Rectangular layouts and contact graphs Abstract. Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding rectangular layouts is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct O(n 2 )-area rectangular layouts for general contact graphs and O(n log n)-area rectangular layouts for trees. (For trees, this is an O(log n)-approximation algorithm.) We also present an infinite family of graphs (respectively, trees) that require (n 2 ) (respectively, (n log n)) area.We derive these results by presenting a new characterization of graphs that admit rectangular layouts, using the related concept of rectangular duals. A corollary to our results relates the class of graphs that admit rectangular layouts to rectangle-of-influence drawings. Given a set of objects in some space, the associated contact graph contains a vertex for each object and an edge implied by each pair of objects that touch in some prescribed fashion. While contact graphs have been extensively studied for objects such as curves, line segments, and even strings (surveyed by Hliněný [1998] and Hliněný and Kratochvíl [2001] ), as has the more general class of intersection graphs (surveyed by Brandstädt et al. [1999] and McKee and McMorris [1999] ), the literature on closed shapes is relatively sparse. Koebe's theorem [1936] states that any planar graph (and only a planar graph) can be expressed as the contact graph of disks in the plane. (This result was lost and recently rediscovered independently by Andreev [1970a Andreev [ , 1970b and Thurston [unpublished] ; Sachs [1994] provides a history.) More recently, de Fraysseix et al. [1994] have shown that any planar graph can be represented as a triangle contact graph, but not vice-versa. In this article, we consider contact graphs of isothetic rectangles.. Contact graphs of rectangles find critical applications in areas including VLSI design [Lai and Leinwand 1988; Bhattacharya 1991, 1988; Tani et al. 1991; Yeap and Sarrafzadeh 1995] , architectural design [Steadman 1976 ], and, in other formulations, computational geometry [de Berg et al. 1992 ; Overmars and Wood 1988] and geographic information systems (GIS) [Gabriel and Sokal 1969] . Previous works considered concepts similar to contact graphs, but used a variety of notions like rectangular duality [He 1993 ; Kinnen 1988, 1985; Lai and Leinwand 1990] and concepts from graph drawing such as orthogonal, rectilinear, visibility, and proximity layouts [Di Battista et al. 1999 , 1994 Rosenstiehl and Tarjan 1986] as well as rectangular drawings [He 2001; Rahman et al. 2004 Rahman et al. , 2000 Rahman et al. , 1998 ] to achieve their results. Our work is the first to deal directly with contact graphs of rectangles, yielding a simpler foundation for study.. We call a collection of rectangles that realizes a given contact graph G a rectangular layout: a set of disjoint, isothetic rectangles corresponding to the vertices of G, such that two rectangles are adjacent if and only if the implied edge exists in G. The associated optimization problem is to find a rectangular layout minimizing some criterion such as area, width, or height. This problem is inherently intriguing, with many enticing subproblems and variations. It arises in practice in the design of an interface to a relational database system used in AT&T to allow customers to model and administer the equipment and accounts in their telecommunications networks. This system allows users to specify their own schemas using entity-relationship models [Ramakrishnan and Gehrke 2000] . The system then presents the database entities as rectangular buttons. Clicking on a button provides information related to the corresponding entity type: for example, detailed descriptions of entity attributes or information about specific records. Experience indicates the benefit of juxtaposing buttons that correspond to related entities. Viewing the database schema as the obvious graph, with entities as vertices and relations as edges, leads to rectangular layouts. Solving the related optimization problems would automate this part of tailoring the interface to the schema, of significant benefit as otherwise each customer's interface must be built manually.. We solve several important problems concerning rectangular layouts. We give a new characterization of planar graphs that admit rectangular layouts in terms of those that can be embedded without filled triangles. En route, we unify a number of different lines of research in this field. We show a suite of results concerning the hardness of finding optimal layouts; design algorithms to construct layouts on graphs, and, with better area bounds, trees; and present some worst-case area lower bounds. We detail these results at the end of this section.. 1.1. RELATIONSHIP TO PRIOR WORK. Rectangular layouts are dual concepts to rectangle drawings of planar graphs, which are straight-line, isothetic embeddings with only rectangular faces. Recent work on rectangular drawings [Rahman et al. 2004 [Rahman et al. , 1998 ] and the related box-rectangular drawings [He 2001; Rahman et al. 2000] culminates with Rahman et al.'s [2004] linear-time algorithm for finding a rectangular drawing of any planar graph if one exists. In a rectangular layout, the rectangles themselves correspond to vertices. While the two concepts are more or less dual to each other, moving between them can be highly technical. In particular, the machinery to find rectangular drawings of graphs that are not three-connected is complex. Our contribution gives a direct method for constructing rectangular layouts, and we handle cases of low connectivity easily.. Rectangular layouts are closely related to rectangular duals, which are like layouts except that a rectangular dual must form a dissection of its enclosing rectangle; that is, it allows no gaps between rectangles. Rectangular duals have a rich history, including much work on characterizing graphs that admit rectangular duals [He 1993 ; Kinnen 1988, 1985; Lai and Leinwand 1990] , transforming those that do not by adding new vertices [Accornero et al. 2000 ; Lai and Leinwand 1988] , and constructing rectangular duals in linear time [Bhasker and Sahni 1988; He 1993; Kant and He 1997] . The proscription of gaps, however, severely limits the class of graphs that admit rectangular duals; for example, paths are the only trees that have such duals. In general, any (necessarily planar) graph admitting a rectangular dual must be internally triangulated, but no such restriction applies to layouts. This simple distinction yields many advantages to layouts over duals. The cleaner, less specified definition of layouts characterizes a class of graphs that is both more general (including all trees, for example) and also much simpler to formalize. When we discuss area, we will also show that while different variations of rectangular layouts have different degrees of area monotonicity under graph augmentation, rectangular duals do not enjoy any such monotonicity: A small graph might require a significantly larger dual than a larger graph. Additionally, there are graphs that admit asymptotically smaller rectangular layouts than duals.. Another closely related area concerns VLSI floorplanning, in which an initial configuration of rectangles (usually a dissection) is given, and the goal is to rearrange and resize the rectangles to minimize the area while preserving some properties of the original layout. While there is an extensive literature on floorplanning, there is much divergence within it as to what criteria must be preserved during minimization. These typically include minimum size constraints on the rectangles plus one of various notions of adjacency