Simple Linear Time Research Articles

The real Schur decomposition estimates Lyapunov characteristic exponents with multiplicity greater than one

Lyapunov characteristic exponents are indicators of the nature and of the stability properties of solutions of differential equations. The estimation of Lyapunov exponents of algebraic multiplicity greater than 1 is troublesome. In this work, we intuitively derive an interpretation of higher multiplicity Lyapunov exponents in forms that occur in simple linear time invariant problems of engineering relevance. We propose a method to determine them from the real Schur decomposition of the state transition matrix of the linear, nonautonomous problem associated with the fiducial trajectory. So far, no practical way has been found to formulate the method as an algorithm capable of mitigating over- or underflow in the numerical computation of the state transition matrix. However, this interesting approach in some practical cases is shown to provide quicker convergence than usual methods like the discrete QR and the continuous QR and Singular Value Decomposition (SVD)methods when Lyapunov exponents with multiplicity greater than one are present.

Strictly Interval Graphs: Characterization and Linear Time Recognition

In this paper we introduce a new subclass of chordal graphs, those which are simultaneously strictly chordal and interval, the strictly interval graphs. We present the characterization of the new class by forbidden subgraphs and a simple linear time recognition obtained directly from the determination of the leafage of strictly chordal graphs. We also show how strictly interval graphs relate to proper interval graphs, presenting the new hierarchy of classes.

Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

A graph is equitably k -colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ = ( G ) . In this paper the problem of determinig χ = for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses χ = ( G ) or χ = ( G ) + 1 colors. Our algorithm is best possible, unless P=NP. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring.

A new LBFS-based algorithm for cocomparability graph recognition

How much can you learn about the structure of a given graph, using a series of successive graph searches? We consider here cocomparability graphs which are the complement of the graphs that admit a transitive orientation (comparability graphs). We study the use of a series of successive LBFS searches in order to capture the structure of these graphs. On cocomparability graphs many problems like coloring, Hamilton path, maximal clique which are NP-complete in the general case can be solved in polynomial time. For all those problems a decisive step is to find a cocomp ordering. We will prove the correctness of an algorithm to find a cocomp ordering using |V(G)| successive LBFS. Our main result answers a question asked in Corneil, Olariu and Stewart [2009] about the existence of a multisweep LBFS algorithm for cocomparability graphs, and lays the foundation of a simple linear time multisweep LBFS algorithm to find a cocomp ordering.

A note on quasi-kernels in digraphs

We describe a simple linear time algorithm to construct a quasi-kernel in a digraph and to find three quasi-kernels in digraphs without a kernel (giving constructive proofs of known results of Chvátal and Lovász, or Jacob and Meyniel). However, we show that it is NP-complete to decide if there is a quasi-kernel containing a specified vertex in a given digraph. The algorithm provides also a simple proof of the characterization of digraphs with only two quasi-kernels given by Gutin, Koh, Tay and Yeo.

Improved approximation algorithms for two variants of the stable marriage problem with ties

We consider the problem of computing a large stable matching in a bipartite graph $$G = (A\cup B, E)$$G=(A?B,E) where each vertex $$u \in A\cup B$$u?A?B ranks its neighbors in an order of preference, perhaps involving ties. Let the matched partner of u in a matching M be M(u). A matching M is said to be stable if there is no edge (a, b) such that a is unmatched or prefers b to M(a) and similarly, b is unmatched or prefers a to M(b). While a stable matching in G can be easily computed in linear time by the Gale---Shapley algorithm, it is known that computing a maximum size stable matching is APX-hard. In this paper we first consider the case when the preference lists of vertices in A are strict while the preference lists of vertices in B may include ties. This case is also APX-hard and the current best approximation ratio known here is 25/17 $$\approx 1.4706$$?1.4706 which relies on solving an LP. We improve this ratio to 22/15 $$\approx 1.4667$$?1.4667 by a simple linear time algorithm. Here we first compute a half-integral stable matching in $$\{0,0.5,1\}^{|E|}$${0,0.5,1}|E| and then round it to an integral stable matching M. The ratio $$|\mathsf {OPT}|/{|M|}$$|OPT|/|M| is bounded via a payment scheme that charges other components in $$\mathsf {OPT}\oplus M$$OPT?M to cover the costs of length-5 augmenting paths. There will be no length-3 augmenting paths here. We next consider the following special case of two-sided ties, where every tie length is 2. This case is known to be UGC-hard to approximate to within 4/3. We show a 10/7 $$\approx 1.4286$$?1.4286 approximation algorithm here that runs in linear time.

