Finding Maximum Edge Bicliques in Convex Bipartite Graphs

A bipartite graph G=(A,B,E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v??A, vertices adjacent to v are consecutive in?B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G=(A,B,E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(nlog?³ nlog?log?n) time and O(n) space, where n=|A|. This improves the current O(n ²) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(n??(n)) and O(n) time, respectively, where n=min?(|A|,|B|) and ??(n) is the slowly growing inverse of the Ackermann function.     

Mining non-redundant time-gap sequential patterns

Mining sequential patterns is to discover sequential purchasing behaviors for most of the customers from a large amount of customer transactions. An example of such a pattern is that most of the customers purchased item B after purchasing item A, and then they purchased item C after using item B. The manager can use this information to promote item B and item C when a customer purchased item A and item B, respectively. However, the manager cannot know what time the customers will need these products if we only discover the sequential patterns without any extra information. In this paper, we develop a new algorithm to discover not only the sequential patterns but also the time interval between any two items in the pattern. We call this information the time-gap sequential patterns. An example of time-gap sequential pattern is that most of the customers purchased item A, and then they bought item B after m to n days, and then after p to q days, they bought item C. When a customer bought item A, the information about item B can be sent to this customer after m to n days, that is, we can provide the product information in which the customer is interested on the appropriate date.     

Bounded DOL languages

For an alphabet A and a morphism h : A¹ → A¹, the set of words w such that the DOL language L(A, h, w) is a BOUNDED language is shown to be B¹, where B is an effectively computable subset of A. Consequently, BOUNDEDNESS is decidable for DOL languages. The result depends on the authors' recent results on periodic DOL languages. Interpretation of the result for polynomially bounded DOL languages is also considered.     

On the problem of general structural assignments of linear systems through sensor/actuator selection

A systematic method is developed for determining an output matrix C for a given matrix pair (A,B) such that the resulting linear system characterized by the matrix triple (A,B,C) has the pre-specified system structural properties, such as the finite and infinite zero structure and the invertibility structures. Since the matrix C describes the locations of the sensors, the procedure of choosing C is often referred to as sensor selection. The method developed in this paper for sensor selection can be applied to the dual problem of actuator selection, where, for a given matrix pair (A,C), a matrix B is to be determined such that the resulting matrix triple (A,B,C) has the pre-specified structural properties.     

Structure-preserving numerical algorithm for solving discrete-time LMI and DARS

We present a numerical algorithm to solve a discrete-time linear matrix inequality (LMI) and discrete-time algebraic Riccati system (DARS). With a given system (A,B,C,D) we associate a para-hermitian matrix pencil. Then we transform it by an orthogonal transformation matrix into a block-triangular para-hermitian form. Under either of the two assumptions (1) matrix pair (A,B) is controllable or (2) matrix pair (A,B) is reachable and (A,B,C,D) is a left invertible system, we extract the solution of LMI and DARS by the entries of the orthogonal transformation matrix.     

MGS: A Rule-Based Programming Language for Complex Objects and Collections

We present the first results in the development of a new declarative programming language called MGS. This language is devoted to the simulation of biological processes, especially those whose state space must be computed jointly with the current state of the system. MGS proposes a unified view on several computational mechanisms initially inspired by biological or chemical processes (Gamma and the CHAM, Lindenmayer systems, Paun systems and cellular automata). The basic computation step in MGS replaces in a collection A of elements, some subcollection B, by another collection C. The collection C only depends on B and its adjacent elements in A. The pasting of C into A – B depends on the shape of the involved collections. This step is called a transformation. The specification of the collection to be substituted can be done in many ways. We propose here a pattern language based on the neighborhood relationship induced by the topology of the collection. Several features to control the transformation applications are then presented.     

A simple method for computing the entropy of the product of general fuzzy intervals

Entropy is generally a measurement of the fuzziness degree for a fuzzy set. This paper studies the entropy calculation of the standard product of two general fuzzy intervals, i.e. C=A×B, and then provides some simple formulas to get the entropy of the product C. Those formulas are composed of the locations of critical points of the fuzzy intervals A and B. By the formulas, the calculation of the membership function of the product C is not needed any more, and the fuzziness estimation of the product of two arbitrary fuzzy intervals can be obtained quickly and effectively.     

On the hilbert transform of bounded bandlimited signals

In this paper we analyze the Hilbert transform and existence of the analytical signal for the space B _?? ^?? of bandlimited signals that are bounded on the real axis. Originally, the theory was developed for signals in L ²(?) and then extended to larger signal spaces. While it is well known that the common integral representation of the Hilbert transform may diverge for some signals in B _?? ^?? and that the Hilbert transform is not a bounded operator on B _?? ^?? , it is nevertheless possible to define the Hilbert transform for the space B _?? ^?? . We use a definition that is based on the H ¹-BMO(?) duality. This abstract definition, which can be used for general bounded signals, gives no constructive procedure to compute the Hilbert transform. However, for the practically important special case of bounded bandlimited signals, we can provide such an explicit procedure by giving a closed-form expression for the Hilbert transform. Further, it is shown that the Hilbert transform of a signal in B _?? ^?? is still bandlimited but not necessarily bounded. With these results we continue the work of [1, 2].     

