Log-Space Algorithms for Paths and Matchings in k-Trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 (Jakoby and Tantau in Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007). In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect matching (decision version), and if so, finding a perfect matching (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs, as shown in Datta et al. (Theory Comput. Syst. 47:737–757, 2010). Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of the divide-and-conquer approach in log-space, along with some ideas from Jakoby and Tantau (Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007) and Limaye et al. (Theory Comput. Syst. 46(3):499–522, 2010).     

Two-stage Robust Network Design with Exponential Scenarios

We study two-stage robust variants of combinatorial optimization problems on undirected graphs, like Steiner tree, Steiner forest, and uncapacitated facility location. Robust optimization problems, previously studied by Dhamdhere et al. (Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05), pp. 367–378, 2005), Golovin et al. (Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006), and Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439–453, 2007), are two-stage planning problems in which the requirements are revealed after some decisions are taken in Stage 1. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust k-Steiner tree problem, for example, one buys some edges in Stage 1. Then k terminals are revealed in Stage 2 and one has to buy more edges, at a higher cost, to complete the Stage 1 solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially many scenarios given implicitly. A scenario consists of any subset of k terminals (for k-Steiner tree), or any subset of k terminal-pairs (for k-Steiner forest), or any subset of k clients (for facility location). Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439–453, 2007) give an LP-based general framework for approximation algorithms for a class of two stage robust problems. Their framework cannot be used for network design problems like k-Steiner tree (see later elaboration). Their framework can be used for the robust facility location problem, but gives only a logarithmic approximation. We present the first constant-factor approximation algorithms for the robust k-Steiner tree (with exponential number of scenarios) and robust uncapacitated facility location problems. Our algorithms are combinatorial and are based on guessing the optimum cost and clustering to aggregate nearby vertices. For the robust k-Steiner forest problem on trees and with uniform multiplicative increase factor for Stage 2 (also known as inflation), we present a constant approximation. We show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question of (Dhamdhere et al. in Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05), pp. 367–378, 2005) and (Golovin et al. in Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006).     

On anonymizing transactions with sensitive items

K-anonymity (Samarati and Sweeny 1998; Samarati, IEEE Trans Knowl Data Eng, 13(6):1010–1027, 2001; Sweeny, Int J Uncertain, Fuzziness Knowl-Based Syst, 10(5):557–570, 2002) and its variants, l-diversity (Machanavajjhala et al., ACM TKDD, 2007) and tcloseness (Li et al. 2007) among others are anonymization techniques for relational data and transaction data, which are used to protect privacy against re-identification attacks. A relational dataset D is k-anonymous if every record in D has at least k-1 other records with identical quasi-identifier attribute values. The combination of released data with external data will never allow the recipient to associate each released record with less than k individuals (Samarati, IEEE Trans Knowl Data Eng, 13(6):1010–1027, 2001). However, the current concept of k-anonymity on transaction data treats all items as quasi-identifiers. The anonymized data set has k identical transactions in groups and suffers from lower data utility (He and Naughton 2009; He et al. 2011; Liu and Wang 2010; Terrovitis et al., VLDB J, 20(1):83–106, 2011; Terrovitis et al. 2008). To improve the utility of anonymized transaction data, this work proposes a novel anonymity concept on transaction data that contain both quasi-identifier items (QID) and sensitive items (SI). A transaction that contains sensitive items must have at least k-1 other identical transactions (Ghinita et al. IEEE TKDE, 33(2):161–174, 2011; Xu et al. 2008). For a transaction that does not contain a sensitive item, no anonymization is required. A transaction dataset that satisfies this property is said to be sensitive k-anonymous. Three algorithms, Sensitive Transaction Neighbors (STN) Gray Sort Clustering (GSC) and Nearest Neighbors for K-anonymization (K-NN), are developed. These algorithms use adding/deleting QID items and only adding SI to achieve sensitive k-anonymity on transaction data. Additionally, a simple “privacy value” is proposed to evaluate the degree of privacy for different types of k-anonymity on transaction data. Extensive numerical simulations were carried out to demonstrate the characteristics of the proposed algorithms and also compared to other types of k-anonymity approaches. The results show that each technique possesses its own advantage under different criteria such as running time, operation, and information loss. The results obtained here can be used as a guideline of the selection of anonymization technique on different data sets and for different applications.     