equivalence. Stockmeyer [1983] presents one of the cleanest definitions of equivalence, preserving the notion of relative placement of rectangles (i.e., whether one appears left of, right of, above, or below another); other criteria include the preservation of relative area and aspect ratios [Wimer et al. 1988 ] and the criterion of relative lengths of abutment [Tani et al. 1991] . These works are further classified by sliceability: A floorplan is sliceable if it can be recursively deconstructed by vertical and horizontal lines extending fully across the bounding box. Minimizing the area of nonsliceable floorplans has been shown to be NP-hard under various constraints [LaPaugh 1980; Pan et al. 1996; Stockmeyer 1983] , whereas area minimization of sliceable floorplans is tractable [Shi 1996; Stockmeyer 1983; Yeap and Sarrafzadeh 1993] . Not all floorplans can be realized by sliceable equivalents Bhattacharya 1991, 1988] , so work exists on isolating and generating sliceable floorplans wherever possible [Yeap and Sarrafzadeh 1995] , as well as minimizing the area of nonsliceable floorplans by various heuristics [Pan et al. 1996; Tani et al. 1991; Wimer et al. 1988] .. Two major facets distinguish floorplanning from area-minimization of rectangular layouts. First, floorplanning seeks to minimize the area of an arrangement of rectangles given a priori by some external process (possibly human design); rectangular layouts themselves are determined by corresponding contact graphs. Second, the notion of equivalence among floorplans differs from context to context, whereas contact graph adjacencies strictly identify the equivalence of rectangular layouts. Thus, there can exist different rectangular layouts of the same graph that do not represent equivalent floorplans, even by Stockmeyer's [1983] definition; and conversely there can exist equivalent floorplans that are layouts for nonisomorphic graphs. Still, much work in floorplanning uses concepts from rectangular duals, so work on rectangular layouts can also contribute to this area.. Finally, also related to rectangular layouts is the idea of proximity drawings [Di Battista et al. 1994; Jaromczyk and Toussaint 1992] , in which a set of objects corresponds to the vertices of a graph, with edges connecting vertices of correspondingly close objects for some definition of proximity. A particularly relevant special case is that of rectangle-of-influence drawings [Liotta et al. 1998 ], which are (not necessarily planar) straight-line embeddings of graphs such that the isothetic rectangles induced by pairs of vertices contain no other vertices if and only if the corresponding edges exist. Using results of Biedl et al. [1999] , we show that graphs that admit rectangular layouts are precisely those that admit a weaker variation of planar rectangle-of-influence drawings, in which induced rectangles may be empty even if the corresponding edges are missing from the graph; namely, that contact graphs of rectangles also express this variation of rectangle-of-influence drawings.. 1.2. OUR RESULTS. The many parallel lines of research (in different communities) in the general area of contact graphs of rectangles have led to overlapping (and in some cases equivalent) definitions and results. Our contributions in this article are twofold: We present algorithms for optimizing rectangular layouts and prove various hardness results, and we also prove various structural results that tie existing work together in a coherent way that produces efficient algorithms.. -We provide a new characterization of the class of graphs that admit rectangular layouts. This characterization is equivalent to an earlier result by Thomassen [1986] and has the added advantage of yielding an O(n)-time algorithm for checking whether a given n-vertex planar graph admits such a layout. Moreover, we can construct a layout for the input graph in O(n) time. We also prove an upper bound of O(n 2 ) on the area of the layout, and we give a matching worst-case lower bound. -We give an O(n)-time algorithm that constructs an O(n log n)-area layout for any tree. We also demonstrate a general class of trees that are flexible: In other words they can be laid out in linear area with any aspect ratio. Finally, we show that, in general, trees cannot have arbitrarily thin layouts: There exists a class of trees such that the minimum dimension must have size (log n). This bound uses solely topological arguments and may be of independent interest. In particular, it leads to an (n log n) worst-case area lower bound, matching the upper bound of our algorithm. -We prove that the problem of optimizing the area of a rectangular layout is NPhard. The proof also shows that optimizing the width, given a fixed height or vice-versa, is NP-hard. -We show that rectangular duals can be much larger than layouts; there exists a class of graphs having O(n)-area layouts but (n 2 )-area duals.. -Our characterization of contact graphs of rectangles in terms of filled triangles establishes a connection between rectangular layouts and rectangle-of-influence drawings. Specifically, a corollary of our characterization is that the class of graphs having rectangular layouts is identical to the class of graphs having planar, weak, closed rectangle-of-influence drawings [Biedl et al. 1999 ].. 1.3. ARTICLE OUTLINE. We continue in Section 2 by introducing necessary definitions, including those for rectangular layouts themselves, and we also state some useful lemmas. In Section 3 we present our new characterization of graphs that admit rectangular layouts. We use this characterization in Section 4 to design our linear-time algorithm for constructing O(n 2 )-area rectangular layouts, and we show matching worst-case lower bounds. In Section 5 we present improved results for trees. We present hardness results in Section 6, and we conclude in Section 7. We have presented a new characterization of contact graphs of isothetic rectangles in terms of those planar graphs that can be embedded with no filled 3-cycles. We have shown the general area and constrained-width (and -height) optimization problems for rectangular layouts to be NP-hard, and provided O(n)-time algorithms to construct O(n 2 )-area rectangular layouts for graphs and O(n log n)-area rectangular layouts for trees.. Many open problems remain. What is the hardness of approximating the minimum-area rectangular layout? Are there better approximation algorithms than the ones we presented here (O(n)-approximation for graphs; O(log n) for trees)? Is approximating the minimum dimension (width or height) easier than approximating the area? This problem is motivated by applications on fixed-width, scrollable displays. Also, does the NP-hardness result extend to rectangular duals? Since graphs that admit such duals are internally triangulated (triangulated except for the outer face), the freedom to place components to satisfy partition-type reductions does not seem to exist.. Can our techniques be applied to study contact graphs on other closed shapes, for example, squares, arbitrary regular polygons, arbitrary convex polygons, and higher-dimensional shapes? Also, allowing corner-touching to imply adjacency changes the class of graphs described by layouts. For example, K 4 can be expressed by four rectangles meeting at a corner, and K 6 , which is not planar, can be expressed by triangles. For k ≥ 4, however, no layout on k-gons can express a nonplanar graph. Allowing corner-touching also opens the question of allowing nonisothetic rectangles, which can represent embeddings with filled triangles. Finally, is there a class of polygonal shapes other than disks whose contact graphs are the planar graphs?

