Information Lower Bounds via Self-Reducibility

We use self-reduction methods to prove strong information lower bounds on two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product (IP). In our first result we affirm the conjecture that the information cost of GHD is linear even under the uniform distribution, which strengthens the Ω(n) bound recently shown by Kerenidis et al. (2012), and answers an open problem from Chakrabarti et al. (2012). In our second result we prove that the information cost of IP_n is arbitrarily close to the trivial upper bound n as the permitted error tends to zero, again strengthening the Ω(n) lower bound recently proved by Braverman and Weinstein (Electronic Colloquium on Computational Complexity (ECCC) 18, 164 2011). Our proofs demonstrate that self-reducibility makes the connection between information complexity and communication complexity lower bounds a two-way connection. Whereas numerous results in the past (Chakrabarti et al. 2001; Bar-Yossef et al. J. Comput. Syst. Sci. 68(4), 702–732 2004; Barak et al. 2010) used information complexity techniques to derive new communication complexity lower bounds, we explore a generic way in which communication complexity lower bounds imply information complexity lower bounds in a black-box manner.     

On Optimally Partitioning a Text to Improve Its Compression

In this paper we investigate the problem of partitioning an input string T in such a way that compressing individually its parts via a base-compressor C gets a compressed output that is shorter than applying C over the entire T at once. This problem was introduced in Buchsbaum et al. (Proc. of 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 175–184, 2000; J. ACM 50(6):825–851, 2003) in the context of table compression, and then further elaborated and extended to strings and trees by Ferragina et al. (J. ACM 52:688–713, 2005; Proc. of 46th IEEE Symposium on Foundations of Computer Science, pp. 184–193, 2005) and Mäkinen and Navarro (Proc. of 14th Symposium on String Processing and Information Retrieval, pp. 229–241, 2007). Unfortunately, the literature offers poor solutions: namely, we know either a cubic-time algorithm for computing the optimal partition based on dynamic programming (Buchsbaum et al. in J. ACM 50(6):825–851, 2003; Giancarlo and Sciortino in Proc. of 14th Symposium on Combinatorial Pattern Matching, pp. 129–143, 2003), or few heuristics that do not guarantee any bounds on the efficacy of their computed partition (Buchsbaum et al. in Proc. of 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 175–184, 2000; J. ACM 50(6):825–851, 2003), or algorithms that are efficient but work in some specific scenarios (such as the Burrows-Wheeler Transform, see e.g. Ferragina et al. in J. ACM 52:688–713, 2005; Mäkinen and Navarro in Proc. of 14th Symposium on String Processing and Information Retrieval, pp. 229–241, 2007) and achieve compression performance that might be worse than the optimal-partitioning by a Ω(log?n/log?log?n) factor. Therefore, computing efficiently the optimal solution is still open (Buchsbaum and Giancarlo in Encyclopedia of Algorithms, pp. 939–942, 2008). In this paper we provide the first algorithm which computes in O(nlog?_1+ε n) time and O(n) space, a partition of T whose compressed output is guaranteed to be no more than (1+ε)-worse the optimal one, where ε may be any positive constant fixed in advance. This result holds for any base-compressor C whose compression performance can be bounded in terms of the zero-th or the k-th order empirical entropy of the text T. We will also discuss extensions of our results to BWT-based compressors and to the compression booster of Ferragina et al. (J. ACM 52:688–713, 2005).     

Quantum and Classical Query Complexities of Local Search Are Polynomially Related

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).     

On Some Optimization Problems for the Rényi Divergence

We consider the problem of determining the maximum and minimum of the Rényi divergence D_λ(P||Q) and D_λ(Q||P) for two probability distribution P and Q of discrete random variables X and Y provided that the probability distribution P and the parameter α of α-coupling between X and Y are fixed, i.e., provided that Pr{X = Y } = α.     

Evaluation Method,Dataset Size or Dataset Content: How to Evaluate Algorithms for Image Matching?

