Dense crystalline dimer packings of regular tetrahedra We present the densest known packing of regular tetrahedra with density φ = 4000 4671 = 0.856347 . . . Like the recently discovered packings of Kallus et al. and Torquato-Jiao, our packing is crystalline with a unit cell of four tetrahedra forming two triangular dipyramids (dimer clusters). We show that our packing has maximal density within a three-parameter family of dimer packings. Numerical compressions starting from random configurations suggest that the packing may be optimal at least for small cells with up to 16 tetrahedra and periodic boundaries. The problem of the packing of tetrahedra, which has modern-day applications to such wide-ranging topics as metamaterials with novel optical properties, nanomaterials [1] [2] [3] , and virus formation, hails back to the early Greeks [4] . Aristotle, in discussing the assignment of geometrical figures to "heavenly bodies", mistakenly believed that regular tetrahedra ("pyramids") tile (Euclidean) space perfectly. Nearly 1800 years later, Johannes Müller (aka Regiomontanus, 1436-1476) contradicted Aristotle's claim [5, 6] . 400 years after Müller, Minkowski showed that the densest lattice packing of any convex body must satisfy certain constraints [7] and mistakenly argued that the densest lattice packing of tetrahedra (i.e. one tetrahedron per unit cell) had density φ = 9 38 [8] . In 1900, Hilbert posed the problem of the densest lattice packing of regular tetrahedra as a special case of the 18th of his famous list of problems [9] . Grömer conjectured [10] , and Hoylman later proved [11] , that the densest lattice packing of a single tetrahedron is φ = 18 49 . In 1972, Stanislaw Ulam conjectured that spheres would have the lowest maximum packing density of all convex bodies, including tetrahedra [12] . That density is φ sphere = π √ 18 = 0.740480 . . . [13] . In 2000, Betke and Henk developed an efficient computer algorithm to compute the densest lattice packing of any convex body, and applied it to the Archimedean solids [14] . Conway and Torquato [15] used Betke and Henk's algorithm to compute the packing density of tetrahedra derived from the densest lattice packing of icosahedra, and also examined other promising packings. All of them packed worse than spheres, with a maximum packing fraction φ ≈ 0.7175. This led them to suggest the tantalizing possibility that the Ulam conjecture might, in fact, fail for tetrahedra. They further proposed that tetrahedra might have the lowest packing density of all convex bodies. Motivated by this, Chaikin and coworkers [16] performed experiments on tetrahedral dice packed in spherical and cylindrical containers. They reported random packings with densities as high as 0.75 ± 0.03 [16] and 0.76 ± 0.02 [17] , with the error arising from the rounded dice vertices and edges [17] .. In 2008, Chen proposed a packing of nonamers (nine tetrahedra forming two pentagonal dipyramids that share one tetrahedron) arranged in layers, whose density (φ ≈ 0.7786) clearly exceeded, for the first time, the maximum packing density of spheres [18] . Since Chen's publication, a flurry of activity has resulted in successively higher and higher packing densities for tetrahedra. It was soon demonstrated that the nonamer crystal could be numerically compressed to slightly higher packing densities (φ = 0.7837 [19] and φ = 0.7820 [20] ). The first disordered (i.e. non-periodic) packing of tetrahedra to exceed the maximum packing density of spheres was reported by Haji-Akbari et al. [19, 21] ; they obtained a packing density of φ = 0.7858 using Monte Carlo simulation of systems of 8000 tetrahedra compressed from random initial conditions. Their disordered packing contains a preponderance of pentagonal dipyramids, arranged randomly and thus differently than in the Chen structure, along with other motifs such as icosahedra.. Previously, all dense ordered packings proposed for tetrahedra were based on analytical construction (or numerical compressions thereof). Haji-Akbari et al. showed, using MC simulations of initially random systems containing up to nearly 22,000 tetrahedra, that at packing densities φ > 0.5 an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to φ = 0.8324 [21] . By numerically constructing and then compressing four unit cells of a periodic quasicrystal approximant with an 82-tetrahedron unit cell, they obtained a packing density as high as φ = 0.8503.. Motivated by a numerical search, Kallus, Elser and Gravel [22] found a one-parameter family of dimer cluster packings obtained via an analytical construction with a density of φ = 100 117 = 0.854700 . . ., exceeding the density of the quasicrystal approximant. Each dimer cluster contains two face-sharing tetrahedra, and is equivalent to a triangular dipyramid (or bipyramid). Two dimers comprise a single unit cell. Torquato . such that p, q, r span the common face. A negative dimer is related to a positive