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


Supporting Mobile Work Processes in Logistics with Wearable Computing Logistics is a very dynamic and heterogeneous application area which generates complex requirements regarding the development of information and communication technologies (ICT). For this area, it is a challenge to support mobile workers on-site in an unobtrusive manner. In this contribution, wearable computing technologies are investigated as basis for a "mobile worker supporting system" for tasks at an automobile terminal. The features of wearable computing technologies are checked against the requirements of the application area to come to an usable and acceptable mobile solution in an user-centred design process. Until now, planning and control of logistic processes are generally executed by centralized logistics systems, which cannot cope with high requirements for flexible order processing due to increasing dynamics and complexity. The use of the RFID (Radio Frequency Identification) technology, wearable computing solutions, and other innovative ICT become more interesting for all participants in logistics. These mobile technologies can increase the transparency of logistical processes, e.g. current stock information in production systems or actual storage positions of vehicles at automobile terminals for a better planning reliability, better access and a higher quality of information. They can support an adapted information flow to the needs of a mobile executive worker, e.g. during the outdoor maintenance of wind turbine generators (WTG), or the indoor maintenance of large passenger aircrafts [1] or for outdoor maintenance in general [2, 3] .. But the development of wearable computing solutions has not reached the level of applicable products until now. One reason is the lack of usability and acceptance of the proposed mobile solutions. To overcome the weaknesses of the available approaches and prototypes, methodological changes of the design process are proposed which focus on the specific features of mobile work processes and the properties of wearable technologies and tools [4] . A "mobile worker supporting system" for logistics has to fulfil several specific requirements.

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A butterfly algorithm for synthetic aperture radar imaging Abstract. In spite of an extensive literature on fast algorithms for synthetic aperture radar (SAR) imaging, it is not currently known if it is possible to accurately form an image from N data points in provable near-linear time complexity. This paper seeks to close this gap by proposing an algorithm which runs in complexity O(N log N log(1/ )) without making the far-field approximation or imposing the beampattern approximation required by time-domain backprojection, with the desired pixelwise accuracy. It is based on the butterfly scheme, which unlike the FFT works for vastly more general oscillatory integrals than the discrete Fourier transform. A complete error analysis is provided: the rigorous complexity bound has additional powers of log N and log(1/ ) that are not observed in practice.Acknowledgment. LD would like to thank Stefan Kunis for early discussions on error propagation analysis in the butterfly algorithm. MF is grateful for AFOSR support from Arje Nachman. MF and NM were partially supported by the AFRL Automatic Target Recognition Center. The ideas of the butterfly scheme are of course very reminiscent of the fast multipole method (FMM) [18, 24] . The authors believe that the butterfly algorithm is the proper way to generalize FMM in an all-purpose way to settings where high-frequency oscillations are present. "All-purpose" means robustness, and applicability to many problems, but it also means that other numerical methods may be faster for certain problems with structure. The fast Fourier transform, and the USFFT [15] , are examples of algorithms which are faster than the butterfly, but which only work for bilinear phases. The table below summarizes ballpark complexity figures and ranges of applicability for the FFT, the USFFT of Dutt and Rokhlin, and the Chebyshev butterfly (B-Cheb) algorithm presented here. The figures are for the one-dimensional transforms, so the complexity multiplier should be squared for the two-dimensional transforms.

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Dynamic resource management considering the real behavior of aggregate traffic Abstract-This paper explains why the theory of effective bandwidth is in general not applicable to characterize realistic traffic sources. Moreover, we show that a static allocation of network resources (bandwidth and/or buffer space) based on large deviation theory can be highly inefficient when the real statistical behavior of traffic is taken into account. As an alternative, we propose a dynamic resource management scheme based on prediction techniques. As a specific example, we apply this scheme to a Differentiated Service (DiffServ) Internet environment, in which the goal, besides policing the incoming traffic, is to optimize the use of network resources, thus minimizing the probability of occurrence of violations of contract guarantees. The performance of the proposed scheme is evaluated via simulations and our results show the superior performance of the new algorithm in terms of buffer overflows, output link utilization, and jitter, as compared to currently used policing and shaping mechanisms.Index Terms-Alpha-stable self-similar processes, dynamic resource management, effective bandwidth theory, prediction techniques, traffic modeling, traffic policing and shaping. We have explained that the theory of effective bandwidth is not applicable to characterize realistic environments due to the long-tailedness of the marginal distribution of the processes that best model the statistical behavior of real traffic sources. We have also shown that a static allocation of network resources (bandwidth and/or buffer space) based on large deviation theory can be highly inefficient since the usual assumption that the probability of buffer overflow decreases exponentially with the buffer size is not applicable. The probability of buffer overflow decreases hyperbolically (much more slowly than exponentially) with either buffer size or allocated bandwidth. As an alternative, we propose a dynamic resource management scheme based on prediction techniques. The prediction technique is explained in detail and a fast algorithm to compute the prediction coefficients is also included. We have applied this scheme to an Internet environment in which DiffServ policies are implemented. The performance of the proposed scheme has been evaluated via simulations and our results show that the system is still capable of providing an acceptable service to real-time applications when the overall load is above 90%. The service provided to nonreal time applications (data) is also within reasonable limits. The system performance is evaluated in terms of buffer overflows, output link utilization, delay, nd jitter.

