Linguistic majorities with difference in support Majorities based on difference of votes and their extension, majorities based on difference in support, were introduced in social choice voting as tools to implement the crisp preference values (votes) and the intensities of preference provided by voters when comparing pairs of alternatives, respectively, with the aim to declare which alternative is socially preferred. Moreover, these rules require the winner alternative to reach a certain positive difference in its social valuation with respect to the one reached by the loser alternative. This paper introduces a new aggregation rule that extends majorities based on difference of votes from the context of crisp preferences to the framework of linguistic preferences. Under linguistic majorities with difference in support, the voters express their intensities of preferences between pairs of alternatives using linguistic labels and an alternative defeats another one if the first one reaches a specific support fixed before the election process. There exist two main representation methodologies of linguistic preferences: the cardinal one based on the use of fuzzy set, and the ordinal one based on the use of the 2-tuples. Linguistic majorities with difference in support are formalised in both representation settings, and conditions are given to guarantee that fuzzy linguistic majorities and 2-tuple linguistic majorities are mathematically isomorphic. Moreover, linguistic majorities with difference in support constitute a class of majority rules because all the possible majority rules can be generalised to the linguistic framework by adjusting the required threshold of support. Finally, linguistic majorities based on difference in support are proved to verify relevant normative properties: anonymity, neutrality, monotonicity, weak Pareto and cancellativeness. Decision making problems deal with the social choice of the best alternative among all the possible alternatives taking into account the views and opinions, i.e. the preferences, of all the individuals of a particular social group [8, 28, 32] . Two approaches are possible to address these problems [15, 17] : a direct approach that derives a social choice from the sole manipulation and processing of the information provided by all the individuals without the intermediate derivation of any kind of collective information using a fusion or aggregation operator, which is characteristic of the indirect approach. Obviously, the type of aggregation rule implemented in the second approach is crucial in designing the corresponding social choice rule, and ultimately in the final social solution to the decision making problem. This paper deals with this specific issue, and it is devoted to the introduction of a new aggregation rule for individual preferences.. A comparison study between different alternative preference elicitation methods is reported in [26] , where it was concluded that pairwise comparison methods are more accurate than non-pairwise methods. The main advantage of pairwise comparison methods is that facilitates individuals expressing their preferences because they focus exclusively on two alternatives at a time. Given two alternatives, an individual either prefers one to the other or is indifferent between them, which can be represented using a preference relation whose elements represent the preference of one alternative over another one. There exist two main mathematical models to represent pairwise comparison of alternatives based on the concept of preference relation [8, 29] : in the first one, a preference relation is defined for each one of the above three possible preference states, which is usually referred to as a preference structure on the set of alternatives; the second one integrates the three possible preference states into a single preference relation. This paper deals with the second type of relations, for which reciprocity of preferences is usually assumed in order to guarantee the following basic rationality properties in making paired comparisons [31] : indifference between any alternative and itself, and asymmetry of preferences, i.e. if an individual prefers alternative x to y, that individual does not simultaneously prefer y to x.. In classical voting systems the set of numerical values {1, 0.5, 0}, or its equivalent {1, 0, −1} [8] , is used to represent when the first alternative is preferred to the second alternative, when both alternatives are considered equally preferred (indifference), and when the second alternative is preferred to the first one, respectively. This classical preference modelling constitutes the simplest numeric discrimination model of preferences, and it proves insufficient in many decision making situations as the following example illustrates: Let {x, y, z} be 3 alternatives of which we know that one individual prefers x to y and y to z, and another individual prefers z to y and y to x; then using the above numerical values it may be difficult or impossible to decide which alternative is the best. As Fishburn points out in [8] , if alternative y is closer to the best alternative than to the worst one for both individuals then it might seem appropriate to 'elect' it as the social choice, whilst if it is closer to the worst than to the best, then it might be excluded from the choice set. Thus, in many cases it might be necessary the implementation of some kind of 'intensity of preference' between alternatives.. The concept of fuzzy set, which extends the classical concept of set, when applied to a classical relation leads to the concept of a fuzzy relation, which in turn allows the implementation of intensity of preferences [36] . In [2] , we can find for the first time the fuzzy interpretation of intensity of preferences via the concept of a reciprocal fuzzy preference relation, which was later reinterpreted by Nurmi in [27] . In this approach, the numeric scale to evaluate intensity of preferences is the whole unit interval [0, 1] instead of {1, 0.5, 0}, which it is argued though to assume unlimited computational abilities and resources from the individuals [4] .. Subjectivity, imprecision and vagueness in the articulation of opinions pervade real world decision applications, and individuals usually find difficult to evaluate their preference using exact numbers [36] . Individuals might feel more comfortable using words by means of linguistic labels or terms to articulate their preferences [37] . Furthermore, hu-mans exhibit a remarkable capability to manipulate perceptions and other characteristics of physical and mental objects, without any exact numerical measurements and complex computations [3, 9, 16, 24, 38] . Therefore, in this paper, the individuals' preferences between pair of alternatives will be assumed to be given in the form of linguistic labels.. It was mentioned before that the type of aggregation rule implemented is crucial in designing the corresponding social choice rule. This paper focuses on the majority voting rules, which are very easy to understand by voters and therefore, when comparing two alternatives, they are seen as very attractive and appropriate to aggregate individual preferences into a collective one. Simple majority rule [25] stands out among the different majority rules. Under this rule, an alternative defeats another one when the number of votes cast for the first one exceeds the number of votes cast for the second one. Simple majority rule states as the most decisive aggregation rule. In fact, the requirement to declare indifference between two