dimer by inversion. The positive dimer with vertices o, p, q, r, s centered at the origin is called +F 2 ; the corresponding negative dimer with vertices −o, −p, −q, −r, −s centered at the origin is called −F 2 . We have found the densest known packing of tetrahedra with density φ = 4000 4671. . This result was obtained as the optimal solution of a three-parameter family of dimer packings, which is a generalization of one-parameter and two-parameter families of packings recently reported with lower maximum densities. Interestingly, the densest packing is not the most symmetric one. Isobaric Monte Carlo simulations with variable cell shape starting from random initial conditions recover the same high packing density within 10 −5 for small systems containing 4, 8, 12, and 16 tetrahedra. We also discovered new candidates for densest packings with 3, 5, 6, and 10 tetrahedra per unit cell. The analytical and numerical results combined suggest that the packing density reported here could be the highest achievable, at least for small N .. The dimer structures are remarkable in the relative simplicity of the 4-tetrahedron unit cell as compared to the 82-tetrahedron unit cell of the quasicrystal approximant [21] , whose density is only slightly less than that of the densest dimer packing. The dodecagonal quasicrystal is the only ordered phase observed to form from random initial configurations of large collections of tetrahedra at moderate densities. It is thus interesting to note that for some certain values of N , when the small systems do not form the dimer lattice packing, they instead prefer clusters (motifs) present in the quasicrystal and its approximant, predominantly pentagonal dipyramids. This suggests that the two types of packings -the dimer crystal and the quasicrystal/approximant -may compete, raising interesting questions about the relative stability of the two very different structures at finite pressure. These questions will be explored in a forthcoming paper.. This appendix contains figures of the double dimer packings (Figures A1-A3) , the threedimensional restricted parameter space ( Figure A4) , and the volume contour function (figure A5), as well as visualizations of the internal symmetry of the restricted parameter space (Figures A6 and A7) . Figure A0 introduces the orientation of the axes for all following figures. We distinguish between the vector coordinate axes x, y, z and the packing coordinate axes u, v, w . In the former case, the viewpoints are 1, 1, 1 , 0, − cos ( Figure A6 ). In the latter case, the viewpoints are cos. ( Figures A4-A5 and A7 ). Figure A1 (4 pages) gives the notation and basic equations, and shows the nearest intersecting neighbors. The colors of the equations match the color of the respective neighbors. Since the double dimer packings are cluster transitive on dimers, we consider only the dimer at the origin. Its positive and negative neighbors are shown for three pairs of special packings at the optimal points, the central points, and the symmetric points in parameter space ( Figure A4 ). The pairs of packings are related by the symmetry T ( Figure A7 ). Notice that the sets of negative neighbors are always the same, whereas the sets of positive neighbors are different. The neighbors for the optimal packings are related by a reflection through the plane +2x − y − z = 0, the neighbors for the central packings by a reflection through the plane +2x − y − z = 0, and the neighbors for the symmetric packings by a reflection through the plane which contains ±a ± b. Figure A2 (2 pages) lists all 20 possible neighbors that can intersect with the dimer at the origin together with their linear incidence conditions. The entire figure is a floor plan diagram of the neighbors, ordered by layer. Notice how the positive layers and the negative layers complement each other. The even layers are related by symmetry, which is broken for the odd layers due to the presence of a small translation by the vector d. Figure A3 (2 pages) shows larger portions of the layers for the optimal packings. The entire figure is a floor plan diagram of the packing, ordered by layer. Notice the almost triangular/hexagonal planar basis mentioned in Remarks 2. We show both optimal packings for comparison to demonstrate the subtle differences. The symmetry T that maps the packings onto each other is a two-fold rotation around an axis running from bottom left to top right in the left columns (view along the 'almost' three-fold axis). Figure A4 (1 page) depicts the restricted three-dimensional parameter space of dense double dimer packings u, v, w together with special planes, lines, and points. The shape is a strongly deformed tetrahedron extended along the u-axis. It is bounded by four boundary planes. Packings with maximal (minimal) density are located on the maximal (minimal) lines ( Figure A5 ). The optimal plane contains the two optimal points, the origin, and the symmetric line.