Characterization of Extremal Antipodal Polygons

Let $$S$$S be a set of $$2n$$2n points on a circle such that for each point $$p \in S$$p?S also its antipodal (mirrored with respect to the circle center) point $$p'$$p? belongs to $$S$$S. A polygon $$P$$P of size $$n$$n is called antipodal if it consists of precisely one point of each antipodal pair $$(p,p')$$(p,p?) of $$S$$S. We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of $$S$$S. Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization of antipodal polygons to higher dimensions we show that a similar characterization does not exist.

NPLs modeling in the banking sector of Bosnia and Herzegovina

We have chosen non-performing loans in Bosnia and Herzegovina's banking sector for the subject of this research. Our goal was to create models for forecasting of their movements. In addition to different simple linear regression models and time series analysis, we have also used the tools of technical analysis, which is major innovation in this area. Variables affecting non-performing loans are: matured receivables, GDP growth rate, and interest rate spread. The most important result of this research is the verification of our hypothesis that the tools of technical analysis can be used for forecasting the trends of non-performing loans. We have proved that different methodologies for prediction and different regression models, with different independent variables, suit different periods of NPLs development.

Post-Adaptation vulnerability of cereals to rainfall and temperature variability in the federal capital territory of Nigeria

This study assessed the vulnerability of cereals yield to climate change using an integrated and multi-scale quantitative approach. The objectives of this study include determining the level of climate variability, assessment of cereals yield sensitivity index, determining climate exposure index, determining adaptive capacity of farmers, assessment of the post adaptation vulnerability of cereals yield to climate change. Socioeconomic data were obtained through administration of questionnaires. Thirty years data of temperature and rainfall as well as fifteen years data of annual cereals yield were used. Mean and standard deviation, standardized coefficient of skewness (Z 1 ) and kurtosis (Z 2 ), simple linear regression and time series statistics analysis were used in this study for the analysis of data. Finding depicts that the exposure index of rainfall is low but high for temperature. Cereals sensitivity index/degree of crop yield failure is more from 2000 – 2010 and significant difference was observed in sensitivity index for all the cereals. Adaptive capacity of farmers to climate change is high in Bwari and AMAC but low in Gwagwalada, Kuje, Abaji and Kwali. Post adaptation vulnerability of maize, rice and millet yield to rainfall and temperature is low in AMAC and Bwari but high in Gwagwalada, Kwali, Abaji and Kuje. Post adaptation vulnerability of sorghum in relation to rainfall is low in all the area councils in the FCT except Abaji. In relation to temperature, vulnerability of sorghum is high in Abaji and Kuje but low in AMAC, Gwagwalada, Kwali and Bwari. It was recommended that there is need to place climate change within the top priority of developmental context, and provision and infrastructure as well as reliable agricultural extension service. Key Words: Cereals, Post-Adaptation, Vulnerability, Rainfall, Temperature, Variability, FCT

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.

