in style, and leavened with exercises for the reader.

Linear Time Solution to Prime Factorization by Tissue P Systems with Cell Division

Prime factorization is useful and crucial for public-key cryptography, and its application in public-key cryptography is possible only because prime factorization has been presumed to be difficult. A polynomial-time algorithm for prime factorization on a quantum computer was given by P. W. Shor in 1997. In this work, it is considered as a function problem, and in the framework of tissue P systems with cell division, a linear-time solution to prime factorization problem is given on biochemical computational devices – tissue P systems with cell division, instead of computational devices based on the laws of quantum physical.