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 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 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 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 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 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 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 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 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 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 B O O O O O O O O O O O O O O O O B 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


Mathematics of the Neural Response We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure between functions which we call the derived kernel. Based on a hierarchical architecture, we give a recursive definition of the neural response and associated derived kernel. The derived kernel can be used in a variety of application domains such as classification of images, strings of text and genomics data. The goal of this paper is to define a distance function on a space of images which reflects how humans see the images. The distance between two images corresponds to how similar they appear to an observer. Most learning algorithms critically depend on a suitably defined similarity measure, though the theory of learning so far provides no general rule to choose such a similarity measure [4, 5, 11, 19] . In practice, problem specific metrics are often used [16] . In this paper we propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The derived kernel is the inner product defined by the neural response and can be used as a similarity measure. The definition of neural response and derived kernel is based on a recursion which defines a hierarchy of local kernels, and can be interpreted as a multi-layer architecture where layers are associated with increasing spatial scales. At each layer (local) derived kernels are built by recursively pooling over previously defined local kernels. Here, pooling is accomplished by taking a max over a set of transformations. This model, while purely mathematical, has a key semantic component: a system of templates which link the mathematical development to real world problems. In the case of images, derived kernels consider sub-patches of images at intermediate layers and whole images at the last layer. Similarly, in the case of derived kernels defined on strings, kernels at some mth layer act on sub-strings. From a learning theory perspective the construction of the derived kernel amounts to an unsupervised learning step and the kernel can ultimately be used to solve supervised as well as unsupervised tasks. The motivating idea is that the unsupervised preprocessing will reduce the sample complexity of a corresponding supervised task.. The work in this paper sets the stage for further developments towards a theory of vision. One might consider especially two complementary directions, one empirical, the other mathematical. The empirical requires numerical experiments starting with databases coming from real world situations. The goal is to test (with various algorithmic parameters) how the similarity derived here is consistent with real world experience. In vision, to what extent does the mathematical similarity correspond to similarity in the way humans view images? In Sect. 6 we show the results of preliminary work towards this end. On the purely mathematical side, the problem is to examine how closely the output response characterizes the input. In other words, does the neural response discriminate well? In the case of strings, it is shown in Theorem 4.1 that if the architecture is rich enough and there are sufficient templates ("neurons") then indeed the answer is a sharp "Yes" (up to reversal and "checkerboard" patterns). We show under quite mild assumptions that the neural response is invariant under rotations, and for strings, is reversal invariant. In Sect. 5 we suggest that the Shannon entropy is a promising tool for obtaining a systematic picture. Note that in this unsupervised context, discrimination refers to the ability to distinguish images of distinct objects in the real world.. Our work seeks to establish a theoretical foundation for recent models designed on the basis of anatomical and physiological data describing the primate visual cortex. These models are beginning to quantitatively account for a host of novel data and to provide human-level performance on rapid categorization of complex imagery (see [13] [14] [15] and references therein). These efforts are the most recent examples of a family of biologically-inspired architectures, see for example [7, 10, 20] , and related computer vision systems [8, 18] . The hierarchical organization of such models-and of the cortex itself-remains a challenge for learning theory as most "learning algorithms", as described in [9] , correspond to one-layer architectures. In this