alternatives is quite strong given that both alternatives have to receive exactly the same number of favourable votes. Furthermore, under the simple majority rule, the support required for an alternative to be the winner is minimum because it is only required to exceed the defeated alternative in just one vote. This characteristic turns out to become a drawback because the collective decision is very unstable, i.e. it could be reverted with the change of just one vote. In an attempt to overcome this shortcoming, tougher requirements for declaring an alternative as the winner have been defined and studied. Among these rules, it is worth mentioning the following: unanimous majority, absolute majority and qualified majority [7, 8, 30] .. Majority based on difference of votes (M k ) [14, 19, 22] constitute a general approach to majority voting rule that generalises all the above majority rules. This rule allows to calibrate the amount of support required for the winner alternative by means of a difference of votes fixed before the election process. At the extreme cases, i.e. no difference and maximum difference of votes, the majority based on difference of votes becomes the simple majority and unanimous majority, respectively. With this rule, indifference between two alternatives is possible to be declared for more cases than under the simple majority rule. In fact, the indifference state could be enlarged as much as desired. The application of the majority based on difference of votes to the case of [0,1]-valued reciprocal fuzzy preference relations is known as the majority based on difference in support ( M k ) [20] .. The aim of this paper is to fill the gap between the majority based on difference of votes and the majority based on difference in support by providing a new majority rule based on difference of support in the linguistic framework. Linguistic majority with difference in support keeps the essence of the former rules in the sense that for an alternative to be declared winner a specific support fixed before the election is to be achieved. The challenge here is to generalise formally the rule to the case of being the preferences linguistic rather than numeric in nature. An additional challenge here is to relate the linguistic majority with difference in support rules that can be obtained when the main two approaches to model and represent linguistic information are applied. On the one hand, linguistic preferences can be modelled using a cardinal approach by means of fuzzy sets and their associated membership functions [36] . On the other hand, an ordinal approach can be used to model and manage linguistic preferences using the 2-tuple symbolic representation [12] . Therefore, two new and different linguistic majority with difference in support rules will be introduced: the linguistic fuzzy majority (LM K ) and the 2-tuple linguistic majority (2T M k ). Figure 1 illustrates the new linguistic majorities in relation with the corresponding ones developed for numerical preferences. The remainder of the paper is structured as follows: The next section introduces concepts essential to the understanding of the rest of the paper. Following that, Section 3 introduces the concept of linguistic majority with difference in support and its mathematical formulation for the main two approaches to model and represent linguistic information: fuzzy set representation (Subsection 3.1) and the 2-tuple symbolic representation (Subsection 3.2). Section 4 proves that both linguistic majority rules are mathematically isomorphic when fuzzy sets are defuzzified into their centroid. In Section 5, the linguistic majority based on difference in support is proved to verify the following relevant normative properties: anonymity, neutrality, monotonicity, weak Pareto and cancellativeness. Lastly, Section 6 concludes the paper. A new aggregation rule that extends the majority based on difference of votes from the context of crisp preferences to the framework of linguistic preferences has been investigated. Linguistic majorities with difference in support have been formalized for the two main representation methodologies of linguistic preferences: the cardinal, based on the use of fuzzy set; and the ordinal, based on the use of the 2-tuples. It has been proven that both representations are mathematically isomorphic when fuzzy numbers are ranked using their respective centroids, and therefore it can be concluded that the cardinal approach constitutes a more general framework to model linguistic majorities with difference in support. Finally, a set of normative properties have been demonstrated to hold for the new linguistic majorities.. Some interesting extensions are left opened. Among them, the study of the collective consistency of the linguistic majority with difference in support when more than two alternatives are compared [23] , and the development of a consistency based selection process seems to be worth further investigation. Also, it seems interesting to explore softer approaches to the linguistic majority with difference in support when the information is represented using fuzzy sets. The use of type-2 fuzzy sets also seems to be a challenging one that deserves future research effort.

B I O B I I O O O O O O O O O O O O B I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B B I O B I I O O O O O O O O O O O O O O B I O O B I O O O O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O B I O B I I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O B I O O O O B I O O O O O O O O O O O O


Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability We continue the study of generalized tractability initiated in our previous paper "Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information", J. Complexity, 23, 262-295 (2007). We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S d given as the d-fold tensor product of a compact linear operator S 1 for d = 1, 2, . . . , with S 1 = 1 and S 1 has at least two positive singular values.Let n(ε, S d ) be the minimal number of information evaluations needed to approximate S d to within ε ∈ [0, 1]. We study generalized tractability by verifying when n(ε, S d ) can be bounded by a multiple of a power ofHere, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T (ε −1 , d) whose multiple bounds n(ε, S d ). We also study weak tractability, i.e., when lim ε −1 +d→∞,(In our previous paper, we studied generalized tractability for proper subsets Ω of [1, ∞)×N, whereas in this paper we take the unrestricted domain Ω unr = [1, ∞)×N.We consider the three cases for which we have only finitely many positive singular values of S 1 , or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S 1 decay slightly faster that logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S 1 and mostly limiting properties of T . The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T (x, y) = x 1+ln y . We also study tractability functions T of product form, T (x, y) = f 1 (x)f 2 (x). Assume that a i = lim inf x→∞ (ln ln f i (x))/(ln ln x) is finite for i = 1, 2. Then generalized tractability takes place iff a i > 1 and (a 1 − 1)(a 2 − 1) ≥ 1, and if (a 1 − 1)(a 2 − 1) = 1 then we need to assume one more condition given in the paper. If (a 1 − 1)(a 2 − 1) > 1 then the exponent of tractability is zero, and if (a 1 − 1)(a 2 − 1) = 1 then the exponent of tractability is finite. It is interesting to add that for T being of the product form, the tractability conditions as well as the exponent of tractability depend only on the second singular eigenvalue of S 1 and they