Flutter shutter (coded exposure) cameras allow to extend indefinitely the exposure time for uniform motion blurs. Recently, Tendero et al. (SIAM J Imaging Sci 6(2):813–847, 2013) proved that for a fixed known velocity v the gain of a flutter shutter in terms of root means square error (RMSE) cannot exceeds a 1.1717 factor compared to an optimal snapshot. The aforementioned bound is optimal in the sense that this 1.1717 factor can be attained. However, this disheartening bound is in direct contradiction with the recent results by Cossairt et al. (IEEE Trans Image Process 22(2), 447–458, 2013). Our first goal in this paper is to resolve mathematically this discrepancy. An interesting question was raised by the authors of reference (IEEE Trans Image Process 22(2), 447–458, 2013). They state that the “gain for computational imaging is significant only when the average signal level J is considerably smaller than the read noise variance \(\sigma _r^2\)” (Cossairt et al., IEEE Trans Image Process 22(2), 447–458, 2013, p. 5). In other words, according to Cossairt et al. (IEEE Trans Image Process 22(2), 447–458, 2013) a flutter shutter would be more efficient when used in low light conditions e.g., indoor scenes or at night. Our second goal is to prove that this statement is based on an incomplete camera model and that a complete mathematical model disproves it. To do so we propose a general flutter shutter camera model that includes photonic, thermal (The amplifier noise may also be mentioned as another source of background noise, which can be included w.l.o.g. in the thermal noise) and additive [The additive (sensor readout) noise may contain other components such as reset noise and quantization noise. We include them w.l.o.g. in the readout.] (sensor readout, quantification) noises of finite variances. Our analysis provides exact formulae for the mean square error of the final deconvolved image. It also allows us to confirm that the gain in terms of RMSE of any flutter shutter camera is bounded from above by 1.1776 when compared to an optimal snapshot. The bound is uniform with respect to the observation conditions and applies for any fixed known velocity. Incidentally, the proposed formalism and its consequences also apply to e.g., the Levin et al. motion-invariant photography (ACM Trans Graphics (TOG), 27(3):71:1–71:9, 2008; Method and apparatus for motion invariant imag- ing, October 1 2009. US Patent 20,090,244,300, 2009) and variant (Cho et al. Motion blur removal with orthogonal parabolic exposures, 2010). In short, we bring mathematical proofs to the effect of contradicting the claims of Cossairt et al. (IEEE Trans Image Process 22(2), 447–458, 2013). Lastly, this paper permits to point out the kind of optimization needed if one wants to turn the flutter shutter into a useful imaging tool.     

A polynomial-time algorithm for a flow-shop batching problem with equal-length operations

A flow-shop batching problem with consistent batches is considered in which the processing times of all jobs on each machine are equal to p and all batch set-up times are equal to s. In such a problem, one has to partition the set of jobs into batches and to schedule the batches on each machine. The processing time of a batch B _i is the sum of processing times of operations in B _i and the earliest start of B _i on a machine is the finishing time of B _i on the previous machine plus the set-up time s. Cheng et al. (Naval Research Logistics 47:128–144, 2000) provided an O(n) pseudopolynomial-time algorithm for solving the special case of the problem with two machines. Mosheiov and Oron (European Journal of Operational Research 161:285–291, 2005) developed an algorithm of the same time complexity for the general case with more than two machines. Ng and Kovalyov (Journal of Scheduling 10:353–364, 2007) improved the pseudopolynomial complexity to \(O(\sqrt{n})\). In this paper, we provide a polynomial-time algorithm of time complexity O(log?³ n).     

Ambiguity and Communication

The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for inputs of size n. For every natural number k we construct a family \((L_{r}^{k}\;|\;r\in \mathbb{N})\) of languages which can be recognized by NFA’s with size k?poly(r) and ambiguity O(n ^k ), but \(L_{r}^{k}\) has only NFA’s with size exponential in r, if ambiguity o(n ^k ) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, SIAM J. Comput. 19:1263–1282, 1989, Leung, SIAM J. Comput. 27:1073–1082, 1998).     

Upper and lower bounds for deterministic broadcast in powerline communication networks

Powerline communication networks assume an interesting position in the communication network space: similarly to wireless networks, powerline networks are based on a shared broadcast medium; unlike wireless networks, however, the signal propagation is constrained to the power lines of the electrical infrastructure, which is essentially a graph. This article presents an algorithmic model to study the design of communication services over powerline communication networks. As a case study, we focus on the fundamental broadcast problem, and present and analyze a distributed algorithm \(\textsc {Color}\textsc {Cast}\) which terminates in at most n communication rounds, where n denotes the network size, even in a model where link qualities are unpredictable and time-varying. For comparison, the achieved broadcast time is lower than what can be achieved by any unknown-topology algorithm (lower bounds \(\varOmega (n\log n / \log (n/D))\) and \(\varOmega (n \log D)\) are proved in Kowalski and Pelc (Distrib Comput 18(1):43–57, 2005) resp. Clementi et al. (2001) where D is the network diameter). Moreover, existing known-topology broadcast algorithms often fail to deliver the broadcast message entirely in this model. This article also presents a general broadcast lower bound for the powerline model.     