paper, we attempt to formalize the basic hierarchy of computations underlying information processing in the visual cortex. Our hope is to ultimately achieve a theory that may explain why such models work as well as they do, and give computational reasons for the hierarchical organization of the cortex.. Some preliminary results appeared in [17] , whereas related developments can be found in [2] . In the Appendix we establish detailed connections with the model in [15] and identify a key difference with the model developed in this paper.. The paper is organized as follows. We begin by introducing the definitions of the neural response and derived kernel in Sect. 2. We study invariance properties of the neural response in Sect. 3 and analyze discrimination properties in a one-dimensional setting in Sect. 4. In Sect. 5 we suggest that Shannon entropy can be used to understand the discrimination properties of the neural response. Finally, we conclude with preliminary experiments in Sect. 6. In this section we discuss simulations in which derived kernels are compared to an L 2 pixel distance baseline in the context of a handwritten digit classification task. Given a small labeled set of images, we use the 1-nearest neighbor (1-NN) classification rule: an unlabeled test example is given the label of the closest training example under the specified distance.. Algorithm 1 Neural response algorithm.. An outline of this section is as follows: We compare a 3-layer architecture to a 2-layer architecture over a range of choices for the patch sizes u and v, and see that for the digit recognition task, there is an optimal architecture. We show that three layers can be better than two layers, and that both architectures improve upon the L 2 baseline. We then illustrate the behavior of the 3-layer derived kernel as compared to the baseline by presenting matrices of pairwise derived distances (as defined in (6)) and pairwise L 2 distances. The block structure that typifies these matrices argues graphically that the derived kernels are separating the different classes of images. Finally, we impose a range of artificial translations on the sets of train and test images and find that the derived kernels are robust to large translations while the L 2 distance deteriorates rapidly with even small translations.. In all experiments we have used Sq = 28 × 28 pixel grayscale images randomly selected from the MNIST dataset of handwritten digits [8] . We consider eight classes of images: 2s through 9s. The digits in this dataset include a small amount of natural translation, rotation, scaling, shearing and other deformations-as one might expect to find in a corpus containing the handwriting of human subjects. Our labeled image sets contain five examples per class, while the out-of-sample test sets contain 30 examples per class. Classification accuracies using the 1-NN classifier are averaged over 50 random test sets, holding the training and template sets fixed. As in the preceding mathematical analysis, the transformations H are restricted to translations.. The template sets are constructed by randomly extracting 500 image patches (of size u and/or v) from images which are not used in the train or test sets. For the digits dataset, templates of size 10 × 10 pixels are large enough to include semi-circles and distinct stroke intersections, while larger templates, closer to 20 × 20, are seen to include nearly full digits where more discriminative structure is present.. In Figs. 3 and 4 we show the effect of different patch size selections on classification accuracy. For this particular task, it is clear that the optimal size for patch u is 12 × 12 pixels for both two and three layer hierarchies. That accuracy levels off for large choices in the case of the 2-layer architecture suggests that the 2-layer derived kernel is approximating a simple local template matching strategy [6] . It is clear, however, from Fig. 4 that an additional layer can improve on such a strategy, and that further position invariance, in the form of eight pixels of translation (since v = 20 × 20 and Sq = 28 × 28) at the last stage, can boost performance. In the experiments that follow, we assume architectures that use the best patch sizes as determined by classification accuracy in Figs. 3 and 4: u = 12 × 12, v = 20 × 20. In practice, the patch size parameters can be chosen via cross validation or on a separate validation set distinct from the test set. Figure 5 illustrates graphically the discrimination ability of the derived kernels when applied to pairs of digits. On the left we show 3-layer derived distances, while the L 2 distances on the raw image intensities are provided for comparison on the right. Both matrices are symmetric. The derived distances are computed from derived The experiments discussed up to this point were conducted using a dataset of images that have been registered so that the digits appear approximately in the center