do not depend on the rate of their decay. Finally, we compare the results obtained in this paper for the unrestricted domain Ω unr with the results from our previous paper obtained for the restricted do-0 )×N with d * ≥ 1 and ε 0 ∈ (0, 1). In general, the tractability results are quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, for instance, for polynomial tractability T (x, y) = xy. We may also have generalized tractability for both domains with different or with the same exponents of tractability. Tractability of multivariate problems has been extensively studied in information-based complexity and the recent account of the tractability research can be found in the forthcoming book [3] . Tractability is the study of approximating operators S d defined on spaces of functions with k d variables with k d proportional to d. Problems with huge d occur in many applications, see [5] . We approximate S d by computing linear information which is given by finitely many, say n, continuous linear functionals, and the error of an algorithm is defined in the worst case setting. Before tractability study, the errors of algorithms were studied as functions of n and the main point was to find the best possible rate of convergence as n tends to infinity. For large d, the errors of algorithms crucially depend also on d, and for some problems this dependence is exponential in d.. Let n(ε, S d ) denote the information complexity of S d which is the minimal number of continuous linear functionals needed to approximate S d to within ε. The main point of tractability is to check whether n(ε, S d ) does not depend exponentially on ε −1 and d. Since there are different ways to measure the lack of exponential behavior, we have different types of tractability. The first type of tractability is polynomial tractability which has been extensively studied in many papers. In this case we want to verify whether n(ε, S d ) can be bounded by a multiple of powers of ε −1 and d for all (ε −1 , d) ∈ [1, ∞)×N. There are many positive and negative results for polynomial tractability. Usually, positive results are for problems for which the successive variables or groups of variables of large cardinality play a diminishing role, and negative results are for problems for which all variables and groups of variables play the same role. The primary example leading to negative results is approximation of linear tensor product problems. In this case, S d is a d-fold tensor product of a compact linear operator S 1 , where S 1 is defined between Hilbert spaces, S 1 = 1 and S 1 has at least two positive singular values. Let { λ j } denote the sequence of the ordered singular values of S 1 , 0 < λ 2 ≤ λ 1 = 1. It is well known, see [4] , that the information complexity of S d is. Clearly, if λ 2 = 1 then n(ε, d) ≥ 2 d for all ε < 1, and we have exponential dependence on d causing intractability of the problem. That is why we need to assume that λ 2 < 1. Still, as long as λ 2 is positive, n(ε, d) goes faster to infinity that any power of d, see [6] , and that is why polynomial tractability does not hold for linear tensor product problems.. In [1] , we propose to study generalized tractability by verifying whether n(ε, S d ) can be bounded by a multiple of a power of T (ε, d) for all (ε. Here T is a tractability function which means that T : [1, ∞) 2 → [1, ∞) is non-decreasing in both variables and grows slower than exponentially to infinity, i.e., lim x+y→∞ ln T (x, y). x + y = 0.. The set Ω is called tractability domain, and can be a proper subset of [1, ∞) × N but at least one of the parameters ε −1 or d is allowed to go to infinity. The exponent of tractability is defined as the smallest (or more precisely as the infimum) power of T (ε −1 , d) whose multiple bounds n(ε, S d ). There is also the notion of weak tractability when. see [2, 3] , and it is a necessary condition on the lack of exponential behavior of n(ε, S d ).. Of course, the hope is that by taking reasonable restricted domains Ω or by allowing tractability functions T that tend to infinity faster than polynomially, we may enlarge the class of tractable problems including linear tensor product problems. Indeed, this is the case. In [1] we showed that polynomial tractability of linear tensor product problems holds if we assume that the singular values tend to zero polynomially fast, and we take the restricted tractability domain. with d * ≥ 1 and ε 0 ∈ (0, 1). In this paper, we study the second option and we take the unrestricted domain. but we allow tractability functions T which go to infinity faster than polynomially. We study linear tensor product problems for three cases depending on the behavior of the singular values of S 1 . In the first case we assume that only finitely many of the singular values are positive, in the second case we assume that they decay exponentially fast, and in the third case that they decay polynomially fast.. For each of these three cases, we have weak tractability. In fact, weak tractability holds if the singular values behave as o((ln(j) ln(ln(j))) −1 ) and it is also "almost" a necessary condition.. We provide necessary and sufficient conditions on T such that generalized tractability holds. These conditions are satisfied if T goes sufficiently fast to infinity. We also provide the formulas for the corresponding exponents of tractability. We illustrate these conditions and formulas for specific tractability functions. For example, take T (x, y) = x 1+ln y . Then we have tractability for the three cases of singular values. For finitely many positive singular values and for exponentially decaying singular values, the exponent of tractability is 2/ ln(λ −1 2 ). Hence it only depends on the second singular value and is independent of how many of them are positive. For polynomially decaying singular values, λ j = Θ(j −β ) for β > 0, the exponent of tractability is max{2/β, 2/ ln(λ −1. 2 )}. We also illustrate our results for tractability functions of product form, that is when T (x, y) = f 1 (x)f 2 (y) with finite a i = lim inf x→∞ (ln ln f i (x))/(ln ln x), i = 1, 2. Then generalized tractability holds iff a i > 1 and (a 1 − 1)(a 2 − 1) ≥ 1, and if (a 1 − 1)(a 2 − 1) = 1 then we need to assume additionally condition (12) for k = 2 which depends only on the second singular value. For (a 1 − 1)(a 2 − 1) > 1, the exponent of tractability is zero, whereas for (a 1 − 1)(a 2 − 1) = 1, the exponent of tractability is positive. In fact, in the last case, depending on specific functions f i for which a i are fixed, the exponent of tractability can be arbitrary. Note that a i only depends on the limiting behavior of f i , and is independent on the behavior of the singular values. Hence, for (a 1 − 1)(a 2 − 1) > 1, we have the zero exponent of tractability independently of the behavior of the singular values, whereas for (a 1 − 1)(a 2 − 1) = 1, the exponent of tractability depends only on the second singular value and is independent of the rest of them.. In the final section, we compare the results obtained in this paper for the unrestricted domain Ω unr with the results from our previous paper obtained for the restricted domain Ω res . The tractability results for the unrestricted and restricted domains may be quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, as we already mentioned, for polynomial tractability T (x, y) = xy. We may also have generalized tractability for both domains, however, the exponents of tractability may depend on the domain and can be much larger for the unrestricted domain than for the restricted domain.

