Linear Time Solution Research Articles

Linear-Time Solution of 3D Magnetostatics With Aggregation-Based Algebraic Multigrid Solvers

Linear-Time Solution of 3D Magnetostatics With Aggregation-Based Algebraic Multigrid Solvers

Watson-crick (D)0 L systems: a survey

AbstractWatson-Crick L systems are string generating formal models obtained by augmenting Lindenmayer systems with the Watson-Crick morphism inspired by the Watson-Crick complementarity principle. The Watson-Crick morphism is controlled by a trigger—a boolean condition eventually met by the generated word. Despite their simplicity, Watson-Crick (D)0 L systems have interesting mathematical properties, a strong generative power, and they can also characterize some open decision problems. Furthermore, networks of Watson-Crick D0L systems are capable of trading space for time, and thus of characterizing linear time solutions to intractable problems. We provide a survey of results in this field since its origin in 1997 to the present, and we conclude with a series of open research problems and ideas.

Two-stage non-submodular maximization

The sheer size of modern datasets has led to an urgent need for summarization techniques that can identify representative elements of the data set. Fortunately, the vast majority of data summarization tasks satisfy an intuitive diminishing returns condition known as submodularity, which allows us to find nearly-optimal solutions in linear time. However, for many applications in practice, including experimental design and sparse Gaussian processes, the objective is in general not submodular. To solve these optimization problems, an important research method is to describe the characteristics of the non-submodular functions. The non-submodular function is a hot research topic in the study of nonlinear combinatorial optimizations. In this paper, we combine and generalize the curvature and the generic submodularity ratio to design an approximation algorithm for two-stage non-submodular maximization under a matroid constraint.

On the Complexity of String Matching for Graphs

Exact string matching in labeled graphs is the problem of searching paths of a graph G=(V, E) such that the concatenation of their node labels is equal to a given pattern string P [1. m ]. This basic problem can be found at the heart of more complex operations on variation graphs in computational biology, of query operations in graph databases, and of analysis operations in heterogeneous networks. We prove a conditional lower bound stating that, for any constant ε > 0, an O (| E | 1 - ε m ) time, or an O (| E | m 1 - ε )time algorithm for exact string matching in graphs, with node labels and pattern drawn from a binary alphabet, cannot be achieved unless the Strong Exponential Time Hypothesis ( SETH ) is false. This holds even if restricted to undirected graphs with maximum node degree 2—that is, to zig-zag matching in bidirectional strings , or to deterministic directed acyclic graphs whose nodes have maximum sum of indegree and outdegree 3. These restricted cases make the lower bound stricter than what can be directly derived from related bounds on regular expression matching (Backurs and Indyk, FOCS’16). In fact, our bounds are tight in the sense that lowering the degree or the alphabet size yields linear time solvable problems. An interesting corollary is that exact and approximate matching are equally hard (i.e., quadratic time) in graphs under SETH . In comparison, the same problems restricted to strings have linear time vs quadratic time solutions, respectively (approximate pattern matching having also a matching SETH lower bound (Backurs and Indyk, STOC’15)).

Ruler Wrapping

In 1985 Hopcroft, Joseph and Whitesides showed it is NP-complete to decide whether a carpenter’s ruler with segments of given positive lengths can be folded into an interval of at most a given length, such that the folded hinges alternate between 180 degrees clockwise and 180 degrees counter-clockwise. At the open-problem session of 33rd Canadian Conference on Computational Geometry (CCCG ’21), O’Rourke proposed a natural variation of this problem called ruler wrapping, in which all folded hinges must be folded the same way. In this paper we show O’Rourke’s variation has a linear-time solution.

A Linear Time Pattern Based Algorithm for N-Queens Problem

The n-queens problem is the placing of n number of queens on an nxn chessboard so that no two queens attack each other. This problem is important due to various usage fields such as VLSI testing, traffic control job scheduling, data routing, dead-lock or blockage prevention, digital image processing and parallel memory storage schemes mentioned in the literature. Besides, this problem has been used as a benchmark for developing new Artificial Intelligence search techniques. However, it is known that backtracking algorithms, one of the most frequently used algorithms to solve this problem, cannot produce all solutions in large n values due to the exponentially growing time complexity. Therefore, various methods have been proposed for producing one or more solutions, not all solutions for each n value. In this study, a pattern based approach that produces at least one unique solution for all n values (n>3) was detected. By using this pattern, a new algorithm that produces at least one unique solution for even very large n values in linear time was developed. The developed algorithm with О(n) time complexity produces quite faster solution to n-queens problem and even in some values, this algorithm produces (n-1)/2 unique solutions in linear time.