of the visual field. Thus the increase in performance when going from two to three layers validates our assumption that objects particular to the task at hand are hierarchically organized, and can be decomposed into parts and parts of parts, and so on. A second aspect of the neural response architecture that warrants empirical confirmation is that of invariance to transformations accounted for in the hierarchy. In particular, translations.. To further explore the translation invariance of the derived kernel, we subjected the labeled and unlabeled sets of images to translations ranging from 0 to 10 pixels in one of eight randomly chosen directions. Figure 6 gives classification accuracies for each of the image translations in the case of 3-and 2-layer derived kernels as well as for the L 2 baseline. As would be expected, the derived kernels are better able to accommodate image translations than L 2 on the whole, and classification accuracy decays more gracefully in the derived kernel cases as we increase the size of the translation. In addition, the 3-layer derived kernel is seen to generally outperform the 2-layer derived kernel for translations up to approximately 20% of the field of view. For very large translations, however, a single layer remains more robust than the particular 2-layer architecture we have simulated. We suspect that this is because large translations cause portions of the digits to be clipped off the edge of the image, whereas templates used by two-layer architectures describe nearly all regions of a class of digits. Lack of a digit part could thus undermine the descriptive advantage of the 3-layer architecture over the 2-layer hierarchy.. On the whole the above experiments confirm that the derived kernels are robust to translations, and provide empirical evidence supporting the claim that the neural response includes mechanisms which can exploit the hierarchical structure of the physical world.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B 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 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 B 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


A general framework for statistical performance comparison of evolutionary computation algorithms This paper proposes a statistical methodology for comparing the performance of evolutionary computation algorithms. A two-fold sampling scheme for collecting performance data is introduced, and this data is assessed using a multiple hypothesis testing framework relying on a bootstrap resampling procedure. The proposed method offers a convenient, flexible, and reliable approach to comparing algorithms in a wide variety of applications. Evolutionary algorithms (EAs) [1, 2] are used to estimate the solution to difficult optimization problems. An EA's result is determined by a stochastic process with two sources of variation: the fitness of the input or initial population (used interchangeably), and the algorithm's improvements on this fitness produced via the mechanisms of selection, reproduction, and mutation. EAs are often hand-crafted to meet the requirements of a particular problem because no single optimization algorithm can solve all optimization problems competitively [3] . When alternative algorithms are proposed, their relative efficacies should be assessed. This paper seeks to provide a general methodology for comparing the performance of evolutionary algorithms based on statistical sampling and hypothesis testing.. In [4] , Christenssen and Wineberg explain the use of appropriate statistics in artificial intelligence and propose non-parametric tests to verify the data's output distribution. In [5] , Flexer proposes many guidelines for statistical evaluation of neural networks that can also be applied to evolutionary computation. Czarn et. al [6] discuss the use of the Analysis of Variance (ANOVA) in comparing the performance of EAs. However, such procedures rely upon distributional assumptions that are not necessarily valid and limit the class of performance metrics that can be used.. An EA's initial population consists of a set of starting values for the evolution process. Most previous EA performance comparisons have only considered results for a single initial population or even provided different inputs for each algorithm studied. Supplying different single inputs to each EA may result in a founder effect, in which a population's initial advantage is continually propagated to successive generations. Furthermore, relying upon a single input can at best determine the plausibility of preferring one candidate EA to another given suitable initial conditions. We can alleviate these issues by assessing relative performance over each of a representative sample of initial populations.. For each particular sampled input, performance differences can be assessed using a hypothesis test. Student's t-statistics [7] are commonly used to test the equality of two