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


Functional Answer Set Programming In this paper we propose an extension of Answer Set Programming (ASP) to deal with (possibly partial) evaluable functions. To this aim, we start from the most general logical counterpart of ASP, Quantified Equilibrium Logic (QEL), and propose a variant QEL=ℱ where the set of functions is partitioned into Herbrand functions (or constructors) and evaluable functions (or operations). We show how this extension has a direct connection to Scott's Logic of Existence, and introduce several useful derived operators, some of them directly borrowed from Scott's formalisation. Using this general framework for arbitrary theories, we proceed to focus on a syntactic subclass that corresponds to normal logic programs with evaluable functions and equality. We provide a translation of this class into function-free normal programs and consider a safety condition so that the resulting program is also safe, under the usual meaning in ASP. Finally, we also establish a formal comparison to Lin and Wang's approach (FASP) dealing with evaluable total functions. Since its introduction two decades ago (Gelfond and Lifschitz 1988) , the paradigm of Answer Set Programming (ASP) (Marek and Truszczyński 1999) has gradually become one of the most successful and practical formalisms for Knowledge Representation due to its flexibility, expressiveness and current availability of efficient solvers. This success can be easily checked by the continuous and plentiful presence of papers on ASP in the main conferences and journals on Logic Programming, Knowledge Representation and Artificial Intelligence during the last years. The declarative semantics of ASP has allowed many syntactic extensions that have simplified the formalisation of complex domains in different application areas like constraint satisfaction problems, planning or diagnosis.. In this paper we consider one more syntactic extension that is an underlying feature in most application domains: the use of (partial) evaluable functions. Most ASP programs include some predicates that are nothing else than relational representations of functions from the original domain being modelled. For instance, when modelling the typical educational example of family relationships, we may use a predicate mother(X, Y ) to express that X's mother is Y , but of course, we must add an additional constraint to ensure that Y is unique wrt X, i.e., that the predicate actually acts as the function mother(X) = Y . In fact, it is quite common that first time Prolog students use this last notation as their first attempt. Functions are not only a natural element for knowledge representation, but can also simplify in a considerable way ASP programs. Apart from avoiding constraints for uniqueness of value, the possibility of nesting functional terms like in W = mother(f ather(mother(X))) allows a more compact and readable representation than the relational version mother(X, Y ), f ather(Y, Z), mother(Z, W ) involving extra variables, which may easily mean a source of formalisation errors.. Similarly, as we will see later, the use of partial functions can also save the programmer from including explicit conditions in the rule bodies to check that the rule head is actually defined.. The addition of functions to ASP is not new at all. In fact, there exist two different ways in which functions are actually understood. The first way of treating functions is followed by most of the existing work in the topic (like the general approaches (Syrjänen 2001; Bonatti 2004; Šimkus and Eiter 2007) or the older use of function Result for Situation Calculus inside ASP (Gelfond and Lifschitz 1993) ). These approaches treat functions in the same way as Prolog, that is, they are just a way for constructing the Herbrand universe, and so they satisfy the unique names assumption -e.g. mother(john) = mary is always false. A second way of treating functions is dealing with them as in Predicate Calculus, as done for instance in Functional Logic Programming (Hanus 1994) . The first and most general approach in this direction is due to the logical characterisation of ASP in terms of Equilibrium Logic (Pearce 1996) and, in particular, to its extension to first order theories, Quantified Equilibrium Logic (QEL) (Pearce and Valverde 2004) . As a result of this characterisation, the concept of stable model is now defined for any theory from predicate calculus with equality. In fact, stable models can be alternatively described by a second-order logic operator (Ferraris et al. 2004 ) quite close to Circumscription (McCarthy 1980) , something that has been already used, for instance, to study strong equivalence for programs with variables (Lifschitz et al. 2007) . Another alternative for ASP with (non-Herbrand) functions has been very recently presented in (Lin and Wang 2008) and, as we will show later, can be seen as a particular case of the current approach, when we restrict to total functions.. As we will explain in the next section, we claim that the exclusive use of Herbrand functions and the currently proposed interpretation of equality in QEL or the requirement for functions to be total, as in (Lin and Wang 2008) , yield some counterintuitive results when introducing functions for knowledge representation. In order to overcome these problems, we propose a variation of QEL that separates Herbrand functions (or constructors) from evaluable functions, as also done in logical characterisations (González-Moreno et al. 1999; Rodríguez-Artalejo 2001; Hanus 2007) of Functional Logic Programming. We further show how our semantics for partial functions has a direct relation to the Logic of Existence (or E-logic) proposed by Scott (Scott 1979) .. The rest of the paper 1 is organized as follows. In the next section, we informally consider some examples of knowledge representation with functions in ASP, commenting the apparently expected behaviour and the problems that arise when using the current proposal for QEL. In Section 3, we introduce our variant called QEL = F . Section 4 defines some useful derived operators, many of them directly extracted from E-logic and showing the same behaviour. In Section 5 we consider a syntac-tic subclass of logic programs with evaluable functions and Herbrand constants, and show how they can be translated into (non-functional) normal logic programs afterwards. This includes a definition of safety that guarantees that the final translation results in a safe program, something crucial for the current ASP grounders. Section 6 establishes a formal comparison showing how (Lin and Wang 2008) can be encoded into our functional logic programs by forcing functions to be total, and also includes a discussion showing that QEL = F is more suitable for nonmontonic reasoning with functions. Finally, Section 7 contains a brief discussion about other related work and Section 8 concludes the paper. We can summarize the main contributions of this paper into the introduction of a new language for dealing with functions in ASP and the discussion about several modelling issues not easily solvable within other ASP modelling paradigms. In this way, the paper has tried to clarify some relevant aspects related to the use of functions in ASP for Knowledge Representation. These aspects include the distinction between Herbrand and evaluable (and possibly partial) functions, the concept of definedness, the treatment of equality, the directionality in function assignments or a new nondeterministic choice operation for selecting a function value.. The functional nature of some predicates is hidden in many ASP domains. When functions are represented in a relational way, we require the continuous addition of constraints for uniqueness of value, and a considerable amount of extra variables to replace the ability of nesting functional terms. All this additional effort may easily become a source for programming errors.. Although, as we have shown, the proposed approach can be translated into relational ASP and merely considered as syntactic sugar, we claim that the use of functions may provide a more natural, compact and readable way of representing many scenarios. The previous experience with a very close language to that of Section 5, implemented in an online interpreter 14 and used for didactic purposes in the past, shows that the functional notation helps the student concentrate on the mathematical definition of the domain to be represented, and forget some of the low level representation tasks, as those commented above, or as the definedness conditions, that must be also considered in the relational representation, but the functional interpreter checks in an automatic way.. We hope that the current approach will help to integrate, in the future, the explicit treatment of arithmetic functions made by some ASP tools, that are currently handled outside the formal setting. For instance, the ASP grounder lparse 15 syntactically accepts a program like p(div(10, X)) ← q(X) but raises a "divide by zero" runtime error if fact q(0) is added to the program. On the other hand, when div is replaced by a non-built-in function symbol, say f , the meaning is quite different, and we get {p(f (10, 0)), q(0)} as a stable model. In this paper we have also identified and separated evaluable and (possibly) partial functions (like div above) from constructors (like f in the previous example).. We have provided a translation of our functional language into normal logic programs to show that: (1) it can be implemented with current ASP solvers; but more important (2) that the proposed semantics is sensible with respect to the way in which we usually program in the existing ASP paradigm. This translation has been implemented in a tool called lppf (logic programs with partial functions). 16 . For future work, we plan to follow (Lin and Wang 2008) work on loop formulas for converting their programs with total functions into Constraint Satisfaction Problems and extend their work for our functional logic programs. As in (Lin and Wang 2008) , we expect to obtain a reduction on the size of ground functional logic programs, with respect to the size of their relational counterparts.. A topic for future study is the implementation of a solver that directly handles the functional semantics. Other open topics are the axiomatisation of the current logical framework, the addition of a second, explicit (or strong) negation, or the extension of lppf to combine evaluable functions with constructors of arity greater than zero, using as a back-end the recently available tool DLV-complex 17 .. The proof for this case is tedious, but follows similar steps to the previous two cases. By analogy, it is not difficult to see that I, w |= (51) iff I * , w |= (22) and that I, w |= (52) iff I * , w |= (23). The present approach has incorporated many of the ideas previously presented in (Cabalar and Lorenzo 2004; Cabalar 2005) . For instance, (Cabalar and Lorenzo 2004) can be seen as the fragment of our FLP-programs where we disable the use of predicates and restrict default negation exclusively for specifying default values of functions.. With respect to other logical characterisations of Functional Programming languages, the closest one is perhaps (González-Moreno et al. 1999) , from where we extracted the separation of constructors and evaluable functions. The main difference is that QEL = F provides a completely logical description of all operators, allowing an arbitrary syntax (including rules with negation, disjunction in the head, negation and disjunction of rules, etc). Another important difference is that QEL = F is constrained to strict functions, while (González-Moreno et al. 1999 ) is based on non-strict functions.. Scott's E-Logic is not the only choice for logical treatment of partial functions. A related approach is the so-called Logic of Partial Functions (LPF) (Barringer et al. 1984) . The main difference is that LPF is a three-valued logic -formulas containing undefined terms have a third, undefined truth value. The relation to (relational) ASP in this way is much more distant than the current approach, since stable models and their logical counterpart, equilibrium models, are two-valued 13 . As for the relation to other approaches exclusively dealing with Herbrand functions (Syrjänen 2001; Bonatti 2004; Šimkus and Eiter 2007) an interesting topic P. Cabalar for future study is analysing to which extent they could be captured by QEL = semantics, i.e., the fragment of QEL = F without evaluable functions.