O B O O O O O O O O O B I O O B O O O O O O B O O O O O O B O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O B B O O O O O O O O B I O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O


Delay, cost and infrastructure tradeoff of epidemic routing in mobile sensor networks This paper studies the delay, cost and infrastructure tradeoff of epidemic routing in mobile sensor networks. We consider a mobile sensor network with mobiles and static base stations. The mobile sensors collect information when moving around and need to report the information to the base stations. Three different epidemic routing schemestarget epidemic routing, uncontrolled epidemic routing and controlled epidemic routing -are analyzed in this paper. For each of the three schemes, we characterize the scaling behaviors of the delay, which is defined to be the average number of time slots required to deliver a message, and the cost, which is defined to be the average number of transmissions required to deliver a message, in terms of the number of mobiles ( ) and the number of base stations ( ). These scaling results reveal the fundamental tradeoff among delay, cost and infrastructure in mobile sensor networks. Wireless technology has provided an infrastructure-free and fast-deployable method to establish communication, and has inspired many emerging applications. One of these applications is mobile sensor network where mobile sensors collect information when moving around and then report the collected information to base stations. For example, Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. IWCMC'10 June 28-July 2, 2010, Caen, France.. Copyright c ⃝2010 ACM 978-1-4503-0062-9/10/06 ...$5.00.. ZebraNet [7] is a mobile sensor network used to monitor and study animal migrations and inter-species interactions, where each zebra is equipped with an wireless antenna and pairwise communication is used to transmit data when two zebras are close to each other. The CarTel project at MIT [4] is another example, where cars equipped with WiFi, Bluetooth and cellular devices collect traffic information and cooperatively deliver the information to a public web site. While mobile sensor networks have many important applications in practice, designing a high performance mobile sensor network is a very challenging task. In mobile sensor networks, the network topology constantly changes over time due to node mobility, so there is no fixed route from a mobile to the base stations. One approach to deliver data to base stations is to use the epidemic routing, i.e., mobile sensors exchange their messages whenever they meet to eventually deliver the messages to the base stations [6] . Under the epidemic routing, a packet in the network can be viewed as a type of infectious disease. A mobile node is said to be infected if it carries the packet in its buffer. An infected mobile may encounter other susceptible mobiles who do not have the packet and further, pass the packet to them, leading to more and more infected mobiles. Due to the similarities between the dissemination of the messages under the epidemic routing and the spread of an infectious disease, Markovian models and ordinary differential equations (ODEs) [2, 3, 5, 8] are adopted from infectious disease spread model [1] to study the characteristics of the epidemic routing. Although ODE is a useful method to analyze the behavior of epidemic routing, it is an fluid-like approximation. In this paper, we propose a Markov chain model which describes the discrete stochastic process of the epidemic routing and study the behavior of epidemic routing based on the discrete Markov chain. We consider a mobile network with mobile wireless nodes and base stations. We are interested in characterizing three important performance metrics of epidemic routing in mobile sensor networks:. • Delay: The time required to deliver a message from a mobile to the base stations.. • Cost: The number of transmissions required to deliver a message to the base stations. Note that the number of transmissions is directly related to the energy consumed to deliver a message.. • Infrastructure: The number of base stations required for guaranteeing certain delay and cost requirements.. It is easy to see that the most cost-efficient strategy is to let the mobiles directly transmit their messages to the base stations, while the most delay-efficient strategy is to allow uncontrolled epidemic routing (i.e., two mobiles exchange their messages whenever they meet). Clearly, there is a tension between delay and cost. More specially, more duplications (transmissions) reduce the delay but increase the cost. So it is imperative to understand the cost and delay tradeoff of epidemic routing so that the algorithm can provide a desired delay performance with a minimum cost. On the other hand, more base stations can reduce both delay and cost. So it is also very important to quantify the benefit of adding more stations to guide the deployment of infrastructure. Motivated by these observations, we are interested in characterizing the delay, cost and infrastructure tradeoff in mobile sensor networks. We note that this paper is different from previous work on epidemic routing by including the infrastructure in the analysis. In practice, many mobile sensor networks include not only mobile sensors but also static base stations (e.g., the CarTel network) to assist data collection process. The infrastructure therefore plays an critical role in many mobile sensor networks. This motivates us to study the threedimensional tradeoff (delay, cost and infrastructure). In the paper, we study three epidemic routing schemes: target epidemic routing, uncontrolled epidemic routing and controlled epidemic routing (these schemes will be defined in a detail in Section 2). We first model the epidemic routing processes as Markov chains and then characterize the scaling behaviors of the three performance metrics based on the Markov chains. In this paper, we studied a mobile sensor network with mobiles and base stations. We modeled the epidemic routing as a Markov chain and characterized the delay-costinfrastructure tradeoff. Our simulations result confirmed the scaling laws obtained hold under a random walk model.

O O O O O O B I O B I I O O O O O O O O O O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B I O O B I O B I O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O


A characterization of entanglement-assisted quantum low-density parity-check codes Abstract-As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.Index Terms-Entanglement-assisted quantum error correction, low-density parity-check code, stabilizer code, Steiner 2-design, pairwise balanced design. T HIS paper addresses the question of how much one pair of qubits in a maximally entangled state can be exploited to import classical sparse graph codes by the entanglementassisted stabilizer formalism proposed by Brun, Devetak, and Hsieh [4] . We will show how quantum error-correcting codes with particular desirable properties under this framework are equivalent to some fundamental objects from combinatorial design theory, known as pairwise balanced designs [3] . While earlier relevant results in the literature give sufficient conditions for the existence of entanglement-assisted quantum codes based on classical sparse graph codes (see [10] , [17] , and references therein), results presented here give necessary and sufficient conditions under the conventional standard assumptions, and mathematically describe the structure of such quantum codes consuming only one ebit.. Low-density parity-check (LDPC) codes are among the best known classical codes in terms of error correction performance and decoding complexity [25] . Extensive efforts have been made to generalize this class of error-correcting schemes in classical coding theory to the quantum setting. Among others, codes obtained by applying the well-known Calderbank-ShorSteane (CSS) construction [5] , [27] to pairs of carefully chosen identical LDPC codes have shown remarkable error correcting performance in simulations (see, for example, [1] , [8] , [15] , [22] for recently proposed combinatorial quantum LDPC codes).. However, the progress on the quantum analogues of LDPC codes has lagged behind their classical counterparts because of special requirements imposed on the code structure in the quantum setting; only a limited class of classical codes can be exploited in a direct manner.. The development of the entanglement-assisted stabilizer formalism is a recent breakthrough in this regard [4] , [16] . This framework allows us to import any binary or quaternary linear codes to the quantum setting by exploiting maximally entangled states. In other words, by taking advantage of ebits, the code designer can turn good classical linear codes into quantum error-correcting codes and expect similar good performance in the quantum setting. In fact, Hsieh, Brun, and Devetak [15] demonstrated this advantage by constructing entanglement-assisted quantum LDPC codes which have notable error correction performance.. However, the entanglement-assisted stabilizer formalism is not a silver bullet; an adequate supply of ebits may not be available. The number of ebits required to import a given pair of classical LDPC codes varies greatly from pair to pair. In fact, as stated by Hsieh, Yen, and Hsu [17] , until recently it was conjectured that the entanglement-assisted stabilizer formalism required an impractically large number of ebits to make use of classical codes with good error correction performance.. Fortunately, Hsieh, Yen, and Hsu [17] and Fujiwara, Clark, Vandendriessche, De Boeck, and Tonchev [10] independently disproved this conjecture by giving examples of codes which require only one ebit while outperforming the previously known quantum LDPC codes in simulations. Given the positive results on quantum error-correcting codes requiring a tiny amount of entanglement and the fact that excessive reliance on ebits is a substantial obstacle to implementation, it is of interest to investigate the characteristics of quantum codes consuming only one ebit.. In this paper, we investigate what kind of quantum LDPC code is obtainable if only one ebit is allowed to import pairs of identical classical LDPC codes. We show the equivalence between such quantum LDPC codes and special classes of combinatorial objects, namely Steiner 2-designs and pairwise balanced designs of index one. (For a thorough introduction to combinatorial design theory, we refer the interested reader to [3] , [7] .) This equivalence provides theoretical insight into the properties and attainable code parameters, and explains why all known high performance quantum LDPC codes requiring only one ebit were derived from combinatorial designs of this kind.. It should be noted that it is also possible to utilize a pair of nonidentical classical error-correcting codes to construct a quantum LDPC code as long as they are of the same length. In fact, very recently quantum LDPC codes with good error correction performance have been found through a clever use of nonidentical ingredients [13] , [20] . However, their methods require a large number of physical qubits to encode, which is at odds with the focus of the current paper, that is, shedding light on more easily implementable quantum LDPC codes. For this reason, we leave the equally interesting case of nonidentical ingredients consuming only a small amount of entanglement to future work.. In the following sections, we will show how the requirement of consuming only one ebit dictates the structure of the exploitable pairs of identical classical LDPC codes. In Section II, we briefly review the entanglement-assisted quantum LDPC codes and related facts from combinatorial design theory and then prove the equivalence between pairwise balanced designs of index one and entanglement-assisted quantum LDPC codes consuming one ebit. Section III provides bounds for the code parameters. Section IV discusses some open questions and directions for future work. We have shown that homogeneous quantum LDPC codes requiring only one ebit and avoiding 4-cycles are equivalent to special classes of fundamental combinatorial designs. Various properties of entanglement-assisted quantum LDPC codes have been revealed by applying known theorems and techniques of combinatorial design theory. Our results also give theoretical insight into the known entanglement-assisted quantum LDPC codes presented as counterexamples to the conjecture on the required amount of entanglement.. We have demonstrated that combinatorial design theory may work as a useful mathematical tool to investigate homogeneous quantum LDPC codes consuming one ebit. It will be interesting to study the case when two or more ebits are allowed and investigate how much information we can extract about the structure of such quantum LDPC codes.. Another important direction would be to investigate quantum LDPC codes obtained from pairs of distinct classical LDPC codes. As we have seen in Section III, if we only allow one ebit, the rate of a homogeneous quantum LDPC code of girth six approaches one as the length becomes larger. Hence, it would be quite interesting to investigate whether heterogeneous quantum LDPC codes can overcome this fundamental limitation while avoiding short cycles and suppressing the number of required ebits. Another possible merit of studying the heterogenous case from the viewpoint of combinatorics would be that combinatorial methods appear to be helpful to design highly structured quantum LDPC codes. One possible direction would be to study how to optimize codes for an asymmetrical quantum channel where the probabilities of bit flips and phase flips are not equal (see [12] , [18] , [26] ). We hope that these questions will be answered in future work.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B I B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Strategies of Loop Recombination in Ciliates Gene assembly in ciliates is an extremely involved DNA transformation process, which transforms a nucleus, the micronucleus, to another functionally different nucleus, the macronucleus. In this paper we characterize which loop recombination operations (one of the three types of molecular operations that accomplish gene assembly) can possibly be applied in the transformation of a given gene from its micronuclear form to its macronuclear form. We also characterize in which order these loop recombination operations are applicable. This is done in the abstract and more general setting of so-called legal strings. Ciliates are a large group of one-cellular organisms having two functionally different nuclei: the micronucleus and the macronucleus. An involved DNA transformation process called gene assembly transforms a micronucleus into a macronucleus. The process is accomplished using three types of DNA transformations, which operate on special DNA sequences called pointers. These three types of operations are called loop recombination, hairpin recombination, and double-loop recombination.. For every gene in its micronuclear form, there can be several sequences of operations to transform this gene to its macronuclear form. We call such a sequence a strategy. For a given micronuclear gene strategies may differ in the number of hairpin and double-loop recombination operations. It has been shown that the number of loop recombination operations is independent of the chosen strategy [4, 3] , and that this number can be efficiently calculated [1] .. In this paper we further investigate the loop recombination operation, called the string negative rule in the string pointer reduction system, a formal model of gene assembly introduced in [2] . We characterize for a given set of pointers D, whether or not there is a strategy that applies loop recombination operations on exactly these pointers. We show that this characterization implies an efficient algorithm that determines this for given D. Also, we characterize the order in which the pointers of D can possibly be applied in strategies. These results are obtained using the reduction graph, a graph similar to the breakpoint graph in the theory of sorting by reversal, introduced in [1] . This paper is organized as follows. In Section 2 we recall basic notions and terminology concerning mainly strings and graphs, and in Section 3 we recall a formal model of the gene assembly process: the string pointer reduction system. In Section 4 we recall the notion of reduction graph and some theorems related to this notion. In Section 5 we define the pointer-component graph, a graph that depends on the reduction graph, and we discuss a natural operation on this graph. In Section 6 we show that spanning trees of the pointer-component graphs reveal interesting properties concerning the string negative rule. Section 7 shows that merging and splitting of vertices in pointer-component graphs relate to the removal of pointers. Using the results of Sections 6 and 7, we characterize in Section 8 for a given set of pointers D, whether or not there is a strategy that applies string negative rules on exactly these pointers. Section 9 strengthens results of Section 8 by also characterizing in which order the string negative rules can be applied on the pointers. We conclude this paper with Section 10. This paper showed that one can efficiently determine the possible sequences of loop recombination operations that can be applied in the transformation of a given gene from its micronuclear to its macronuclear form. Formally, one can determine which string negative rules can be applied in which order to a legal string u, given only the reduction graph of u. This is characterized in terms of graphs defined on the reduction graphs. Future research could focus on similar characterizations for the string positive rules and the string double rules.