population means. However, the parametric t-test assumes that the data are normally distributed. If this assumption is not valid, the resulting inference may not be meaningful. Therefore, we require a more general and objective framework for statistical comparison of evolutionary computation algorithms. This paper proposes a two-fold sampling scheme to perform repeated EA trials at each of a representative sample of possible inputs. The candidate EAs' efficacies will be assessed in a multiple hypothesis testing framework that relies upon bootstrap resampling [8] to estimate the joint distribution of the test statistics. This methodology will establish a procedure for fair comparison of EAs that can be considered general in the following aspects: first, the results do not rely heavily on a single advantageous input. Second, the bootstrap-based testing procedure is applicable to any data-generating distribution and requires no a priori model assumptions. Finally, this methodology can be applied to essentially any function of the data collected, so the researcher is free to choose how performance should be evaluated.. The paper is organized as follows: Section 2 describes the two-fold sampling scheme for data collection. Section 3 introduces performance comparison in a multiple hypothesis testing framework. Section 4 shows how to use the bootstrap to estimate the test statistic's underlying distribution. Section 5 introduces a variety of multiple testing procedures. Section 6 provides an example comparing the performance of two EAs seeking to minimize Ackley's function. Section 7 discusses further applications of statistics in EAs and concludes the paper. Although this paper's methodology provides a general approach to EA performance comparison, the reader should be cautioned that issues of sample size cannot be neglected. In particular, the bootstrap approximation of (4)'s joint distribution grows more accurate as the values B and n a increase. In practice, researchers may choose to collect as much data as a pre-specified time limit will allow. Data-adaptive study designs may also be implemented to halt data collection once a pre-specified level of statistical power is achieved. The framework proposed in this paper allows the researcher to choose the parameter of interest in an EA comparison. When parameters other than the expected optimum fitness are used (such as the median, 75 th percentile, or other quantile estimates), our methodology is applicable provided that the necessary data are collected and appropriate estimating statistics (1), null hypotheses (3), and test statistics (4) are chosen. In crafting an EA for a particular optimization problem, this paper's methodology can be used iteratively to select the best among a set of candidate parameter values for quantities such as the mutation rate, population size, and selection proportion. When three or more EAs are simultaneously compared, a null hypothesis of equality in means may be tested using F statistics. If competing algorithms draw from different input sets, then testing average results from representative input samples in a single (M=1) test may be considered. When the input sets are identical, an alternative to the approach of this paper may choose to average all trials in a single hypothesis test provided that all inputs are i.i.d. The choice of which approach to use is philosophical: this paper assumes that EAs should be compared using the same input sample. In this setting, the parameter of interest is the expected performance given the initial population. This allows the algorithm to be assessed solely on its own merits without any possibility of a founder effect. However, if one views the input generation and resulting evolution as inextricably linked in the same algorithm, then a single hypothesis testing framework may be more appropriate, and this paper's methodology is otherwise applicable. In this scenario, the parameter of interest shifts to the unconditional expectation of performance, and each EA should generate inputs independently for comparison trials. Though a single test may simplify the interpretation of performance differences, this approach is not applicable when inputs are dependently generated and lacks the appeal of direct performance comparison on the same trial inputs.. The researcher may also wish to compare EAs as a function of time by collecting data at regular generational intervals. Displaying performance curves and confidence regions graphically may allow one to quickly determine decision criteria and search for clues about an algorithm's rate of convergence and asymptotic result. Finally, an EA's efficacy should be considered in terms of both performance and computational complexity. Researchers may consider performing a comparison in which each candidate algorithm is allowed to run for the same amount of time instead of the same number of generations to satisfy both objectives simultaneously.