Generalising Unit-Refutation Completeness and SLUR via Nested Input Resolution

The class ${\mathcal{SLUR}}$ (Single Lookahead Unit Resolution) was introduced in Schlipf et al. (Inf Process Lett 54:133–137, 1995) as an umbrella class for efficient (poly-time) SAT solving, with linear-time SAT decision, while the recognition problem was not considered. ?epek et al. (2012) and Balyo et al. (2012) extended this class in various ways to hierarchies covering all of CNF (all clause-sets). We introduce a hierarchy ${\mathcal{SLUR}}_k$ which we argue is the natural “limit” of such approaches. The second source for our investigations is the class ${\mathcal{UC}}$ of unit-refutation complete clause-sets, introduced in del Val (1994) as a target class for knowledge compilation. Via the theory of “hardness” of clause-sets as developed in Kullmann (1999), Kullmann (Ann Math Artif Intell 40(3–4):303–352, 2004) and Ansótegui et al. (2008) we obtain a natural generalisation ${\mathcal{UC}}_k$ , containing those clause-sets which are “unit-refutation complete of level k”, which is the same as having hardness at most k. Utilising the strong connections to (tree-)resolution complexity and (nested) input resolution, we develop basic methods for the determination of hardness (the level k in ${\mathcal{UC}}_k$ ). A fundamental insight now is that ${\mathcal{SLUR}}_k = {\mathcal{UC}}_k$ holds for all k. We can thus exploit both streams of intuitions and methods for the investigations of these hierarchies. As an application we can easily show that the hierarchies from ?epek et al. (2012) and Balyo et al. (2012) are strongly subsumed by ${\mathcal{SLUR}}_k$ . Finally we consider the problem of “irredundant” clause-sets in ${\mathcal{UC}}_k$ . For 2-CNF we show that strong minimisations are possible in polynomial time, while already for (very special) Horn clause-sets minimisation is NP-complete. We conclude with an extensive discussion of open problems and future directions. We envisage the concepts investigated here to be the starting point for a theory of good SAT translations, which brings together the good SAT-solving aspects from ${\mathcal{SLUR}}$ together with the knowledge-representation aspects from ${\mathcal{UC}}$ , and expands this combination via notions of “hardness”.     

An Improved FPT Algorithm and a Quadratic Kernel for Pathwidth One Vertex Deletion

The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph?G and an integer k, one can delete at most k vertices from?G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied Feedback Vertex Set (FVS) problem, where the deletion of at most k vertices has to result in the remaining graph having treewidth at most 1 (i.e., being a forest). Recently Philip et?al. (WG, Lecture Notes in Computer Science, vol.?6410, pp.?196?C207, 2010) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7 ^k n ^O(1). In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65 ^k n ^O(1), thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp.?251?C260, ACM, New York, 2010). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomassé (ACM Trans. Algorithms 6(2), 2010).     

Functional un|unparsing

Danvy??s functional unparsing problem (Danvy in J. Funct. Program. 8(6), 621?C625, 1998) is to implement a type-safe ??printf?? function, which converts a sequence of heterogeneous arguments to a string according to a given format. The dual problem is to implement a type-safe ??scanf?? function, which extracts a sequence of heterogeneous arguments from a string by interpreting (Friedman and Wand in LFP, pp. 348?C355, 1984 and in Essentials of Programming Languages, MIT Press, 2008) the same format as an equally heterogeneous sequence of patterns that binds zero or more variables. We derive multiple solutions to both problems (Wand in J. ACM 27(1), 164?C180, 1980) from their formal specifications (Wand in Theor. Comput. Sci. 20(1), 3?C32, 1982). On one hand, our solutions show how the Hindley-Milner type system, unextended, permits accessing heterogeneous sequences with the static assurance of type safety. On the other hand, our solutions demonstrate the use of control operators (Felleisen et al. in Proceedings of the 1988 ACM Conference on Lisp and Functional Programming, pp. 52?C62, ACM Press, New York, 1988; Wand in POPL 85: Conference Record of the Annual ACM Symposium on Principles of Programming Languages, vol. 16, ACM Press, New York, 1985; Meyer and Wand in Logics of Programs, Lecture Notes in Computer Science, vol. 193, pp. 219?C224, Springer, Berlin, 1985) to communicate with formats as coroutines (Wand in Proceedings of the 1980 ACM Conference on Lisp and Functional Programming, vol. 12, pp. 285?C299, ACM Press, New York, 1980 and Haynes et al. in LFP, pp. 293?C298, 1984).     