A Linear Time Solution to the Labeled Robinson-Foulds Distance Problem.

A large variety of pairwise measures of similarity or dissimilarity have been developed for comparing phylogenetic trees, for example, species trees or gene trees. Due to its intuitive definition in terms of tree clades and bipartitions and its computational efficiency, the Robinson–Foulds (RF) distance is the most widely used for trees with unweighted edges and labels restricted to leaves (representing the genetic elements being compared). However, in the case of gene trees, an important information revealing the nature of the homologous relation between gene pairs (orthologs, paralogs, and xenologs) is the type of event associated to each internal node of the tree, typically speciations or duplications, but other types of events may also be considered, such as horizontal gene transfers. This labeling of internal nodes is usually inferred from a gene tree/species tree reconciliation method. Here, we address the problem of comparing such event-labeled trees. The problem differs from the classical problem of comparing uniformly labeled trees (all labels belonging to the same alphabet) that may be done using the Tree Edit Distance (TED) mainly due to the fact that, in our case, two different alphabets are considered for the leaves and internal nodes of the tree, and leaves are not affected by edit operations. We propose an extension of the RF distance to event-labeled trees, based on edit operations comparable to those considered for TED: node insertion, node deletion, and label substitution. We show that this new Labeled Robinson–Foulds (LRF) distance can be computed in linear time, in addition of maintaining other desirable properties: being a metric, reducing to RF for trees with no labels on internal nodes and maintaining an intuitive interpretation. The algorithm for computing the LRF distance enables novel analyses on event-label trees such as reconciled gene trees. Here, we use it to study the impact of taxon sampling on labeled gene tree inference and conclude that denser taxon sampling yields trees with better topology but worse labeling. [Algorithms; combinatorics; gene trees; phylogenetics; Robinson–Foulds; tree distance.]

A Comparison of Different Folding Models in Variations of the Map Folding Problem

In this paper, we compare the performance of three different folding models when they are applied to three different map folding settings. Precisely, the three folding models include the simple folding model, the simple folding–unfolding model, and the general folding model. The different map folding settings are discussed by comparing different folded states, i.e., different overlapping orders on the set of the squares of 1 × n maps, the squares of m × n maps, and the squares lying on the boundary of m × n maps. These folding models are abstracts of manual works and robotics. We clarify the relationship between their reachable final folded states under different settings and give proof of all the inclusion relationships between every two of these sets. In addition, there are nine distinct problems with the three folding models applied to three folding settings. We give the optimal linear time solutions to all the unsolved ones: the valid total overlapping order problems of 1 × n maps, m × n maps, as well as the valid boundary overlapping order problems of m × n maps with the three different folding models. Our work gives the conclusion of the research field where the folding models and the overlapping orders of map folding are concerned.

Analysis of influence contribution in social advertising

Online Social Network (OSN) providers usually conduct advertising campaigns by inserting social ads into promoted posts. Whenever a user engages in a promoted ad, she may further propagate the promoted ad to her followers recursively and the propagation process is known as the word-of-mouth effect. In order to spread the promotion cascade widely and efficiently, the OSN provider often tends to select the influencers, who normally have large audiences over the social network, to initiate the advertising campaign. This marketing model, also termed as influencer marketing, has been gaining increasing traction and investment and is rapidly becoming one of the most widely-used channels in digital marketing. In this paper, we formulate the problem for the OSN provider to derive the influence contributions of influencers given the campaign result, considering the viral propagation of the ads, namely influence contribution allocation (ICA) . We make a connection between ICA and the concept of Shapley value in cooperative game theory to reveal the rationale behind ICA. A naive method to obtain the solution to ICA is to enumerate all possible cascades delivering the campaign result, resulting in an exponential number of potential cascades, which is computationally intractable. Moreover, generating a cascade producing the exact campaign result is non-trivial. Facing the challenges, we develop an exact solution in linear time under the linear threshold (LT) model, and devise a fully polynomial-time randomized approximation scheme (FPRAS) under the independent cascade (IC) model. Specifically, under the IC model, we propose an efficient approach to estimate the expected influence contribution in probabilistic graphs modeling OSNs by designing a scalable sampling method with provable accuracy guarantees. We conduct extensive experiments and show that our algorithms yield solutions with remarkably higher quality over several baselines and improve the sampling efficiency significantly.