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


An exact method for computing the area moments of wavelet and spline curves AbstractÐWe present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis. Using Green's Theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product. The multidimensional filter coefficients are precomputed exactly as the solution of a two-scale relation. To demonstrate the performance improvement of the new method, we compare it with existing methods such as pixel-based approaches and approximation of the region by a polygon. We also propose an alternate scheme when the scaling function is sinx. M OMENTS are standard descriptors of the shape of an object [1] , [2] , [3] ; they easily yield features that are invariant to translation and rotation [4] or, more generally, to affine transformations, which makes them useful tools for pattern recognition. In the standard formulation, they are computed as surface integrals which requires raster scanning through the image. However, there are many instances where the boundaries of objects are described by parametric curves. This is the case, for example, when the objects are detected using parametric snakes which are represented using B-spline [5] , [6] , [7] , [8] or wavelet basis functions [9] , [10] . Another simple case is when the region is described as a polygon [11] .. In this paper, we address the problem of computing the area moments of objects described by such parametric curves when the basis functions are scaling functions. The popular wavelet curve descriptors also fall into this class. The originality of our approach is that the computation is exact and also more direct than the conventional pixelbased method which requires an explicit labeling of the inner region of the curve prior to computation. Moreover, the pixel-based schemes suffer a low accuracy due to the loss of subpixel details in the rasterizing process. Also, the error in the area-based computation of moments is dependent on the orientation of the shape.. Since a polygon can be represented in terms of linear splines, the computation of moments by approximating the shape as a polygon [11] , [12] , [13] is a particular case of our approach. While the polygon method can be made as accurate as desired by increasing the number of segments, the convergence is slow because of the low approximation order of linear splines. Moreover, it is not suitable for computing the curvature, which is an interesting shape feature as it is invariant to rotation and translations, and can be easily normalized to scale changes. This motivates us to investigate higher order schemes where the curve is represented by smoother basis functions such as B-splines and other scaling functions that appear in wavelet theory [14] , [15] . These type of basis functions also occur naturally when one seeks multiresolution representation of curves which are well suited for pattern recognition and shape simplification [16] , [10] .. The paper is organized as follows: In Section 2, we show how Green's Theorem can be used for the computation of the area moments of a parametric curve. In Section 3, we consider the computation of the moments of such a curve represented in spline or wavelet bases. Here, we also discuss the properties of the multidimensional kernel used in the computation of moments. In Section 4, we give the implementation details of the moment computation. In the following section, we deal with the precomputation of the kernel. In Section 6, we present an alternate implementation that works for any order moments, but it is rigorously exact only when the scaling function is sinx. This is especially interesting because it makes our method applicable to the Fourier representation of curves as well. In the last section, we compare the new method with the existing schemes such as approximation using polygons and rasterizing. In this paper, we have presented a new approach for the computation of the moments of a curve described in a wavelet or scaling function basis. It is especially useful for objects detected using parametric snakes. The main advantages of the proposed scheme over the conventional methods are:. . the exactness of the computation, . its independence of the orientation of the shape, and . the consistency with the snake model and the fact that it is the most direct method available. In addition, the method is reasonably fast and easy to implement.. We recommend using our exact kernel-based approach for computing the lower order moments (typically m n P) for which the kernels are available. For higher order moments, we have proposed a quadrature formula that approximates the continuous integrals with Riemann sums. The latter method is exact for the sinc basis functions; otherwise, it can be made as accurate as desirable by resampling the model at a finer rate ( sufficiently large). In this section, we analyze equation (21) and simplify it for faster computation. We start with the simplest case: the area of the region.. The area bounded by the curve (cf. (15) where g H is given by (17) . The sequences p k and d p k are wEperiodized versions of the coefficients k and d k with respect to the period w. This is simply because convolving a nonperiodized sequence with a periodized kernel is equivalent to convolving a periodized sequence with a nonperiodized kernel. We have also reduced the range of summation of the inner sum to Àx I to x À P, which is typically much less than the range H to w À I. Similarly, for the higher order moments all the summations, except the outer one, are in the range Àx I to x À P.. From (45), we see that the computation of the area involves just a filtering operation by gÀl g l, followed by an inner product. This can be written as,. where hXY Xi stands for the inner product hY di wÀI kH kdk. With a similar notation, the computation of the other moments are given as. As the m n IEh sequence is separable, the filtering operation is much simpler than the usual m n IEdimensionl filtering. The complexity in the computation of the moment s mYn is wXPx À P mnP , without taking the symmetries into account. Thus, for basis functions with small support and reasonable m and n, the complexity is quite managable.