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Characterizing software architecture changes: A systematic review Abstract Software change is inevitable. All software systems must evolve to meet the ever-expanding needs of its users. Therefore, it is vital for organizations to perform software maintenance in such a way as to reduce complications arising from changes and the potential for new bugs to be introduced by the change. Software developers need a comprehensive solution that helps them understand changes and their impact. This understanding is important because, as changes are made architectural complexity tends to increase, which will likely result in an increase in the number of bugs introduced. A complex system is potentially less understandable for developers resulting in decreased quality. Due to the number and frequency of changes that mature systems undergo, software maintenance has been regarded as the most expensive phase of the software lifecycle.. Late-lifecycle changes are changes that occur after at least one cycle of the development process has been completed and a working version of the system exists. These unavoidable changes pose an especially high risk for developers. Understanding late-lifecycle changes is important because of their high cost, both in money and effort, especially when they are due to requirements changes. Furthermore, these late-lifecycle changes tend to be the most crucial changes because they are the result of better understood customer and end-user needs.. Implementation of these changes often reduces the flexibility of a system by drawing it away from its original design. There are many sources of late-lifecycle changes including: defect repair, adapting to changing market conditions or software environments, and evolving user requirements. Due to the time pressure resulting from these crucial late-lifecycle changes, developers often cannot fully evaluate the architectural impact of each change. As a result, the architecture degrades, increasing the likelihood of faults and the difficulty of making future changes [47, 79, 107] .. When dealing with late-lifecycle changes, it is important to focus on the architecture, which defines the structure and interactions of the system. When a change affects the architecture, the original architectural model must be updated to ensure that the system remains flexible and continues to function as originally designed. When an architectural change causes the interactions to become more complex, the architecture is degenerating. Architectural degeneration is a mismatch between the actual functions of the system and its original design. Because architectural degeneration is confusing for developers, the system must undergo either a major reengineering effort or face early retirement [61] .. To address these problems, developers need a way to better understand the effects of a change prior to making it. This paper presents a systematic literature review that identifies and characterizes software changes. A systematic literature review is a formalized, repeatable process in which researchers systematically search a body of literature to document the state of knowledge on a particular subject. A systematic review provides the researchers with more confidence in their conclusions compared with an ad-hoc review. The high-level goal of this review is to: This systematic review to identify and classify architecture changes was conducted to assist in performing software maintenance. A needs assessment indicated several key areas that must be addressed to improve the software change process.. • Change Understanding and Architecture Analysis: Prior to making a change, it is important for a software developer to understand how it will impact the architecture. A change analysis tool should allow the developer to analyze a change prior to implementation to understand the change, the architecture, and how the change fits with the architecture [10, 22, 50, 106] .. • Build Historical Baseline of Software Change Data: The ability to compare a change request to a system's history will provide insight into the change impact, difficulty, and required effort. Recording information about the change type, its impact on the architecture and the effort required will provide insight into future changes [80] .. • Group Changes Based on Impact/Difficulty: Change requests can be grouped based on their characteristics. Similar changes should exhibit a similar impact on the system. Heuristics can be developed to handle certain types of changes [105] .. • Facilitate Discussion Amongst Developers: Methods that facilitate discussion among the development team are useful in achieving consensus on the implementation approach. A characterization scheme should facilitate consensus building by providing a list of the items to discuss to prevent the change from violating the planned architectural structure [60] .. • Facilitate Change Difficulty/Complexity Estimation: The characterization scheme should allow a developer to determine change complexity as a function of type and size. Characterizing the context of the change request (i.e., a description of influencing factors external to the request) should also help facilitate difficulty estimation because certain types of changes may be more difficult in certain domains than others [72] .. An output of this review is the creation of the Software Architecture Change Characterization Scheme (SACCS), which describes the effects that changes can have on architecture. The attributes of the proposed scheme were extracted from change taxonomies and associated change characteristics identified during the review.. The first characteristic is the motivation for the change ( Table 7) . The change can be motivated by either the need for an enhancement (i.e., to improve the system) or in response to a defect (i.e., resulting from an error, fault, or failure) [120] . The relative frequency of defects vs. enhancements will, over time, provide insight into system maintenance. An increase in the relative number of defects over time may suggest that the system quality is declining because more defects are introduced during the maintenance process [18] . Enhancement [69, 120] Defect [69, 120] This section discusses previous work about software change, late changes and change classifications. Then it provides the motivation for conducting this systematic review.