Efficient computation of the oriented chromatic number of recursively defined digraphs

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G=(V,E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

A greedy algorithm for dropping digits

Abstract Consider the following puzzle: given a number, remove k digits such that the resulting number is as large as possible. Various techniques are employed to derive a linear-time solution to the puzzle: we justify the structure of a greedy algorithm by predicate logic, give a constructive proof of the greedy condition using a dependently typed proof assistant and calculate the greedy step as well as the final, linear-time optimisation by equational reasoning.

A multi-start local search algorithm for the Hamiltonian completion problem on undirected graphs

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian completion problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

An Optimal Recovery Approach for Liberation Codes in Distributed Storage Systems

To reduce the storage cost, distributed storage systems are gradually using erasure codes to ensure data reliability. Liberation codes, which satisfy the maximum distance separable (MDS) property and provide optimal modification overhead, are a class of popular two fault tolerant erasure codes. However, erasure codes need to read from surviving nodes and transfer across the network large amounts of data when recovering from single node failures. Existing single node failure recovery approaches for Liberation codes are either time-consuming or suboptimal. In this article, firstly, we prove the minimum number of symbols required to recover one failed node for a Liberation coded system. Then we derive the conditions that optimal recovery solutions need to satisfy. Finally, we propose an algorithm, called Disk Read Optimal Recovery (DROR), which can determine an optimal recovery solution in linear time and recover the failed node reading the minimum amount of data. We have implemented DROR in a real-world storage system Ceph and evaluated DROR on a cluster of Amazon EC2 instances. We show that DROR reduces the reconstruction time by up to 23.6% compared to that of the recovery approach in Ceph.

A Linear Time Solution for N-Queens Problem Using Generalized Networks of Evolutionary Polarized Processors

In this paper we consider a new variant of Networks of Polarized Evolutionary Processors (NPEP) named Generalized Networks of Evolutionary Polarized Processors (GNPEP) and propose them as solvers of combinatorial optimization problems. Unlike the NPEP model, GNPEP uses its numerical evaluation over the processed data from a quantitative perspective, hence this model might be more suitable to solve specific hard problems in a more efficient and economic way. In particular, we propose a GNPEP network to solve a well-known NP-hard problem, namely the [Formula: see text]-queens. We prove that this GNPEP algorithm requires a linear time in the size of a given instance. This result suggests that the GNPEP model is more suitable to address problems related to combinatorial optimization in which integer restrictions have a relevant role.

Governing convergence of Max-sum on DCOPs through damping and splitting

Max-sum is a version of Belief Propagation, used for solving DCOPs. In tree-structured problems, Max-sum converges to the optimal solution in linear time. Unfortunately, when the constraint graph representing the problem includes multiple cycles (as in many standard DCOP benchmarks), Max-sum does not converge and explores low quality solutions. Recent attempts to address this limitation proposed versions of Max-sum that guarantee convergence, while ignoring some of the problem's constraints. Damping is a method that is often used for increasing the chances that Belief Propagation will converge. That being said, it has not been suggested for inclusion in the algorithms that propose Max-sum for solving DCOPs.In this paper we advance the research on incomplete-inference DCOP algorithms by: 1) investigating the effect of damping on Max-sum. We prove that, while damping slows down the propagation of information among agents, on tree-structured graphs, Max-sum with damping is guaranteed to converge to the optimal solution in weakly polynomial time; and 2) proposing a novel method for adjusting the level of asymmetry in the factor graph, in order to achieve a balance between exploitation and exploration, when using Max-sum for solving DCOPs. By converting a standard factor graph to an equivalent split constraint factor graph (SCFG), in which each function-node is split into two function-nodes, we can control the level of asymmetry for each constraint.Our empirical results demonstrate a drastic improvement in the performance of Max-sum when using damping (referred to herein as Damped Max-sum, DMS). However, in contrast to the common assumption that Max-sum performs best when converging, we demonstrate that non converging versions perform efficient exploration, and produce high quality results, when implemented within an anytime framework. On most standard benchmarks, the best results were achieved using versions with a high damping factor, which outperformed existing incomplete DCOP algorithms. In addition, our results imply that by applying DMS to SCFGs with a minor level of asymmetry, we can find high quality solutions within a small number of iterations, even without using an anytime framework. We prove that for a factor graph with a single constraint, if this constraint is split symmetrically, Max-sum applied to the resulting cycle is guaranteed to converge to the optimal solution. We further demonstrate that for an asymmetric split, convergence is not guaranteed.