Reductions between disjoint NP-Pairs

Disjoint NP-pairs are pairs (A, B) of nonempty, disjoint sets in NP. We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs (A, B) and (C, D) is a Turing reduction with the additional property that if the input belongs to A B, then all queries belong to C D. We prove under the reasonable assumption that UP ∩ co-UP has a P-bi-immune set that there exist disjoint NP-pairs (A, B) and (C, D) such that (A, B) is truth-table reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DistNP has a truth-table-complete disjoint NP-pair, but has no many-one-complete disjoint NP-pair.     

Further results on structural assignment of linear systems via sensor selection

The problem of assigning structural properties of a linear system through sensor selection is, for a given pair (A,B), to find an output pair (C,D) such that the resulting system (A,B,C,D) has the pre-specified structural properties, such as the finite and infinite zero structures and the invertibility properties. In this paper, by introducing the notion of infinite zero assignable sets for the pair (A,B), we establish necessary and sufficient conditions for the assignability of a given set of infinite zeros and a set of structural properties which includes the left invertibility property. In establishing these conditions, we develop a numerical algorithm for the construction of the required (C,D).     

Birman–Wenzl–Murakami algebra, topological parameter and Berry phase

In this paper, a 3?×?3-matrix representation of Birman?CWenzl?CMurakami (BWM) algebra has been presented. Based on which, unitary matrices A(??, ?? ₁, ?? ₂) and B(??, ?? ₁, ?? ₂) are generated via Yang?CBaxterization approach. A Hamiltonian is constructed from the unitary B(??, ??) matrix. Then we study Berry phase of the Yang?CBaxter system, and obtain the relationship between topological parameter and Berry phase.     

Time-Bounded Kolmogorov Complexity and Solovay Functions

A Solovay function is an upper bound g for prefix-free Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n. We obtain natural examples of computable Solovay functions by showing that for some constant c ₀ and all computable functions t such that c ₀ n??t(n), the time-bounded version K ^t of K is a Solovay function. By unifying results of Bienvenu and Downey and of Miller, we show that a right-computable upper bound g of K is a Solovay function if and only if ?? _g =??2^?g(n) is Martin-Löf random. We obtain as a corollary that the Martin-Löf randomness of the various variants of Chaitin??s ?? extends to the time-bounded case in so far as $\Omega _{ \textnormal{K}^{t}}$ is Martin-Löf random for any t as above. As a step in the direction of a characterization of K-triviality in terms of jump-traceability, we demonstrate that a set A is K-trivial if and only if A is O(g(n)?K(n))-jump traceable for all computable Solovay functions g. Furthermore, this equivalence remains true when the universal quantification over all computable Solovay functions in the second statement is restricted either to all functions of the form K ^t for some function t as above or to a single function K ^t of this form. Finally, we investigate into the plain Kolmogorov complexity C and its time-bounded variant C ^t of initial segments of computably enumerable sets. Our main theorem here asserts that every high c.e. Turing degree contains a c.e. set B such that for any computable function t there is a constant c _t >0 such that for all m it holds that C ^t (B?m)??c _t ?m, whereas for any nonhigh c.e. set A there is a computable time bound t and a constant c such that for infinitely many m it holds that C ^t (A?m)??logm+c. By similar methods it can be shown that any high degree contains a set B such that C ^t (B?m)??⁺ m/4. The constructed sets B have low unbounded but high time-bounded Kolmogorov complexity, and accordingly we obtain an alternative proof of the result due to Juedes et al. (Theor. Comput. Sci. 132(1?C2):37?C70, 1994) that every high degree contains a strongly deep set.     

Sequential estimation of a threshold crossing time for a Gaussian random walk through correlated observations

Given a Gaussian random walk X with drift, we consider the problem of estimating its first-passage time ?? _A for a given level A from an observation process Y correlated to X. Estimators may be any stopping times ?? with respect to the observation process Y. Two cases of the process Y are considered: a noisy version of X and a process X with delay d. For a given loss function f(x), in both cases we find exact asymptotics of the minimal possible risk E f((?? ? ?? _A )/r) as A, d ?? ??, where r is a normalizing coefficient. The results are extended to the corresponding continuous-time setting where X and Y are Brownian motions with drift.     

Priorities on context conditions in rewriting systems

This paper is concerned with letter-rewriting systems in which context-free rewriting productions are equipped with context conditions. Such a production, π = A → α can be used to rewrite an occurrence of A in a string x only if x satisfies the context conditions attached to π. In particular, we are concerned with situations where the context conditions of several productions are satisfied by a given occurrence of A. Deciding which of these productions can be applied to rewrite this occurrence is done by (once a priori) fixed priorities among all context conditions.     