<Emphasis Type="Italic">δ</Emphasis>-equality of intuitionistic fuzzy sets: a new proximity measure and applications in medical diagnosis

Intuitionistic fuzzy set is capable of handling uncertainty with counterpart falsities which exist in nature. Proximity measure is a convenient way to demonstrate impractical significance of values of memberships in the intuitionistic fuzzy set. However, the related works of Pappis (Fuzzy Sets Syst 39(1):111–115, 1991), Hong and Hwang (Fuzzy Sets Syst 66(3):383–386, 1994), Virant (2000) and Cai (IEEE Trans Fuzzy Syst 9(5):738–750, 2001) did not model the measure in the context of the intuitionistic fuzzy set but in the Zadeh’s fuzzy set instead. In this paper, we examine this problem and propose new notions of δ-equalities for the intuitionistic fuzzy set and δ-equalities for intuitionistic fuzzy relations. Two fuzzy sets are said to be δ-equal if they are equal to an extent of δ. The applications of δ-equalities are important to fuzzy statistics and fuzzy reasoning. Several characteristics of δ-equalities that were not discussed in the previous works are also investigated. We apply the δ-equalities to the application of medical diagnosis to investigate a patient’s diseases from symptoms. The idea is using δ-equalities for intuitionistic fuzzy relations to find groups of intuitionistic fuzzified set with certain equality or similar degrees then combining them. Numerical examples are given to illustrate validity of the proposed algorithm. Further, we conduct experiments on real medical datasets to check the efficiency and applicability on real-world problems. The results obtained are also better in comparison with 10 existing diagnosis methods namely De et al. (Fuzzy Sets Syst 117:209–213, 2001), Samuel and Balamurugan (Appl Math Sci 6(35):1741–1746, 2012), Szmidt and Kacprzyk (2004), Zhang et al. (Procedia Eng 29:4336–4342, 2012), Hung and Yang (Pattern Recogn Lett 25:1603–1611, 2004), Wang and Xin (Pattern Recogn Lett 26:2063–2069, 2005), Vlachos and Sergiadis (Pattern Recogn Lett 28(2):197–206, 2007), Zhang and Jiang (Inf Sci 178(6):4184–4191, 2008), Maheshwari and Srivastava (J Appl Anal Comput 6(3):772–789, 2016) and Support Vector Machine (SVM).     

On variable-weighted exact satisfiability problems

We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(||C||2^0.2441n ) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n ²·||C||?+?2^0.40567n ) time. In recent years only the unweighted counterparts of these problems have been studied (Dahllöf and Jonsson, An algorithm for counting maximum weighted independent sets and its applications. In: Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms, pp. 292–298, 2002; Dahllöf et al., Theor Comp Sci 320: 373–394, 2004; Porschen, On some weighted satisfiability and graph problems. In: Proceedings of the 31st Conference on Current Trends in Theory and Practice of Informatics (SOFSEM 2005). Lecture Notes in Comp. Science, vol. 3381, pp. 278–287. Springer, 2005).     

Normality and Finite-State Dimension of Liouville Numbers

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano (1993) and Bugeaud (2002) have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k ≥ 2, we give a new construction of a Liouville number which is normal to the base k. This construction is combinatorial, and is based on de Bruijn sequences.     

On the effect of the deployment setting on broadcasting in Euclidean radio networks

The paper studies broadcasting in radio networks whose stations are represented by points in the Euclidean plane (each station knows its own coordinates). In any given time step, a station can either receive or transmit. A message transmitted from station v is delivered to every station u at distance at most 1 from v, but u successfully hears the message if and only if v is the only station at distance at most 1 from u that transmitted in this time step. A designated source station has a message that should be disseminated throughout the network. All stations other than the source are initially idle and wake up upon the first time they hear the source message. It is shown in [17] that the time complexity of deterministic broadcasting algorithms depends on two parameters of the network, namely, its diameter (in hops) D and a lower bound d on the Euclidean distance between any two stations. The inverse of d is called the granularity of the network, denoted by g. Specifically, the authors of [17] present a deterministic broadcasting algorithm that works in time O (Dg) and prove that every broadcasting algorithm requires \(\varOmega \left( D \sqrt{g} \right) \) time. In this paper, we distinguish between the arbitrary deployment setting, originally studied in [17], in which stations can be placed everywhere in the plane, and the new grid deployment setting, in which stations are only allowed to be placed on a d-spaced grid. Does the latter (more restricted) setting provide any speedup in broadcasting time complexity? Although the O (Dg) broadcasting algorithm of [17] works under the (original) arbitrary deployment setting, it turns out that the \(\varOmega \left( D \sqrt{g} \right) \) lower bound remains valid under the grid deployment setting. Still, the above question is left unanswered. The current paper answers this question affirmatively by presenting a provable separation between the two deployment settings. We establish a tight lower bound on the time complexity of deterministic broadcasting algorithms under the arbitrary deployment setting proving that broadcasting cannot be completed in less than \(\varOmega (D g)\) time. For the grid deployment setting, we develop a deterministic broadcasting algorithm that runs in time \(O \left( D g^{5 / 6} \log g \right) \), thus breaking the linear dependency on g.     