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


Detecting anomalous records in categorical datasets We consider the problem of detecting anomalies in high arity categorical datasets. In most applications, anomalies are defined as data points that are 'abnormal'. Quite often we have access to data which consists mostly of normal records, along with a small percentage of unlabelled anomalous records. We are interested in the problem of unsupervised anomaly detection, where we use the unlabelled data for training, and detect records that do not follow the definition of normality.A standard approach is to create a model of normal data, and compare test records against it. A probabilistic approach builds a likelihood model from the training data. Records are tested for anomalousness based on the complete record likelihood given the probability model. For categorical attributes, bayes nets give a standard representation of the likelihood. While this approach is good at finding outliers in the dataset, it often tends to detect records with attribute values that are rare. Sometimes, just detecting rare values of an attribute is not desired and such outliers are not considered as anomalies in that context. We present an alternative definition of anomalies, and propose an approach of comparing against marginal distributions of attribute subsets. We show that this is a more meaningful way of detecting anomalies, and has a better performance over semi-synthetic as well as real world datasets. With the ever increasing amount of data being collected universally, it gets more important and challenging to spot This work is funded in part by CDC under award 8-R01-HK000020-02 and by NSF under award IIS-0325581. unusual or unexpected observations. Anomaly detection aims to address this issue from a statistical datamining framework.. Anomalies can be defined as anything that is 'different' from 'normal' behavior. Hence, we need to first define 'normality'. Usually this is specified as a probability model from which observations are assumed to be drawn. We also need a similarity measure to compare the observations with respect to the model.. Detecting anomalies in observations is important in many different respects. A traditional form of anomaly detection is in industrial process control. Time-series data from various sensors are monitored to detect out of control processes. In most applications, we are interested in the detection of emergence of new phenomena unexplained by the previous model. For example, in astronomical datasets, astronomers want to detect new interesting space objects. In applications such as bio-surveillance and customs monitoring, detecting suspicious activity in high dimensional data is the goal of anomaly detection. Anomaly detection can also be used to automatically detect data entry errors, which can then be corrected.. However, forming a general framework for anomaly detection is a difficult challenge as the definition of normality is typically very domain specific. This has led to independent efforts for various domains.. First, we give a summary of the related work. We then present the problem statement, along with our algorithms for anomaly detection. We show ways of speeding up the computation and making it more memory efficient. This is followed by the experimental setup, where we describe the datasets used, and the evaluation procedure. The results of our algorithms on the datasets are presented next. We conclude with a discussion of possible extensions of the current work. The current work focuses on finding single records that are anomalous. Sometimes in real world applications we are more interested in detecting groups of unusual records that deviate from the norm, rather than detecting the records separately. For example, in astronomical datasets, we might be more interested in an unusual phenomenon if it keeps repeating at some interval. Just observing one such instance may not be significant, as it could be attributed to some measurement error. In biosurveillance, we might be interested in the emergence of a new disease by detecting a group of unusual but similar cases. It is especially relevant in network security monitoring, as we can detect a new pattern of user behavior from a group of records. This can signal possible malicious behavior.. An important challenge here is to define what can be considered as a group. We need to specify a similarity measure, and group records on the basis of it. If the data has temporal and/or spatial components, they provide a natural measure for grouping. For temporal analysis, we propose to group records on the basis of a temporal unit such as a day. We can extend the method of comparison of marginal and conditional probability distributions of all records in the current day, to the historical data. Similarly, for a spatial analysis, grouping can be predetermined such as by zip code or area. It can also be computed dynamically similar to spatial scan. Apart from this, we can also use the association based dissimilarity measures such as methods presented in [20, 17] for grouping records.. Another possible improvement is the way we deal with real valued attributes. Since we deal with actual probability values P rather than probability densities p, all the real values are discretized to perform the analysis. But by discretizing the values we lose some information, such as the ordering of values. While estimating counts to determine the P values, we can borrow information from neighboring bins in case of discretized attributes.. Currently, we have a fixed number