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


Finite transducers for divisibility monoids Divisibility monoids are a natural lattice-theoretical generalization of Mazurkiewicz trace monoids, namely monoids in which the distributivity of the involved divisibility lattices is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be commutations. Here, we show that every divisibility monoid admits an explicit finite transducer which allows to compute normal forms in quadratic time. In addition, we prove that every divisibility monoid is biautomatic. The goal of this paper is to establish the following result :. Main Theorem. Every left divisibility monoid admits an explicit finite transducer which allows to compute right normal forms in quadratic time.. Mazurkiewicz's trace theory provides a well-investigated mathematical model for the sequential behavior of a parallel system in which the order of two independent actions is regarded as irrelevant. This is achieved by considering a free partially commutative monoid, namely, the free monoid of all words over a fixed alphabet modulo the congruence generated by equations of the form ab = ba for pairs of independent actions (a, b). Roughly speaking, a letter corresponds to an event and two letters commute when the corresponding events can occur simultaneously.. However, there are several areas in computer science where one would like to consider more general equations of the form ab = cd, rather than just ab = ba as in trace theory. Left divisibility monoids have been introduced as a natural algebraic generalization of Mazurkiewicz's trace monoids, namely monoids in which the distributivity of the underlying left divisibility lattices is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be commutations.. The purpose of this paper is to study how to compute efficiently normal forms in left divisibility monoids. Following Thurston's original idea about the automaticity of the braid groups (see [11] ), we shall construct an explicit finite transducerthat is a finite automaton with output-allowing to compute normal forms in every left divisibility monoid. Since a standard transducer reads words from the left to the right, we need to define a right normal form, even if a left normal form for elements in a left divisibility monoid-like the normal form defined by Kuske [18] and generalizing the Cartier-Foata normal form known from the theory of Mazurkiewicz traces [6] -seems to be a priori the most pertinent choice.. Before describing the transduction machinary, we exhibit several nice properties of this new normal form with, in particular, a deep geometric property concerning the associated so-called Cayley graph. Our work provides a detailed and complete proof to Kuske's claim that every left divisibility monoid is automatic [18] . Furthermore, we show that every left divisibility monoid is (both left and right) biautomatic, according to Hoffmann's terminology [12] .. The rest of this paper is organized as follows. In Section 2, we recall several definitions about automaticity for monoids. In Section 3, we gather the needed basic properties of left divisibility monoids. Section 4 introduces the right normal form. We show that the language of right normal forms has good properties, preparing the proof of how it provides a biautomatic structure to every left divisibility monoid. In Section 5, we then state the main results of this paper (Theorems 27 and 28), discuss and illustrate them.

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How many candidates are needed to make elections hard to manipulate? In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied [4], but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never.

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Induction of M-convex Functions by Linking Systems Induction (or transformation) by bipartite graphs is one of the most important operations on matroids, and it is well known that the induction of a matroid by a bipartite graph is again a matroid. As an abstract form of this fact, the induction of a matroid by a linking system is known to be a matroid. M-convex functions are quantitative extensions of matroidal structures, and they are known as discrete convex functions. As with matroids, it is known that the induction of an M-convex function by networks generates an M-convex function. As an abstract form of this fact, this paper shows that the induction of an M-convex function by linking systems generates an M-convex function. Furthermore, we show that this result also holds for M-convex functions on constant-parity jump systems. Previously known operations such as aggregation, splitting, and induction by networks can be understood as special cases of this construction. As a generalization of matroids, many matroidal structures and its quantitative extensions are studied. The concept of M-convex functions on base polyhedra introduced in [13] is a generalization of valuated matroids [4] , [6] , which are quantitative extensions of matroids. M-convex functions are known as discrete convex functions, and play a central role in discrete convex analysis [15] . Recently, the concept of M-convex functions on constant-parity jump systems is introduced in [16] as a generalization of M-convex functions on base polyhedra. The definitions of M-convex functions and jump systems are to be described later. To distinguish between M-convex functions on base polyhedra and those on constant-parity jump systems, we sometimes refer to the former as M Bconvex functions and the latter as M J -convex functions. For M-convex functions global minimality is equivalent to local minimality defined in an appropriate manner. In addition, discrete duality theorems such as discrete separation and min-max formula hold for M B -convex functions, whereas they fail for M J -convex functions.. In the study of matroidal structures and their extensions, a number of natural operations and transformations are considered. To be specific, induction by bipartite graphs is one of the most important operations on matroids. Given a bipartite graph G = (S, T ; E) and a matroid M = (S, I) with the collection of independent sets I ⊆ 2 S , the induction of M by G is defined as M = (T, I ) with. where ∂M is the set of vertices covered by M . It is known that M forms a matroid [17] . As an abstract form of this fact, the induction of a matroid by a linking system, which is introduced as a generalization of bipartite matchings, is known to be a matroid [18] .. Operations on M B -convex functions and M J -convex functions are studied in [13] and [10] , respectively. It is known that the induction of an M B -convex function (respectively, an M J -convex function) by a network is an M B -convex function (respectively, an M J -convex function). As a natural generalization of this operation, we introduce a new operation called induction of M-convex functions by integral poly-linking systems, where integral poly-linking systems are generalizations of integral flows in networks. Our main results (Theorems 5 and 6) show that the induction of an M-convex function by an integral poly-linking system is M-convex. We mention the relation between composition and induction by integral poly-linking systems. By the equation (

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Minimum-link paths among obstacles in the plane Abstract. Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in time O(Ect(n) log 2 n) (and space O(E)), where n is the total number of edges of the obstacles, E is the size of the visibility graph, and c~(n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted at s) of minimum-link paths from s to all obstacle vertices. This leads to a method of solving the query version of our problem (for query points t).