From MDD to BDD and Arc consistency

In this paper, we present a new conversion of multivalued decision diagrams (MDD) to binary decision diagrams (BDD) which can be used to improve MDD-based fil- tering algorithms such as MDDC or MDD-4R. We also propose BDDF, an algorithm that copies modified parts of the BDD “on the fly” during the search of a solution, and yields a better incrementality than a pure MDDC-like approach. MDDC is not very efficient when used to represent poorly structured positive table constraints. Our new representation combined with BDDF retains the properties of the MDD representation and has comparable performances to the STR2 algorithm by Ullmann (2007) and Lecoutre (Constraints, 16.4, 341–371 2011).     

A commentary on “Hybrid search for minimal perturbation in Dynamic CSPs”

The Hybrid Search for Minimal Perturbation Problems algorithm in Dynamic CSP (HS_MPP) (Zivan, Constraints, 16(3), 228–249, 2011) ensures for a given dynamic problem and its solution to the previous CSP, to find the optimal solution to the newly generated CSP. This proposed method exploits the fact that its reported solution must satisfy two requirements. First, that it is a solution for T complete assignment for the derived CSP and second, that it is as close as possible to the solution of the former CSP. Unfortunately, the pseudo-code of the algorithm in Zivan (Constraints, 16(3), 228–249, 2011) is confusing and may lead to an implementation in which HS_MPP may not perform the expected outcomes of a given instance of Dynamic CSPs correctly. In this erratum, we demonstrate the possible undesired outcomes and give corrections in HS_MPP’s pseudo-code.     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

Some properties of the disjunctive languages contained in <Emphasis Type="Italic">Q</Emphasis>

The set of all primitive words Q over an alphabet X was first defined and studied by Shyr and Thierrin (Proceedings of the 1977 Inter. FCT-Conference, Poznan, Poland, Lecture Notes in Computer Science 56. pp. 171–176 (1977)). It showed that for the case |X| ≥ 2, the set along with \({Q^{(i)} = \{f^i\,|\,f \in Q\}, i\geq 2}\) are all disjunctive. Since then these disjunctive sets are often be quoted. Following Shyr and Thierrin showed that the half sets \({Q_{ev} = \{f \in Q\,|\,|f| = {\rm even}\}}\) and Q _od = Q \ Q _ev of Q are disjunctive, Chien proved that each of the set \({Q_{p,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,p) \},\,0\leq r < p}\) is disjunctive, where p is a prime number. In this paper, we generalize this property to that all the languages \({Q_{n,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,n) \},\, 0\leq r < n}\) are disjunctive languages, where n is any positive integer. We proved that for any n ≥ 1, k ≥ 2, (Q _n,0) ^k are all regular languages. Some algebraic properties related to the family of languages {Q _n,r | n ≥ 2, 0 ≤ r < n } are investigated.     

