We investigate the classical Morris-Pratt and Knuth-Morris-Pratt pattern matching algorithms from the perspective of computer architecture, focusing on the effects of incorporating a simple branch prediction mechanism into the computational model. Assuming a fixed pattern and a random text, we derive precise estimates for the number of branch mispredictions incurred by these algorithms when using local predictors. Our analysis relies on tools from automata theory and Markov chains, offering a theoretical framework that can be extended to other text processing algorithms and more sophisticated branch prediction strategies.

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A d -interval is the union of d disjoint intervals on the real line, and a graph is a d -interval graph if it is the intersection graph of d -intervals. In particular, it is a unit d -interval graph if it admits a d -interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP -complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP -complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit d -interval graphs for any d ≥ 2 , which does not follow directly in graph recognition problems — as an example, it took almost 20 years to close the gap between d = 2 and d > 2 for the recognition of d -track interval graphs. Our result has several implications, including that for every d ≥ 2 , recognizing ( x , … , x ) d -interval graphs and depth r unit d -interval graphs is NP -complete for every x ≥ 11 and every r ≥ 4 .

For any natural number d, a graph G is a (disjoint) d-interval graph if it is the intersection graph of (disjoint) d-intervals, the union of d (disjoint) intervals on the real line. Two important subclasses of d-interval graphs are unit and balanced d-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for d-interval graphs. In particular, we prove that for any d ⩾ 2, if G is a K_{1,2d+1}-free interval graph, then G is a unit d-interval graph. However, somehow surprisingly, under the same assumptions, G is not always a disjoint unit d-interval graph. This implies that the class of disjoint unit d-interval graphs is strictly included in the class of unit d-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint d-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for d > 2.

We study the Maximum Zero-Sum Partition problem (or MZSP ), defined as follows: given a multiset S = { a 1 , a 2 , … , a n } of integers a i ∈ Z ⁎ (where Z ⁎ denotes the set of non-zero integers) such that ∑ i = 1 n a i = 0 , find a maximum cardinality partition { S 1 , S 2 , … , S k } of S such that, for every 1 ≤ i ≤ k , ∑ a j ∈ S i a j = 0 . Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP , in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S .

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A $d$-interval is the union of $d$ intervals on the real line, and a graph is a $d$-interval graph if it is the intersection graph of $d$-intervals. In particular, it is a unit $d$-interval graph if it admits a $d$-interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit $d$-interval graphs for any $d\geq 2$, which does not follow directly in graph recognition problems --as an example, it took almost 20 years to close the gap between $d=2$ and $d> 2$ for the recognition of $d$-track interval graphs. Our result has several implications, including that recognizing $(x, \dots, x)$ $d$-interval graphs and depth $r$ unit 2-interval graphs is NP-complete for every $x\geq 11$ and every $r\geq 4$.

We study in this paper the Doubly Partially Ordered Pattern Matching (or DPOP Matching) problem, a natural extension of the Permutation Pattern Matching problem. Permutation Pattern Matching takes as input two permutations σ and π, and asks whether there exists an occurrence of σ in π; whereas DPOP Matching takes two partial orders P_v and P_p defined on the same set X and a permutation π, and asks whether there exist |X| elements in π whose values (resp., positions) are in accordance with P_v (resp., P_p). Posets P_v and P_p aim at relaxing the conditions formerly imposed by the permutation σ, since σ yields a total order on both positions and values. Our problem being NP-hard in general (as Permutation Pattern Matching is), we consider restrictions on several parameters/properties of the input, e.g., bounding the size of the pattern, assuming symmetry of the posets (i.e., P_v and P_p are identical), assuming that one partial order is a total (resp., weak) order, bounding the length of the longest chain/anti-chain in the posets, or forbidding specific patterns in π. For each such restriction, we provide results which together give a(n almost) complete landscape for the algorithmic complexity of the problem.

The additive x-disorder of a permutation is the sum of the absolute differences of all pairs of consecutive elements. We show that the additive x-disorder of a permutation of S(n), n ≥ 2, ranges from n-1 to ⌊n²/2⌋ - 1, and we give a complete characterization of permutations having extreme such values. Moreover, for any positive integers n and d such that n ≥ 2 and n-1 ≤ d ≤ ⌊n²/2⌋ - 1, we propose a linear-time algorithm to compute a permutation π ∈ S(n) with additive x-disorder d.

A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph $(G,c)$ has a colourful partition of size at most $k$. This problem is closely related to the COLOURFUL COMPONENTS problem, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most $p$ edges. Nevertheless we show that COLOURFUL PARTITION and COLOURFUL COMPONENTS may have different complexities for restricted instances. We tighten known NP-hardness results for both problems and in addition we prove new hardness and tractability results for COLOURFUL PARTITION. Using these results we complete our paper with a thorough parameterized study of COLOURFUL PARTITION.

Massive biological datasets are available in public databases and can be queried using portals with keyword queries. Ranked lists of answers are obtained by users. However, properly querying such portals remains difficult since various formulations of the same query can be considered (e.g., using synonyms). Consequently, users have to manually combine several lists of hundreds of answers into one list. Rank aggregation techniques are particularly well-fitted to this context as they take in a set of ranked elements (rankings) and provide a consensus, that is, a single ranking which is the "closest" to the input rankings. However, the problem of rank aggregation is NP-hard in most cases. Using an exact algorithm is currently not possible for more than a few dozens of elements. A plethora of heuristics have thus been proposed which behaviour are, by essence, difficult to anticipate: given a set of input rankings, one cannot guarantee how far from an exact solution the consensus ranking provided by an heuristic will be. The two challenges we want to tackle in this paper are the following: (i) providing an approach based on a pre-process to decompose large data sets into smaller ones where high-quality algorithms can be run and (ii) providing information to users on the robustness of the positions of elements in the consensus ranking produced. Our approach not only lies in mathematical bases, offering guarantees on the result computed but it has also been implemented in a real system available to life science community and tested on various real use cases.