Set graphs. IV. Further connections with claw-freeness

A graph G is said to be a set graph if it admits an acyclic orientation that is also extensional, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, set graphs are the underlying graphs of digraph representations of hereditarily finite sets, where a set is hereditarily finite if it is finite and has only hereditarily finite sets as members.It is known that every connected K1,3-free (or claw-free) graph is a set graph, and that an extensional acyclic orientation of such a graph can be found in polynomial time. In this paper, we generalize this result in three different directions. First, we identify the largest hereditary class of graphs G such that for every connected induced subgraph H of G, it holds that H is a set graph if and only if it is claw-free. Second, we consider graphs in which no two distinct induced claws have equal or adjacent centers, and prove that in this class of graphs set graphs can be equivalently characterized in terms of a property related to successive vertex removal. Finally, we show that for every r⩾1, connected K1,r+2-free graphs admit an acyclic orientation that is r-extensional, in the sense that at most r distinct vertices can have the same out-neighborhood. This also leads to a simple linear time algorithm for finding an extensional acyclic orientation of a given connected claw-free graph, thus improving over the previous algorithm.

A linear time algorithm for consecutive permutation pattern matching

We say that two sequences x and w of length m are order-isomorphic (of the same “shape”) if w[i]⩽w[j] if and only if x[i]⩽x[j] for each i,j∈[1,m]. We present a simple linear time algorithm for checking if a given sequence y of length n contains a factor which is order-isomorphic to a given pattern x. A factor is a subsequence of consecutive symbols of y, so we call our problem the consecutive permutation pattern matching. The (general) permutation pattern matching problem is related to general subsequences and is known to be NP-complete. We show that the situation for consecutive subsequences is significantly different and present an O(n+m) time algorithm under a natural assumption that the symbols of x can be sorted in O(m) time, otherwise the time is O(n+mlogm). In our algorithm we use a modification of the classical Knuth–Morris–Pratt string matching algorithm.

Simple linear time approximation algorithm for betweenness

We study the Betweenness problem. We are given a set of vertices and betweenness constraints. Each betweenness constraint of the form x⇝{y,z} requires that vertex x lies between vertices y and z. Our goal is to find a vertex ordering that maximizes the number of satisfied constraints. In 1995, Chor and Sudan designed an SDP algorithm that satisfies half of all constraints in a satisfiable instance. We present a simple combinatorial linear time algorithm with the same approximation guarantee.

Direct evidence for continuous linear kinetics in the low-temperature degradation of Y-TZP

The kinetics of the tetragonal to monoclinic (t–m) transformation of zirconia in a hydrous environment at 134°C and 3bar pressure was studied. As surface X-ray diffraction, which is conventionally used to explore the progress, has a very limited depth of information, it distorts the quantitative results in a layer-on-layer situation and by itself is ill suited for this reason. Analyzing cross sections is more suitable; therefore, focused ion beam techniques were used to prepare artifact-free cuts. The material was subsequently investigated by scanning electron microscopy, electron backscatter diffraction and Raman spectroscopy. Only the combination of methods makes it possible to resolve the quantifiable details of the process. The transformation starts in the near-surface areas, forms a layer, and the growth of this layer proceeds into the bulk material following a simple linear time law (0.0624μmh−1 for material in the chosen condition), without apparent retardation or limit. The progress yields a gradientless layer with a fixed amount of residual tetragonal zirconia (∼27% for 3Y-TZP in the present conditions) separated from unaffected material by a boundary, which has a roughness only in the grain size range. The kinetics indicates a reaction rate control, where the hydration reaction is the key factor, but is modified by the stepwise access of water to the reaction front opened by the autocatalytic transformation of zirconia with a critical hydration level.