The syntactic monoid of hairpin-free languages

The study of hairpin-free words has been initiated in the context of DNA computing. DNA strands that, theoretically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is complementary to T and G to C, DNA single strands that are complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: if some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper continues the theoretical study of hairpin-free languages. We study algebraic properties of hairpin-free words and hairpins. We also give a complete characterization of the syntactic monoid of the language consisting of all hairpin-free words over a given alphabet and illustrate it with an example using the DNA alphabet.     

Shorthand Universal Cycles for Permutations

The set of permutations of ??n??={1,??,n} in one-line notation is ??(n). The shorthand encoding of a ₁?a _n ????(n) is a ₁?a _n?1. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n! whose substrings of length n?1 are the shorthand encodings of ??(n). When an SP-cycle is decoded, the order of ??(n) is a Gray code in which successive permutations differ by the prefix-rotation ?? _i =(1 2 ? i) for i??{n?1,n}. Thus, SP-cycles can be represented by n! bits. We investigate SP-cycles with maximum and minimum ??weight?? (number of ?? _n?1s in the Gray code). An SP-cycle n a n b?n z is ??periodic?? if its ??sub-permutations?? a,b,??,z equal ??(n?1). We prove that periodic min-weight SP-cycles correspond to spanning trees of the (n?1)-permutohedron. We provide two constructions: B(n) and C(n). In B(n) the spanning trees use ??half-hunts?? from bell-ringing, and in C(n) the sub-permutations use cool-lex order by Williams (SODA, 987?C996, 2009). Algorithmic results are: (1)?memoryless decoding of B(n) and C(n), (2)?O((n?1)!)-time generation of B(n) and C(n) using sub-permutations, (3)?loopless generation of B(n)??s binary representation n bits at a time, and (4)?O(n+??(n))-time ranking of B(n)??s permutations where ??(n) is the cost of computing a permutation??s inversion vector. Results (1)?C(4) improve on those for the previous SP-cycle construction D(n) by Ruskey and Williams (ACM Trans. Algorithms 6(3):Art.?45, 2010), which we characterize here using ??recycling??.     

Measuring the understanding between two agents through concept similarity

Two agents previously unknown to each other cannot communicate by exchanging concepts (nodes of their own ontology): they need to use a common communication language. If they do not use a standard protocol, most likely they use a natural language. The ambiguities of it, and the different concepts the agents possess, give rise to imperfect understanding among them: How closely concepts in ontology O_A map¹ to which of O_B? Can we measure these mismatches?Given a concept from ontology O_A, a method is provided to find the most similar concept in O_B, and to measure the similarity between both concepts. The paper also gives an algorithm to gauge du(A, B), the degree of understanding that agent A has about the ontology of B. The procedures use word comparison, since no agent (except the Very Wise Creature, VWC) can measure du directly. Examples are given.     

Deciding word neighborhood with universal neighborhood automata

Given some form of distance between words, a fundamental operation is to decide whether the distance between two given words w and v is within a given bound. In earlier work, we introduced the concept of a universal Levenshtein automaton for a given distance bound n. This deterministic automaton takes as input a sequence χ of bitvectors computed from w and v. The sequence χ is accepted iff the Levenshtein distance between w and v does not exceed n. The automaton is called universal since the same automaton can be used for arbitrary input words w and v, regardless of the underlying input alphabet. Here, we extend this picture. After introducing a large abstract family of generalized word distances, we exactly characterize those members where word neighborhood can be decided using universal neighborhood automata similar to universal Levenshtein automata. Our theoretical results establish several bridges to the theory of synchronized finite-state transducers and dynamic programming. For small neighborhood bounds, universal neighborhood automata can be held in main memory. This leads to very efficient algorithms for the above decision problem. Evaluation results show that these algorithms are much faster than those based on dynamic programming.     

Low-level dichotomy for quantified constraint satisfaction problems

Building on a result of Larose and Tesson for constraint satisfaction problems (CSPs), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime.We show that the class of B such that CSP(B) is first-order expressible (indeed trivial) is a microcosm for all QCSPs. Specifically, for any B there exists a C — generally not a core — such that CSP(C) is trivial, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions.     

Categorical aspects of data type constructors

Sketches are introduced as presentations of many-sorted algebraic theories and data types are described as initial algebras for such sketches. A construction A⁺ofB is described where A⁺ is a suitably structured sketch and B is a sketch. In this construction each operation from A⁺ operates on all the sorts of B. Many examples are given. The two main theoretical results are the theorem about the tensor product ⊗ for structured sketches such that A⁺of(B⁺of C)≃(A⁺⊗B⁺)of C (Theorem 3.17), and the “homomorphism” theorem 4.2 which describes the operation of structured sketches on the fibred category of all models of all sketches.