Multi-player epistemic games: Guiding the enactment of classroom knowledge-building communities

Teachers and students face many challenges in shifting from traditional classroom cultures to enacting the Knowledge-Building Communities model (KBC model) supported by the CSCL environment, Knowledge Forum (Bereiter, 2002; Bereiter & Scardamalia, 1993; Scardamalia, 2002; Scardamalia & Bereiter, 2006). Enacting the model involves socializing students into knowledge work, similar to disciplinary communities. A useful construct in the field of the Learning Sciences for understanding knowledge work is “epistemic games” (Collins & Ferguson, 1993; Morrison & Collins 1995; Perkins, 1997). We propose that a powerful means for supporting classroom enactments of the KBC model entails conceptualizing Knowledge Forum as a collective space for playing multi-player epistemic games. Participation in knowledge-building communities is then scaffolded through learning the moves of such games. We have designed scaffolding tools that highlight particular knowledge-building moves for practice and reflection as a means of supporting students and teachers in coming to understand how to collectively work together toward the progressive improvement of ideas. In order to examine our design theories in practice, we present research on Ideas First, a design-based research program involving enactments of the KBC model in Singaporean primary science classrooms (Bielaczyc & Ow, 2007, 2010; Ow & Bielaczyc, 2007; 2008).     

A Uniform Paradigm to Succinctly Encode Various Families of Trees

We propose a uniform method to encode various types of trees succinctly. These families include ordered (ordinal), k-ary (cardinal), and unordered (free) trees. We will show the approach is intrinsically suitable for obtaining entropy-based encodings of trees (such as the degree-distribution entropy). Previously-existing succinct encodings of trees use ad hoc techniques to encode each particular family of trees. Additionally, the succinct encodings obtained using the uniform approach improve upon the existing succinct encodings of each family of trees; in the case of ordered trees, it simplifies the encoding while supporting the full set of navigational operations. It also simplifies the implementation of many supported operations. The approach applied to k-ary trees yields a succinct encoding that supports both cardinal-type operations (e.g. determining the child label i) as well as the full set of ordinal-type operations (e.g. reporting the number of siblings to the left of a node). Previous work on succinct encodings of k-ary trees does not support both types of operations simultaneously (Benoit et al. in Algorithmica 43(4):275–292, 2005; Raman et al. in ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 233–242, 2002). For unordered trees, the approach achieves the first succinct encoding. The approach is based on two recursive decompositions of trees into subtrees. Recursive decomposition of a structure into substructures is a common technique in succinct encodings and has even been used to encode (ordered) trees (Geary et al. in ACM Trans. Algorithms 2(4):510–534, 2006; He et al. in ICALP, pp. 509–520, 2007) and dynamic binary trees (Munro et al. in ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 529–536, 2001; Storm in Representing dynamic binary trees succinctly, Master’s thesis, 2000). The main distinction of the approach in this paper is that a tree is decomposed into subtrees in a manner that the subtrees are maximally isolated from each other. This intermediate decomposition result is interesting in its own right and has proved useful in other applications (Farzan et al. in ICALP (1), pp. 451–462, 2009; Farzan and Munro in ICALP (1), pp. 439–450, 2009; Farzan and Kamali in ICALP, 2011).     