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Acoustic super models for large scale video event detection Given the exponential growth of videos published on the Internet, mechanisms for clustering, searching, and browsing large numbers of videos have become a major research area. More importantly, there is a demand for event detectors that go beyond the simple finding of objects but rather detect more abstract concepts, such as "feeding an animal" or a "wedding ceremony". This article presents an approach for event classification that enables searching for arbitrary events, including more abstract concepts, in found video collections based on the analysis of the audio track. The approach does not rely on speech processing, and is language-indepent, instead it generates models for a set of example query videos using a mixture of two types of audio features: Linear-Frequency Cepstral Coefficients and Modulation Spectrogram Features. This approach can be used in complement with video analysis and requires no domain specific tagging. Application of the approach to the TRECVid MED 2011 development set, which consists of more than 4000 random "wild" videos from the Internet, has shown a detection accuracy of 64 % including those videos which do not contain an audio track. Data-driven event classification can be considered more flexible than symbolic or semi-symbolic approaches that require manual crafting of event descriptions. In most cases, symbolic event descriptions also require manual tagging of vast amounts of training data (e.g., in platforms like LabelMe [5] ) in order to learn how to detect the elements found in a symbolic event description.. Consider for instance a description of a wedding, one event class found in the TRECVid MED 2011 data set. A simple description would probably include a couple (two people -in most cases one male "groom" and one female "bride") being wed by a third person (priest or official) in the presence of an audience (group of people watching). In order to enable a system to search for the elements of this description in a video, it must first be able to detect elements fitting the concept of bride, groom, priest, etc. This in turn requires these concepts to be somehow modeled in the system, usually requiring a vast amount of manual tagging. The situation is aggravated by the fact that tagging is mostly domain specific. Hence a new kind of event description often requires a significant amount of manual tagging in order to be usable for a search query. This problem can also arise within descriptions of the same event as a single concept might be represented by two or three different topics (e.g. Surfing, skateboarding and skiing all are classified as board tricks).. Another drawback of using manual event descriptions is that they are designed by humans but are to be used by computer algorithms. The point in making this distinction is that computer algorithms might work in a way that fundamentally differs from the way that human perception works. Hence two video or, for that matter, two audio files mthat look alike to a human might be rated as completely differ-ent by an algorithm. The same holds true in the opposite case, as two video or audio files rated by an algorithm as similar might show different concepts to a human. However, as the actual search is conducted by a computer, the feature values that discriminate events best in a computer algorithm should be focused on. An important advantage of data-driven approaches in this context is that multiple examples for one event class are fed into a system and that the algorithm filters out those features that best discriminate the instances with respect to the given classes, thus in a way translating the human concept into a specification that can be interpreted by a machine. One should keep in mind, however, that this translation just coincides with concepts as represented by the training data. So selection of the training data is crucial. Selecting the right training data still requires less effort than does hand coding event descriptions since hand coding involves coding at multiple levels starting at the semantic level down to connecting semantic concepts with feature level concepts. While these arguments speak in favor of data-driven approaches for event detection, their applicability still has to be proven. Moreover, current computational constraints make it difficult to extract and compare visual features for 25 frames or more per second in a selection of thousands of high definition videos.. This article describes the first approach of acoustic datadriven event detection for large scale retrieval tested on the TRECVid MED 2011 development data set. While approaches for earlier TRECVid challenges have in some instances used speech recognition or detection of preselected audio sub-concepts (see Section 2), audio features have never been used for TRECVid in a purely data-driven way. The results presented in this paper show that learning based on arbitrary low level audio features can be used to detect high level concepts like those given in TRECVid MED 2011. This insight is not only valuable in itself, it also suggests a role for low level audio features as an option in multimodal concept detection thus augmenting present concept detection systems. The paper is organized as follows. Section 2 gives an overview of related work in both the field of acoustic and visual multimedia analysis as well as multimedia event detection. Section 3 discusses the TRECVid MED 2011 data set before Section 4 describes the algorithms employed. An evaluation of the approach is presented in Section 5 before Section 6 concludes the paper with a discussion of evaluation results as well as future work. The two obvious media that can assist in video event detection are the visual and acoustic modalities. Traditional vision-based methods, however, mostly rely on analysis techniques that involve object recognition. This fact makes visual approaches highly domain dependent as the trained classifiers need to be trained based on human annotation. This is impractical with ever-changing sample queries over found consumer-produced data. Data-driven approaches for image analysis, such as "bag of words" approaches are highly computationally inefficient as they require the analysis of two to three orders of magnitude more data than audiobased analysis techniques 1 . Therefore, in order to explore the applicability of data-driven approaches to event detection, the approach presented in this paper is based on audio features only. The features used are LFCC and MSG. They are combined on a model level to combine the classification strengths of both features. The evaluation discussed in this paper has shown that a purely data driven approach for video concept detection can yield results that are comparable to visual approaches. Application of the approach to the TRECVid MED 2011 development set, which consists of more than 4000 random "wild" videos from the Internet, has shown a detection accuracy of 64 % which includes videos that do not contain an audio track. This is even more surprising as these results have been achieved with a purely acoustic approach and the challenge calls for multimodal approaches. The system presented in this paper is designed to be part of a multimodal event detection framework that will incorporate visual concept detection, OCR, audio transcripts and semantic analysis. It is expected that the combination of these different kinds of classifiers will lead to even better results.. Future work will include experiments with other feature sets and different frequency ranges. We also plan to address the fact that different classifiers perform differently on an event-class by event-class basis. In order to tackle the shortcoming that results can not be explained intuitively to the user, we explore the use of higher level audio features such as described in [11] . 1 Remember that content analysis is usually performed by extracting features from raw, i.e. uncompressed, video. A second of mono audio sampled at 44.1 kHz with 16 bit resolution is 88 kB while a second of low-resolution video, say (320 × 240 RGB-pixels at 25 frames per second) is 5.76 MB. Our system achieves the most promising results in the "attempting a board trick" and "working on a woodworking project" categories. Therefore, we decided to perform a more in-depth human analysis of these videos, with the hope of understanding what allowed the system to distinguish them. The five most common acoustic events in the "attempting a board trick" category according to human annotators were: Post-production music, speech, thumps (of a board landing on a surface), skateboard wheels on concrete, and slow motion distortion. A human analysis of the successfully categorized and miscategorized "board trick" videos revealed that the great majority of successfully categorized videos contained high tempo post-production music, and the majority of miscategorized videos had virtually no music at all. The sounds most commonly associated with board tricks (by humans) seem to have little bearing on successful categorization by our system. Similarly, in the "Woodworking" category our top-five acoustic events were as follows: Speech, Knife Scraping, Power Lathe, Hammer, and Table Saw . Speech was by far the most common sound in these videos, occurring in approximately 2/3 of the videos which we studied. Woodworking videos often have an individual clearly speaking to the camera, explaining technique. Our analysis of successfully classified videos suggests that the system is detecting speech of this type, as most of them have this layout. The actual sounds of woodworking seem to have little weight in our current system's classification of this type of video. An analysis of the top-ten videos classified as woodworking from the "other" category showed rather unsurprising results based upon our previous analysis: they were mostly "instructional" videos in which speakers used an instructional voice, consisting of a single person speaking directly to the camera. A few amusing outliers occurred, notably HVC328564, which was of people breaking wooden boards in a karate class, which one could argue is actually "woodworking" (albeit unusual). Overall, however, we interpret this as further indication that machines find other features discriminative than humans, which is again supporting the hypothesis for a data-driven approach. There is a wealth of related work in multimedia content analysis, especially video analysis. A comprehensive description of the related work would easily exceed the page limits of this paper. Therefore, we concentrate on surveying only parts of the most relevant work. Very few papers take into account audio, and most importantly most work concentrates on finding small events or objects rather than entire concepts. Very good summaries are provided by a 2006 article by Lew, Sebe, Djeraba, and Jain [6] and a 2009 article by Snoek and Worring [12] . In [9] , for example, the authors present a content-adaptive audio texture based method to segment video into audio scenes. A similar idea using a different approach was proposed in [1] , where an unsupervised method for repeated sequence detection in TV broadcast streams was presented. The approaches would be very interesting to try on the data we used. However, the work mainly concentrates on scene transition markers and thus works mostly for professionally produced content. In [6] the possibilities of using multimodal analysis including audio are emphasized as a major research challenge.. The TRECVid evaluation [6] , organized on a year-by-year basis by the US National Institute of Standards and Technologies (NIST), investigates mostly visual event detection on broadcast videos [8] . The task is to detect concepts like "a person applauding" or "a person riding a bicycle". However, while approaches for earlier TRECVid challenges have in some instances used speech recognition, audio has mostly been ignored, with the exception of the detection of preselected audio sub-concepts such as "outdoor urban", "outdoor rural" or "indoor quiet" described in [4] . Many visual methods developed in the community are related to the research presented here and could be used to complement it. The Informedia project's [15] basic goal is to achieve machine understanding of video and film media, including all aspects of search, retrieval, visualization and secularization in both contemporaneous and archival content collections. However, again, audio is not explored beyond speech recognition. TRECVid's counterpart, the NIST Rich Transcription (RT) [7] evaluation, focuses on acoustic methods for transcribing multimedia content. The evaluation is currently focusing on meeting data, but previous evaluations included broadcast news from radio and television.