Two‐stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms

AbstractWe introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is ‐hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max‐flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max‐flow structure of the subproblems.

Ranked document selection

Let D be a collection of string documents of n characters in total. The top-k document retrieval problem is to preprocess D into a data structure that, given a query (P,k), can return the k documents of D most relevant to pattern P. The relevance of a document d for a pattern P is given by a predefined ranking function w(P,d). Linear space and optimal query time solutions already exist for this problem. In this paper we consider a novel problem, document selection, in which a query (P,k) aims to report the kth document most relevant to P (instead of reporting all top-k documents). We present a data structure using O(nlogϵ⁡n) space, for any constant ϵ>0, answering selection queries in time O(log⁡k/log⁡log⁡n), and a linear-space data structure answering queries in time O(log⁡k), given the locus node of P in a (generalized) suffix tree of D. We also prove that it is unlikely that a succinct-space solution for this problem exists with poly-logarithmic query time, and that O(log⁡k/log⁡log⁡n) is indeed optimal within O(npolylogn) space for most text families. Finally, we present some additional space-time trade-offs exploring the extremes of those lower bounds.

Remarkable problem-solving ability of unicellular amoeboid organism and its mechanism.

Choosing a better move correctly and quickly is a fundamental skill of living organisms that corresponds to solving a computationally demanding problem. A unicellular plasmodium of Physarum polycephalum searches for a solution to the travelling salesman problem (TSP) by changing its shape to minimize the risk of being exposed to aversive light stimuli. In our previous studies, we reported the results on the eight-city TSP solution. In this study, we show that the time taken by plasmodium to find a reasonably high-quality TSP solution grows linearly as the problem size increases from four to eight. Interestingly, the quality of the solution does not degrade despite the explosive expansion of the search space. Formulating a computational model, we show that the linear-time solution can be achieved if the intrinsic dynamics could allocate intracellular resources to grow the plasmodium terminals with a constant rate, even while responding to the stimuli. These results may lead to the development of novel analogue computers enabling approximate solutions of complex optimization problems in linear time.

Enumerating consistent sub-graphs of directed acyclic graphs: an insight into biomedical ontologies.

MotivationModern problems of concept annotation associate an object of interest (gene, individual, text document) with a set of interrelated textual descriptors (functions, diseases, topics), often organized in concept hierarchies or ontologies. Most ontology can be seen as directed acyclic graphs (DAGs), where nodes represent concepts and edges represent relational ties between these concepts. Given an ontology graph, each object can only be annotated by a consistent sub-graph; that is, a sub-graph such that if an object is annotated by a particular concept, it must also be annotated by all other concepts that generalize it. Ontologies therefore provide a compact representation of a large space of possible consistent sub-graphs; however, until now we have not been aware of a practical algorithm that can enumerate such annotation spaces for a given ontology.ResultsWe propose an algorithm for enumerating consistent sub-graphs of DAGs. The algorithm recursively partitions the graph into strictly smaller graphs until the resulting graph becomes a rooted tree (forest), for which a linear-time solution is computed. It then combines the tallies from graphs created in the recursion to obtain the final count. We prove the correctness of this algorithm, propose several practical accelerations, evaluate it on random graphs and then apply it to characterize four major biomedical ontologies. We believe this work provides valuable insights into the complexity of concept annotation spaces and its potential influence on the predictability of ontological annotation.Availability and implementation https://github.com/shawn-peng/counting-consistent-sub-DAG Supplementary information Supplementary data are available at Bioinformatics online.

Galerkin Differences for acoustic and elastic wave equations in two space dimensions

The Galerkin Difference method, a finite element method built using standard Galerkin projection techniques but employing nonstandard basis functions, was originally developed for one space dimension in [1]. Here the method is extended to two space dimensions using a tensor product construction. Theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation. For the elastic wave equation, the approximations are found to be at least as accurate as predicted, but with a free surface the scheme may exhibit an unexpected superconvergence. Extension to curvilinear mapped grids is also considered for acoustics. In all cases, the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix, which implies optimal linear time solutions with respect to the number of degrees of freedom.