Expected Optimal Feedback with Time-Varying Parameters

In this paper we derive the closed loop form of the Expected Optimal Feedback rule, sometimes called passive learning stochastic control, with time varying parameters. As such this paper extends the work of Kendrick (Stochastic control for economic models, 1981; Stochastic control for economic models, 2002, Chap. 6) where parameters are assumed to vary randomly around a known constant mean. Furthermore, we show that the cautionary myopic rule in Beck and Wieland (J Econ Dyn Control 26:1359–1377, 2002) model, a test bed for comparing various stochastic optimizations approaches, can be cast into this framework and can be treated as a special case of this solution.     

A Combinatorial Analysis for the Critical Clause Tree

In Paturi, Pudlák, Saks, and Zane (Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS1998), pp. 628–637, 1998) proposed a simple randomized algorithm for finding a satisfying assignment of a k-CNF formula. The main lemma of the paper is as follows: Given a satisfiable k-CNF formula that has a d-isolated satisfying assignment z, the randomized algorithm finds z with probability at least $2^{-(1-\mu_{k}/(k-1)+\epsilon_{k}(d))n}$ , where $\mu_{k}/(k-1)=\sum_{i=1}^{\infty}1/(i((k-1)i+1))$ , and ? _k (d)=o _d (1). They estimated the lower bound of the probability in an analytical way, and used some asymptotics. In this paper, we analyze the same randomized algorithm, and estimate the probability in a combinatorial way. The lower bound we obtain is a little simpler: $2^{-(1-\mu_{k}(d)/(k-1))n}$ , where $\mu_{k}(d)/(k-1)=\sum_{i=1}^{d}1/(i((k-1)i+1))$ . This value is a little bit larger (i.e., better) than that of Paturi et al. (Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS1998), pp. 628–637, 1998) although the two values are asymptotically equal when d=ω(1).     

The Parameterized Complexity of Unique Coverage and Its Variants

In this paper we study the parameterized complexity of the Unique Coverage problem, a variant of the classic Set Cover problem. This problem admits several parameterizations and we show that all, except the standard parameterization and a generalization of it, are unlikely to be fixed-parameter tractable. We use results from extremal combinatorics to obtain the best-known kernel for Unique Coverage and the well-known color-coding technique of Alon et al. (J. ACM 42(4), 844–856, 1995) to show that a weighted version of this problem is fixed-parameter tractable. Our application of color-coding uses an interesting variation of s-perfect hash families called (k,s)-hash families which were studied by Alon et al. (J. Comb. Theory Ser. A 104(1), 207–215, 2003) in the context of a class of codes called parent identifying codes (Barg et al. in SIAM J. Discrete Math. 14(3), 423–431, 2001). To the best of our knowledge, this is the first application of (k,s)-hash families outside the domain of coding theory. We prove the existence of such families of size smaller than the best-known s-perfect hash families using the probabilistic method (Alon and Spencer in The Probabilistic Method, Wiley, New York, 2000). Explicit constructions of such families of size promised by the probabilistic method is open.     

A Simple D 2-Sampling Based PTAS for k-Means and Other Clustering Problems

Given a set of points \(P \subset\mathbb{R}^{d}\) , the k-means clustering problem is to find a set of k centers \(C = \{ c_{1},\ldots,c_{k}\}, c_{i} \in\mathbb{R}^{d}\) , such that the objective function ∑ _x∈P e(x,C)², where e(x,C) denotes the Euclidean distance between x and the closest center in C, is minimized. This is one of the most prominent objective functions that has been studied with respect to clustering. D ²-sampling (Arthur and Vassilvitskii, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points \(P \subset\mathbb{R}^{d}\) , the first point is chosen uniformly at random from P. Subsequently, a point from P is chosen as the next sample with probability proportional to the square of the distance of this point to the nearest previously sampled point. D ²-sampling has been shown to have nice properties with respect to the k-means clustering problem. Arthur and Vassilvitskii (Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) show that k points chosen as centers from P using D ²-sampling give an O(logk) approximation in expectation. Ailon et al. (NIPS, pp. 10–18, 2009) and Aggarwal et al. (Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 15–28, Springer, Berlin, 2009) extended results of Arthur and Vassilvitskii (Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) to show that O(k) points chosen as centers using D ²-sampling give an O(1) approximation to the k-means objective function with high probability. In this paper, we further demonstrate the power of D ²-sampling by giving a simple randomized (1+?)-approximation algorithm that uses the D ²-sampling in its core.     

Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

In this paper we provide improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems. Given a graph G=(V,E) with weights w:E→?⁺, a set T ₁,T ₂,…,T _k of subtrees of G is called a tree cover of G if $V=\bigcup_{i=1}^{k} V(T_{i})$ . In the Min-Max k-tree Cover problem we are given graph G and a positive integer k and the goal is to find a tree cover with k trees, such that the weight of the largest tree in the cover is minimized. We present a 3-approximation algorithm for this improving the two different approximation algorithms presented in Arkin et al. (J. Algorithms 59:1–18, 2006) and Even et al. (Oper. Res. Lett. 32(4):309–315, 2004) with ratio 4. The problem is known to have an APX-hardness lower bound of $\frac{3}{2}$ (Xu and Wen in Oper. Res. Lett. 38:169–173, 2010). In the Bounded Tree Cover problem we are given graph G and a bound λ and the goal is to find a tree cover with minimum number of trees such that each tree has weight at most λ. We present a 2.5-approximation algorithm for this, improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).     

On sunflowers and matrix multiplication

We present several variants of the sunflower conjecture of Erd?s & Rado (J Lond Math Soc 35:85–90, 1960) and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erd?s–Rado sunflower conjecture (if true) implies a negative answer to the “no three disjoint equivoluminous subsets” question of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990); we also formulate a “multicolored” sunflower conjecture in ${\mathbb{Z}_3^n}$ and show that (if true) it implies a negative answer to the “strong USP” conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith–Winograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in ${\mathbb{Z}_3^n}$ is a strengthening of the well-known (ordinary) sunflower conjecture in ${\mathbb{Z}_3^n}$ , and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51 . . .) ⁿ on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21 . . . ) ⁿ Edel (2004) on the size of the largest 3-sunflower-free set in ${\mathbb{Z}_3^n}$ .     

pth Moment Exponential Stability of Stochastic Recurrent Neural Networks with Markovian Switching

This paper investigates the problem of the pth moment exponential stability for a class of stochastic recurrent neural networks with Markovian jump parameters. With the help of Lyapunov function, stochastic analysis technique, generalized Halanay inequality and Hardy inequality, some novel sufficient conditions on the pth moment exponential stability of the considered system are derived. The results obtained in this paper are completely new and complement and improve some of the previously known results (Liao and Mao, Stoch Anal Appl, 14:165–185, 1996; Wan and Sun, Phys Lett A, 343:306–318, 2005; Hu et al., Chao Solitions Fractals, 27:1006–1010, 2006; Sun and Cao, Nonlinear Anal Real, 8:1171–1185, 2007; Huang et al., Inf Sci, 178:2194–2203, 2008; Wang et al., Phys Lett A, 356:346–352, 2006; Peng and Liu, Neural Comput Appl, 20:543–547, 2011). Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

Finding and Counting Vertex-Colored Subtrees

The problems studied in this article originate from the Graph Motif problem introduced by Lacroix et al. (IEEE/ACM Trans. Comput. Biol. Bioinform. 3(4):360–368, 2006) in the context of biological networks. The problem is to decide if a vertex-colored graph has a connected subgraph whose colors equal a given multiset of colors M. It is a graph pattern-matching problem variant, where the structure of the occurrence of the pattern is not of interest but the only requirement is the connectedness. Using an algebraic framework recently introduced by Koutis (Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 5125, pp. 575–586, 2008) and Koutis and Williams (Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 5555, pp. 653–664, 2009), we obtain new FPT algorithms for Graph Motif and variants, with improved running times. We also obtain results on the counting versions of this problem, proving that the counting problem is FPT if M is a set, but becomes #W[1]-hard if M is a multiset with two colors. Finally, we present an experimental evaluation of this approach on real datasets, showing that its performance compares favorably with existing software.     