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Parallel smoothing of quad meshes Abstract For use in real-time applications, we present a fast algorithm for converting a quad mesh to a smooth, piecewise polynomial surface on the Graphics Processing Unit (GPU). The surface has well-defined normals everywhere and closely mimics the shape of Catmull-Clark subdivision surfaces. It consists of bicubic splines wherever possible, and a new class of patches-c-patches-where a vertex has a valence different from 4. The algorithm fits well into parallel streams so that meshes with 12,000 input quads, of which 60% have one or more non-4-valent vertices, are converted, evaluated and rendered with 9 × 9 resolution per quad at 50 frames per second. The GPU computations are ordered so that evaluation avoids pixel dropout. Smoothing quad meshes on the GPU offers an alternative to highly refined facet representations transmitted to the GPU and is preferable for interactive graphics and integration with complex morphing and displacement. The separation into vertex and patch construction stages isolates the computation on arbitrary valences from the final patch construction, simplifying the vertex and geometry shaders. Moreover, the data transfer between passes in the 2-pass conver-sion is low since only 4 × 6 control points are intermittently generated.. Since we only compute and evaluate in terms of the 24 cpatch coefficients, the computation of the cubic boundaries shared by a bicubic and a c-patch is mathematically identical. An explicit 'if'-statement in the evaluation guarantees the exact same ordering of computations since boundary coefficients are only computed once, in the vertex shader, according to Table 1 . That is, there is no pixel drop out or gaps in the rendered surface. The resulting surface is watertight.. We advertised a 2-pass scheme, since, as we argued, the DX10 geometry shader is not well suited for data amplification and evaluation after conversion. The 1-pass scheme outlined in Sect. 4 may become more valuable with the availability of a dedicated hardware tessellator [8] . Such a tessellator will make amplification more efficient and support watertight adaptive tessellation (which is why we only discussed uniform tessellation in Sect. 4). Such a hardware amplification will also benefit the 2-pass approach in that the (u, v) domain tessellation, fed into the second pass will be replaced by the amplification unit. We compiled and executed the implementation on the latest graphics cards of both major vendors under DirectX10 and tested the performance for several industry-sized models. Two surface models and models with displacement mapping are shown in Figs. 2 and 1 respectively. Table 4 summarizes the performance of the 2-pass algorithm for different granularities of evaluation. The (rocket) Frog model, in particular, provides a challenge due to the large number of extraordinary patches. The Frog Party shown in Fig. 18 currently renders at 50 fps for uniform evaluation of nine frogs for N = 9, i.e. on a 9 × 9 grid. That is, the implementation converts nine times 1292 coarse input quads, of which 59% are extraordinary, and renders nearly 1.5 million polygons 50 times per second. Additionally, our method scales well to higher tes- Table 3 . The first pass converts, the second renders. Note that the geometry shader only computes at most 24 coefficients per patch and does not evaluate Fig. 12 At present, the 1-pass conversion-and-rendering must place patch assembly and evaluation on the geometry shader. This is not efficient Table 4 Frames per second for some standard test meshes with each patch evaluated on a grid of size N × N ; eqs = percentage of extraordinary quads. Sword and Frog are shown in Fig. 2 , Head in Fig. 12 of [11] . For the smallest object, Sword, at low resolution, rendering rather than evaluation is the bottleneck. The measurements were made on an NVidia GeForce 8800 GTX graphics card sellation levels, since the patch creation time does not increase for larger evaluation grids. On the same hardware, we measured Bunnell's efficient implementation (distribution accompanying [2] ) featuring the single frog model, i.e. 1/9th of the work of the Frog Party, running at 44 fps with three subdivisions (equivalent to tessellation factor N = 9). That is, GPU smoothing of quad meshes is an order of magnitude faster. Compared to [18] , the speedup is even more dramatic. While the comparison is not among equals since both [18] and [2] implement recursive Catmull-Clark subdivision, it is nevertheless fair to observe that the speedup is at least partially due to our avoiding stream back after amplification (data explosion due to refinement). Figure 13 , right, visualizes the approximation to a densely refined Catmull-Clark mesh. Both geometric distance, as percent of the local quad size, and normal distance, in degrees of variation, are measured. Large models Fig. 13 Comparison to Catmull-Clark. Position (distance) and normal (angle) difference to the limit surface of Catmull-Clark subdivision for (left) the scheme in [10] and (right) the c-patch surface (see also Table 6 and models with a large percentage of regular quads appear visually indistinguishable when rendered by subdivision or c-patch smoothing. We therefore chose small, predictable models with many extraordinary input quads (and without displacement). Table 6 quantifies and summarizes these distances. Since we have been asked to compare c-patch surfaces to the non-smooth approximation [10] , Fig. 13 and Table 6 juxtapose the measurements. The more subtle effect of not creating smooth surfaces is evident from Fig. 14 . Fig. 18 Asynchronous animation of nine Frogs [12] Despite the lower total degree and internal C 1 join, the visual appearance of c-patches is remarkably similar to that of bicubic patches. In particular, the close-up in Fig. 17 illustrates our observation that c-patches do not create shape problems compared to a single bicubic patch. As is generally the recommendation for quad meshes, adjacent high-valent vertices in the input model should be avoided (see the sin terms in the denominator of the formulas of Table 2 ). The video [12] (see screen shots in Figs. 15, 16, 17, 18 ) illustrates real-time displacement and animation. Fig. 19 Comparison. (Input) quad mesh, (PN)-quad and (PN-lim) PN-Quad using Catmull-Clark limit points and normals [15] , (ACC) [10] (c-patch) this paper, (CC) Catmull-Clark subdivision

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A new technique to reduce cross terms in the Wigner distribution We present a new method for time-frequency representation (TFR), which combines a Fourier Bessel (FB) series expansion and the Wigner-Ville distribution (WVD In many engineering applications such as speech analysis, speech synthesis, radar, sonar and telecommunications, the signals under considerations are known to be non-stationary, for which the signal parameters are time-varying. For spectral analysis of such type of signals, discrete Fourier transform (DFT) can not be employed. Time-frequency analysis [1] , among other methods, was proposed to deal with such signals. The short time Fourier transform (STFT) is one of the earliest methods used for time-frequency analysis. A moving window cuts out a slice of the signal, and the Fourier transform of this slice gives the local properties of the signal. The STFT can be interpreted as the output signals from a band pass filter bank. The spectrogram, which is the squared magnitude of the STFT, is used for the analysis of non-stationary signals. The result depends strongly on the choice of the window function and this necessitates a trade off between time localization and frequency resolution [1] . Another commonly used TFD is the Wigner-Ville distribution (WVD) [1, 2] . Theoretically the WVD has an infinite resolution in time due to the absence of averaging over a finite time interval. Also for infinite lag length, it has infinite frequency resolution. The WVD being quadratic in nature introduces cross terms, for a multi-component signal. The cross terms can have significant amplitudes and they can corrupt the transform spaces. A particular application where this effect might have serious implications in speech analysis, since speech can be modeled as a sum of amplitude modulated (AM) and frequency modulated (FM) signals corresponding to formant frequencies [3, 4] . In the last two decades, the research has been aimed at effective suppression of cross terms and improvement of the frequency resolution, preserving the desired properties of the TFD [1] . A denoising approach [5] based on shift invariant wavelet packet decomposition has been proposed for adaptive suppression of cross terms. Recently, a WVD based on signal decomposition approach, realized by a perfect reconstruction filter bank (PRFB), has been proposed [6] . The PRFB decomposes the multi-component signal into its components. The summation of their WVDs, results in a WVD whose cross terms and noise, are significantly reduced [6] . In this paper , a new technique based on FB series expansion has been proposed. This combines a FB series expansion and the WVD. The FB series decomposes the multicomponent signal into mono-component signals like filter bank approach in [6] . The technique presented here is conceptually simpler and free from cross terms. In this paper, we have shown, a new method based on FB expansion for decomposing a multi-component signal into its mono-components. and cross terms in the Wigner distribution can be removed effectively. We have used this method in a variety of cases with success.