Managing variants of a personalized product

A product variant table is a table that lists legal combinations of product features. Variant tables can be used to constrain the variability offered for a personalized product. The concept of such a table is easy to understand. Hence, variant tables are natural to use when ensuring the completeness and correctness of a quote/order for a customizable product. They are also used to filter out inadmissible choices for features in an interactive specification (configuration) process. Variant tables can be maintained as relational (database) tables, using spreadsheets, or in proprietary ways offered by the product modeling environment. Variant tables can become quite large. A way of compressing them is then sought that supports a space-efficient representation and a time-efficient evaluation. The motivation of this work is to develop a simple approach to compress/compile a variant table into an easy to read, but possibly hard to write form that can be deployed in a business setting at acceptable cost and risk in a similar manner as a database. The main result is a simple compression and evaluation scheme for an individual variant table called a Variant Decomposition Diagram (VDD). A VDD supports efficient consistency checks, the filtering of inadmissible features, and iteration over the table. A simple static heuristic for decomposition order is proposed that suggests itself from a “column oriented viewpoint”. This heuristic is not always optimal, but it has the advantage of allowing fast compilation of a variant table into a VDD. Compression results for a publicly available model of a Renault Megane are given. With the proposed heuristic the VDD is a specialization of a Zero-suppressed (binary) Decision Diagram (ZDD) (Knuth 2011) and also maps to a Multi-valued Decision Diagram (MDD) (Andersen et al. 2007; Berndt et al. 2012).     

Cryptanalysis of an MOR cryptosystem based on a finite associative algebra基于有限结合代数的 MOR 公钥密码安全性分析

The Shor algorithm is effective for public-key cryptosystems based on an abelian group. At CRYPTO 2001, Paeng (2001) presented a MOR cryptosystem using a non-abelian group, which can be considered as a candidate scheme for post-quantum attack. This paper analyses the security of a MOR cryptosystem based on a finite associative algebra using a quantum algorithm. Specifically, let L be a finite associative algebra over a finite field F. Consider a homomorphism φ: Aut(L) → Aut(H)×Aut(I), where I is an ideal of L and H ? L/I. We compute dim Im(φ) and dim Ker(φ), and combine them by dim Aut(L) = dim Im(φ)+dim Ker(φ). We prove that Im(φ) = Stab_Comp(H,I)(μ + B²(H, I)) and Ker(φ) ? Z¹(H, I). Thus, we can obtain dim Im(φ), since the algorithm for the stabilizer is a standard algorithm among abelian hidden subgroup algorithms. In addition, Z¹(H, I) is equivalent to the solution space of the linear equation group over the Galois fields GF(p), and it is possible to obtain dim Ker(φ) by the enumeration theorem. Furthermore, we can obtain the dimension of the automorphism group Aut(L). When the map ? ∈ Aut(L), it is possible to effectively compute the cyclic group 〈?〉 and recover the private key a. Therefore, the MOR scheme is insecure when based on a finite associative algebra in quantum computation.     

A rough <Emphasis Type="Italic">ν</Emphasis>-twin support vector regression machine

After combining the ν-Twin Support Vector Regression (ν-TWSVR) with the rough set theory, we propose an efficient Rough ν-Twin Support Vector Regression, called Rough ν-TWSVR for short. We construct a pair of optimization problems which are motivated by and mathematically derived from a related ν-TWSVR Rastogi et al. (Appl Intell 46(3):670–683 2017) and Rough ν-SVR Zhao et al. (Expert Syst Appl 36(6):9793–9798 2009). Rough ν-TWSVR not only utilizes more data information rather than the extreme data points in the ν-TWSVR, but also makes different points having different effects on the regressor depending on their positions. This method can implement the structural risk minimization and automatically control accuracies according to the structure of the data sets. In addition, the double ε s are utilized to construct the rough tube for upper(lower)-bound Rough ν-TWSVR instead of a single ε in the upper(lower)-bound ν-TWSVR. Moreover, This rough tube consisting of positive region, boundary region, and negative region yields the feasible set of the Rough ν-TWSVR larger than that of the ν-TWSVR, which makes the objective function of the Rough ν-TWSVR no more than that of ν-TWSVR. The Rough ν-TWSVR improves the generalization performance of the ν-TWSVR, especially for the data sets with outliers. Experimental results on toy examples and benchmark data sets confirm the validation and applicability of our proposed Rough ν-TWSVR.     

The Number of Tree Stars Is <Emphasis Type="Italic">O</Emphasis><Superscript>*</Superscript>(1.357<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>)

Every rectilinear Steiner tree problem admits an optimal tree T ^* which is composed of tree stars. Moreover, the currently fastest algorithms for the rectilinear Steiner tree problem proceed by composing an optimum tree T ^* from tree star components in the cheapest way. The efficiency of such algorithms depends heavily on the number of tree stars (candidate components). Fößmeier and Kaufmann (Algorithmica 26, 68–99, 2000) showed that any problem instance with k terminals has a number of tree stars in between 1.32 ^k and 1.38 ^k (modulo polynomial factors) in the worst case. We determine the exact bound O ^*(ρ ^k ) where ρ≈1.357 and mention some consequences of this result.