Multi-letter quantum finite automata: decidability of the equivalence and minimization of states

Multi-letter quantum finite automata (QFAs) can be thought of quantum variants of the one-way multi-head finite automata (Hromkovi?, Acta Informatica 19:377?C384, 1983). It has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages, for example (a?+?b)*b, that are not acceptable by QFAs of Moore and Crutchfield (Theor Comput Sci 237:275?C306, 2000) as well as Kondacs and Watrous (66?C75, 1997; Observe that 1-letter QFAs are exactly measure-once QFAs (MO-1QFAs) of Moore and Crutchfield (Theor Comput Sci 237:275?C306, 2000)). In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Three new results presented in this paper are the following ones: (1) Given a k ₁-letter QFA ${{\mathcal A}_1}$ and a k ₂-letter QFA ${{\mathcal A}_2}$ over the same input alphabet ??, they are equivalent if and only if they are (n ² m ^k-1?m ^k-1?+?k)-equivalent, where m =?|??| is the cardinality of ??, k =?max(k ₁,k ₂), and n =?n ₁?+?n ₂, with n ₁ and n ₂ being numbers of states of ${{\mathcal A}_{1}}$ and ${{\mathcal A}_{2}}$ , respectively. When k =?1, this result implies the decidability of equivalence of measure-once QFAs (Moore and Crutchfield in Theor Comput Sci 237:275?C306, 2000). (It is worth mentioning that our technical method is essentially different from the previous ones used in the literature.) (2) A polynomial-time O(m ^2k-1 n ⁸?+?km ^k n ⁶) algorithm is designed to determine the equivalence of any two multi-letter QFAs (see Theorems 2 and 3; Observe that if a brute force algorithm to determine equivalence would be used, as suggested by the decidability outcome of the point (1), the worst case time complexity would be exponential). Observe also that time complexity is expressed here in terms of the number of states of the multi-letter QFAs and k can be seen as a constant. (3) It is shown that the states minimization problem of multi-letter QFAs is solvable in EXPSPACE. This implies also that the state minimization problem of MO-1QFAs (see Moore and Crutchfield in Theor Comput Sci 237:275?C306, 2000, page 304, Problem 5), an open problem stated in that paper, is also solvable in EXPSPACE.     

Comments and further improvements on “pth moment stability of stochastic neural networks with Markov volatilities”

In a very recent paper, Peng and Liu (Neural Comput Appl 20:543–547, 2011) investigated the pth moment stability of the stochastic Grossberg–Hopfield neural networks with Markov volatilities by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). We should point out that Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) investigated the pth moment exponentially stable for a class of stochastic dynamical systems with constant delay; however, this theorem cannot apply to the case of variable time delays. It is also worthy to emphasize that Peng and Liu (Neural Comput Appl 20:543–547, 2011) discussed by Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1) the pth moment exponentially stable for the Grossberg–Hopfield neural networks with variable delays, and therefore, there are some gaps between Peng and Liu (Neural Comput Appl 20:543–547, 2011, Theorem 1) and Mao et al. (Bernoulli 6:73–90, 2000, Theorem 4.1). In this paper, we fill up this gap. Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.     

A novel sparse representation method based on virtual samples for face recognition

Though sparse representation (Wagner et al. in IEEE Trans Pattern Anal Mach Intell 34(2):372–386, 2012, CVPR 597–604, 2009) can perform very well in face recognition (FR), it still can be improved. To improve the performance of FR, a novel sparse representation method based on virtual samples is proposed in this paper. The proposed method first extends the training samples to form a new training set by adding random noise to them and then performs FR. As the testing samples can be represented better with the new training set, the ultimate classification obtained using the proposed method is more accurate than the classification based on the original training samples. A number of FR experiments show that the classification accuracy obtained using our method is usually 2–5 % greater than that obtained using the method mentioned in Xu and Zhu (Neural Comput Appl, 2012).