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An Effective GA-Based Scheduling Algorithm for FlexRay Systems SUMMARY An advanced communication system, the FlexRay system, has been developed for future automotive applications. It consists of timetriggered clusters, such as drive-by-wire in cars, in order to meet different requirements and constraints between various sensors, processors, and actuators. In this paper, an approach to static scheduling for FlexRay systems is proposed. Our experimental results show that the proposed scheduling method significantly reduces up to 36.3% of the network traffic compared with a past approach. Microprocessor-controlled electro-mechanical systems have been used to replace mechanically linked hydraulic steering and braking of cars from over a decade [1] . Some other computerized vehicle-control applications such as adaptive cruise control, collision avoidance, and autonomous driving are also being developed. These applications will be realized as real-time distributed systems requiring dependable interaction among sensors, processors and actuators.. The FlexRay system [2] is a communication system developed for the next generations of automobiles by a consortium founded in 2000 by BMW, DaimlerChrysler, Motorola, and Philips Semiconductors. The core of the FlexRay system is the FlexRay communication protocol. It has been designed for the high data transmission rates required by advanced automotive control systems. It consists of timetriggered clusters, such as drive-by-wire in cars, in order to meet different requirements and constraints between various sensors, processors, and actuators.. The FlexRay system is a time-triggered architecture providing a computing infrastructure for the design and implementation of dependable distributed embedded systems. The communication in this architecture is based on a fault-tolerant time-triggered protocol (TTP) [3] . Pop et al have been developed scheduling strategies using TTP as a communication protocol for distributed real-time systems [4], [5] . There are two basic approaches for handling tasks in real-time applications [6] . In the event-triggered approach, activities are initiated whenever a particular event is noted. In the time-triggered approach, activities are initiated at predetermined points in time. Two approaches can be used together inside a few certain applications. Pop et al [7] have proposed an approach for multicluster distributed embedded systems consisting of time-triggered and eventtriggered clusters, interconnected via gateways by using the worst-case response time analysis of the application for the controller area network (CAN).. In [7] , the schedulability analysis is outlined in Fig. 1(a) . In first step, the application is partitioned on the time-triggered cluster (TTC) and event-triggered cluster (ETC), and processes are mapped to the nodes of the architecture (i.e., mapping). In second step, the mapping of messages are combined into a frame in order to be transmitted to the bus(i.e., frame packing). In the last step, the released time of tasks and the timing of sending messages are determined (i.e., time scheduling). For each step, a given set of parameters is leading to find out if a system is schedulable, that is, all the time constraints are met. If application is unschedulable in one of these steps, the set of parameters must be changed until obtaining the approximate optimal solution. If after these steps the application is unschedulable, the approach conclude that no satisfactory implementation could be found with the available amount of resources.. In general, based the mechanism of the above approach, it tends to find only locally optimal solutions. In The author is with College of Information Science and Engineering, Northeastern University, Shenyang, P.R. China.. The authors are with the Graduate School of Information Science, Nagoya University, Nagoya-shi, 464-8603 Japan.. *This paper was presented at EMSoft 2005 .. addition to the characteristics of objects, more time is necessary to be carried out searching. Pop et al have supposed that all the processes belonging a process graph have the same period. Moreover,,such an optimization problem is NP complete, thus obtaining the optimal solution is no feasible. Pop et al have proposed two frame-packing optimization strategies, one based on a simulated annealing (SA) approach, the other is based on a greedy heuristic to explore the design space. The main limitation of SA-based methods is the difficulty in tuning the control parameters. Since the complexity of a system on which implements several difference kinds of applications, this model is hard to apply in the real world. An optimization procedure based on a SA has been proposed by Murakami et al [8] . For various sensors, processors, and actuators with different execution period, it is necessary to develop an efficient scheduling method for static segment of communication cycle in the FlexRay system. In this paper, a more practical and flexible approach to static scheduling method for the distributed automotive control system which integrate FlexRay and CAN systems has been proposed. They supposed that all the processes belonging a process graph can have difference periods. Moreover, they also took into account the influence of system load on the scheduling method.. In this paper, we propose an effective GA-based scheduling algorithm for FlexRay Systems that can include CAN system which be treated as a FlexRay node. The advantages of a GA-based approach depends heavily on how well the various components of GA incorporate the salient features of the problem under consideration [9] . To evaluate the effectiveness of this approach, we have chosen a representative safety-critical application to simulate as case study. Our experiments show the GA can be applied to such kinds of application to find a schedule better than SA approach.. There has been much attention surrounding hardwaresoftware co-design techniques. In [10], they studied the configuration problem which consists of two sections: specifying the hardware capacities of the processing elements and statically allocating the software tasks to them. This technology can be applied in allocating clusters of tasks to any processor for optimization of scheduling subtasks and communication.. The remainder of this paper is organized as follows: in Sect. 2 the application model and system architecture are presented. In Sect. 3 the problem is defined. Section 4 discusses in detail our GA approach. In Sect. 5, experimental tests are carried out. Some concluding remarks follow in Sect. 6.

O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O