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Fuzzy inequational logic We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if-then rules which is obtained as particular case of the general result. In this paper, we introduce a general logic for approximate reasoning about atomic predicate formulas which take form of inequalities between terms. Such formulas, called inequalities, are essential in the classic theory of varieties of ordered algebras [10] since the varieties are exactly the classes of ordered algebras which are definable by sets of inequalities. We extend the classic logic for reasoning about inequalities by considering degrees to which one considers the inequalities valid. We would like to stress that our approach is truth-functional and the degrees we use are interpreted as the degrees of truth and they should not be confused or interchanged with degrees that appear in other formalisms and uncertainty theories (like the degrees of belief ). We assume that the degrees come from general structures of truth degrees. In particular, we use complete residuated lattices [4, 19, 22] .. In the proposed logic, we introduce two types of entailment: First, a semantic entailment which is based on evaluating inequalities in particular fuzzy structures called algebras with fuzzy orders [33] . Using algebras with fuzzy orders as models, we are able to introduce degrees to which inequalities semantically follow from collections of partially valid inequalities. Second, we introduce a graded notion of provability (syntactic entailment) which allows us to infer partially valid conclusions from collections of partially valid inequalities. The notion of graded provability is defined using a specific deductive system which consists of axioms and three deduction rules. We prove that our logic is complete in that the degrees of semantic entailment coincide with the degrees of provability. This type of graded completeness is called Pavelka-style completeness [25] after J. Pavelka who, inspired by the influential paper by J. A. Goguen [22] , presented the general concept in [29, 30, 31] and studied Pavelka-style complete propositional logics. A thorough and general treatment of logics with this style of completeness is presented in [21] .. We consider the completeness result to be the main result of this paper.. In addition to that, we present an application of the result showing a complete axiomatization of a logic for reasoning about graded if-then rules called attribute implications. Such rules, sometimes used under different names, are formulas which play important roles in several disciplines concerned with data analysis and management such as the formal concept analysis [20] and relational databases [28] . We show in the paper that the rules can be treated as particular inequalities and that using our general result we may obtain a complete logic for approximate reasoning with such inequalities. By making this observation, we contribute to the area of reasoning with graded if-then rules and present an alternative to the approaches in [9, 32] which may further be explored.. Previous results which are related to our paper include the fuzzy equational logic [3] which introduced Pavelka-style logic for reasoning about graded equalities and fuzzy Horn logic dealing with implications between graded equalities [7] .. Our logic can be seen as a generalization of the fuzzy equational logic. Indeed, from the syntactic point of view, it is a logic which results from the fuzzy equational logic by omitting the deduction rule of symmetry. From the semantic point of view, the present logic uses more general models-algebras with fuzzy orders [33] instead of algebras with fuzzy equalities [6] . A survey of results on fuzzy equational logic can be found in [5] . This paper is organized as follows. In Section 2, we present preliminaries from residuated structures of truth degrees and algebras with fuzzy orders. In Section 3, we introduce our logic and present the central notions of semantic and syntactic entailments. In Section 4, we show that our logic is syntacticosemantically complete in Pavelka style. In Section 5, we present an application of the general completeness result provided in Section 4 by showing a general logic of attribute implications with a complete Pavelka-style axiomatization. We showed a Pavelka-style complete logic for reasoning with graded inequalities using any complete residuated lattices as the structure of truth degrees. The results generalize the previous results on completeness of fuzzy equational logic by considering more general semantics given by algebras with fuzzy orders and omitting the deduction rule of symmetry. In addition, we showed an application of the general completeness result showing a way to generalize the ordinary attribute implications in a graded setting with a general semantics and Pavelkastyle complete inference system which generalizes the well-known Armstrong system of inference rules.

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