Approximate probabilistic analysis of biopathway dynamics

Biopathways are often modeled as systems of ordinary differential equations (ODEs). Such systems will usually have many unknown parameters and hence will be difficult to calibrate. Since the data available for calibration will have limited precision, an approximate representation of the ODEs dynamics should suffice. One must, however, be able to efficiently construct such approximations for large models and perform model calibration and subsequent analysis. We present a graphical processing unit (GPU) based scheme by which a system of ODEs is approximated as a dynamic Bayesian network (DBN). We then construct a model checking procedure for DBNs based on a simple probabilistic linear time temporal logic. The GPU implementation considerably extends the reach of our previous PC-cluster-based implementation (Liu et al., 2011b). Further, the key components of our algorithm can serve as the GPU kernel for other Monte Carlo simulations-based analysis of biopathway dynamics. Similarly, our model checking framework is a generic one and can be applied in other systems biology settings. We have tested our methods on three ODE models of bio-pathways: the epidermal growth factor-nerve growth factor pathway, the segmentation clock network and the MLC-phosphorylation pathway models. The GPU implementation shows significant gains in performance and scalability whereas the model checking framework turns out to be convenient and efficient for specifying and verifying interesting pathways properties. The source code is freely available at http://www.comp.nus.edu.sg/~rpsysbio/pada-gpu/

Linear Time Split Decomposition Revisited

Given a family $\mathcal{F}$ of subsets of a ground set V, its orthogonal is defined to be the family of subsets that do not overlap any element of $\mathcal{F}$. Using this tool we revisit the problem of designing a simple linear time algorithm for undirected graph split (also known as 1-join) decomposition.

Estimating the final cost of a DoD acquisition contract

The most common technique to determine the predicted final cost of a Department of Defense (DoD) acquisition contract, or the Estimate at Completion (EAC), involves the use of performance indices to adjust the EAC. Other methods including simple linear regression and time series analysis have been developed to predict the final cost, but these methods are not widely publicized or have limited applicability. As a potential remedy, this research utilizes the historical contract data reported in the Defense Acquisition Executive Summary database and provides to the analyst a set of five working multiple regression models. Useful over the life of the contract, they accurately predict the final cost of the average major weapons system contract using contractor Cost Performance Report data.

Analysis of the dial-a-ride problem of Hunsaker and Savelsbergh

Hunsaker and Savelsbergh [B. Hunsaker, M. Savelsbergh, Efficient feasibility testing for dial-a-ride problems, Operations Research Letters 30 (2002) 169–173] discussed feasibility testing for a dial-a-ride problem under maximum wait time and maximum ride time constraints. We show that this feasibility test can be expressed as a shortest path problem in vertex-weighted interval graphs, which leads to a simple linear time algorithm.

A Simple Linear-Time Recognition Algorithm for Weakly Quasi-Threshold Graphs

Weakly quasi-threshold graphs form a proper subclass of the well-known class of cographs by restricting the join operation. In this paper we characterize weakly quasi-threshold graphs by a finite set of forbidden subgraphs: the class of weakly quasi-threshold graphs coincides with the class of {P 4, co-(2P 3)}-free graphs. Moreover we give the first linear-time algorithm to decide whether a given graph belongs to the class of weakly quasi-threshold graphs, improving the previously known running time. Based on the simplicity of our recognition algorithm, we can provide certificates of membership (a structure that characterizes weakly quasi-threshold graphs) or non-membership (forbidden induced subgraphs) in additional \({{\mathcal O}(n)}\) time. Furthermore we give a linear-time algorithm for finding the largest induced weakly quasi-threshold subgraph in a cograph.

Matchability and [formula omitted]-maximal matchings

We present a collection of new structural, algorithmic, and complexity results for matching problems of two types. The first problem involves the computation of k -maximal matchings, where a matching is k -maximal if it admits no augmenting path with ≤ 2 k vertices. The second involves finding a maximal set of vertices that is matchable — comprising one side of the edges in some matching. Among our results, we prove that the minimum cardinality β 2 of a 2-maximal matching is at most the minimum cardinality μ of a maximal matchable set, with equality attained for triangle-free graphs. We show that the parameters β 2 and μ are NP-hard to compute in bipartite and chordal graphs, but can be computed in linear time on a tree. Finally, we also give a simple linear-time algorithm for finding a 3-maximal matching, a consequence of which is a simple linear-time 3 / 4 -approximation algorithm for the maximum-cardinality matching problem in a general graph.