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square ( 213 , 231 ) -avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on directed perfect matchings, we prove that recognizing square permutations is NP -complete.

Recent techniques increase rapidly the amount of our knowledge on interactions between proteins. The interpretation of these new information depends on our ability to retrieve known substructures in the data, the Protein-Protein Interactions (PPIs) networks. In an algorithmic point of view, it is an hard task since it often leads to NP-hard problems. To overcome this difficulty, many authors have provided tools for querying patterns with a restricted topology, i.e., paths or trees in PPI networks. Such restriction leads to the development of fixed parameter tractable (FPT) algorithms, which can be practicable for restricted sizes of queries. Unfortunately, Graph Homomorphism is a W[1]-hard problem, and hence, no FPT algorithm can be found when patterns are in the shape of general graphs. However, Dost et al. gave an algorithm (which is not implemented) to query graphs with a bounded treewidth in PPI networks (the treewidth of the query being involved in the time complexity). In this paper, we propose another algorithm for querying pattern in the shape of graphs, also based on dynamic programming and the color-coding technique. To transform graphs queries into trees without loss of informations, we use feedback vertex set coupled to a node duplication mechanism. Hence, our algorithm is FPT for querying graphs with a bounded size of their feedback vertex set. It gives an alternative to the treewidth parameter, which can be better or worst for a given query. We provide a python implementation which allows us to validate our implementation on real data. Especially, we retrieve some human queries in the shape of graphs into the fly PPI network.

In this paper, we are interested in the computational complexity of computing (dis)similarity measures between two genomes when they contain duplicated genes or genomic markers, a problem that happens frequently when comparing whole nuclear genomes. Recently, several methods ( [1], [2]) have been proposed that are based on two steps to compute a given (dis)similarity measure M between two genomes G_1 and G_2: first, one establishes a oneto- one correspondence between genes of G_1 and genes of G_2 ; second, once this correspondence is established, it defines explicitly a permutation and it is then possible to quantify their similarity using classical measures defined for permutations, like the number of breakpoints. Hence these methods rely on two elements: a way to establish a one-to-one correspondence between genes of a pair of genomes, and a (dis)similarity measure for permutations. The problem is then, given a (dis)similarity measure for permutations, to compute a correspondence that defines an optimal permutation for this measure. We are interested here in two models to compute a one-to-one correspondence: the exemplar model, where all but one copy are deleted in both genomes for each gene family, and the matching model, that computes a maximal correspondence for each gene family. We show that for these two models, and for three (dis)similarity measures on permutations, namely the number of common intervals, the maximum adjacency disruption (MAD) number and the summed adjacency disruption (SAD) number, the problem of computing an optimal correspondence is NP-complete, and even APXhard for the MAD number and SAD number.

In this paper, we investigate the computational and approximation complexity of the Exemplar Longest Common Subsequence of a set of sequences (ELCS problem), a generalization of the Longest Common Subsequence problem, where the input sequences are over the union of two disjoint sets of symbols, a set of mandatory symbols and a set of optional symbols. We show that different versions of the problem are APX-hard even for instances with two sequences. Moreover, we show that the related problem of determining the existence of a feasible solution of the Exemplar Longest Common Subsequence of two sequences is NP-hard. On the positive side, we first present an efficient algorithm for the ELCS problem over instances of two sequences where each mandatory symbol can appear in total at most three times in the sequences. Furthermore, we present two fixed-parameter algorithms for the ELCS problem over instances of two sequences where the parameter is the number of mandatory symbols.

MOTIVATION: Molecular evolution, which is classically assessed by comparison of individual proteins or genes between species, can now be studied by comparing co-expressed functional groups of genes. This approach, which better reflects the functional constraints on the evolution of organisms, can exploit the large amount of data generated by genome-wide expression analyses. However, it requires new methodologies to represent the data in a more accessible way for cross-species comparisons. RESULTS: In this work, we present an approach based on Multi-dimensional Scaling techniques, to compare the conformation of two gene expression networks, represented in a multi-dimensional space. The expression networks are optimally superimposed, taking into account two criteria: (1) inter-organism orthologous gene pairs have to be nearby points in the final multi-dimensional space and (2) the distortion of the gene expression networks, the organization of which reflects the similarities between the gene expression measurements, has to be circumscribed. Using this approach, we compared the transcriptional programs that drive sporulation in budding and fission yeasts, extracting some common properties and differences between the two species.

BACKGROUND: Information obtained by DNA microarray technology gives a rough snapshot of the transcriptome state, i.e., the expression level of all the genes expressed in a cell population at any given time. One of the challenging questions raised by the tremendous amount of microarray data is to identify groups of co-regulated genes and to understand their role in cell functions. RESULTS: MiCoViTo (Microarray Comparison Visualization Tool) is a set of biologists' tools for exploring, comparing and visualizing changes in the yeast transcriptome by a gene-centric approach. A relational database includes data linked to genome expression and graphical output makes it easy to visualize clusters of co-expressed genes in the context of available biological information. To this aim, upload of personal data is possible and microarray data from fifty publications dedicated to S. cerevisiae are provided on-line. A web interface guides the biologist during the usage of this tool and is freely accessible at http://www.transcriptome.ens.fr/micovito/. CONCLUSIONS: MiCoViTo offers an easy-to-read picture of local transcriptional changes connected to current biological knowledge. This should help biologists to mine yeast microarray data and better understand the underlying biology. We plan to add functional annotations from other organisms. That would allow inter-species comparison of transcriptomes via orthology tables.