of levels for discretization. It is conceivable that different real attributes have varying characteristics, and discretizing into the same number of levels is not the best solution. We can use different clustering techniques to determine appropriate levels. Anomaly detection applied to network intrusion detection has been an active area of research since it was proposed by Denning [28] . Traditional anomaly detection approaches build models of normal data and detect deviations from the normal model in observed data. A survey of these techniques is given in [31] . One approach is to use sequence analysis to determine anomalies. A method of modeling normal sequences using look ahead pairs and contiguous sequences is presented in [16] , and a statistical method to determine frequent sequences in intrusion data is presented in [15] . Lee et al. [21] uses a decision tree model over normal data and Ghosh et al. [13] uses a neural network to obtain the model. Eskin [12] uses a probability distribution model from the training data to determine anomalous data. They use a mixture model to explain the presence of anomalies. A clustering based approach to detecting anomalies in a dataset is used in [22] and [11] . One-class SVMs [23, 14] and Genetic Algorithms [29] have also been used to classify anomalies in this context.. Anomaly detection is also commonly applied in time series data to detect unusual fluctuations compared to past data points [4, 18, 7, 26] . Another area of considerable recent interest is spatial anomaly detection [19] .. The methods described so far apply to real valued data or work in a supervised setting when we have labeled training data. We now describe methods that apply to the problem of interest, i.e., on categorical datasets in an unsupervised setting. Computing the qval(at) of an attribute set A in test record t is somewhat more complicated than calculating the r-value. To calculate qval(at), we not only need to know C(at), but also the counts for all other possible values ai of A such that, C(ai) ≤ C(at). When dealing with composite attributes, the number of possible values it can have becomes exponentially large. Even if all the counts are cached, going through each of them for every test becomes prohibitive.. Instead, for every composite attribute A, we store the histogram h of the number of times different values occur in the training dataset. For example we precompute the fact that A has h(1) values occurring only once, h(2) values occurring twice and in general, h(i) values occurring i times. When testing attribute set A in record t, we compute C(at), and compare that to the precomputed histogram. We compute the quantity Crarer = P i≤C(at) i * h(i). Normalizing with respect to the number of data-points N , gives the desired qval(at). We still need to get the count C(at), and unlike the conditional method, we are especially interested in rare values. Hence, we cannot reuse the AD Tree constructed for the conditional method. We construct another AD Tree without any reduction of arity from the original dataset. We call this the marginal AD Tree. We use a bigger leaf-list size to keep the size of the tree manageable [24] .. Note that all the information in the conditional AD Tree is also contained in the marginal AD Tree. But, we still maintain the conditional AD Tree separately as it is faster to query from the smaller tree for the conditional method.

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


Modeling RPS and Evaluating Video Repair With VQM Abstract-Reference Picture Selection (RPS) is a promising repair technique in lossy networks for delay-sensitive video, whereby the video encoder uses one of several previous frames as a reference frame for predictive encoding. RPS can operate in two different modes: an optimistic policy that uses negative acknowledgements (NACKs) and a more conservative policy that relies upon positive acknowledgements (ACKs). This paper compares RPS NACK and RPS ACK under various network conditions and video contents using two analytical models. The two models characterize RPS NACK and RPS ACK by incorporating the impact of reference distance on video quality, prediction dependency among video frames and Group of Pictures (GOP) length. Given packet-loss rate, round-trip time and capacity constraints, the models predict average video quality for videos with RPS ACK and RPS NACK using the Video Quality Metric (VQM). Using these two models, a series of experiments are conducted to investigate RPS performance under various conditions. The insights derived from our models can help determine appropriate choices for RPS NACK and RPS ACK under various network conditions and video contents.

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Stratified Analysis of AOL Query Log Characterizing user's intent and behaviour while using a retrieval information tool (e.g a search engine) is a key question on web research, as it hold the keys to know how the users interact, what they are expecting and how we can provide them information in the most beneficial way. Previous research has focused on identifying the average characteristics of user interactions. This paper proposes a stratified method for analyzing query logs that groups queries and sessions according to their hit frequency and analyzes the characteristics of each group in order to find how representative the average values are. Findings show that behaviours typically associated with the average user do not fit in most of the aforementioned groups. An accepted idea in the HCI area is that 'there is no average user ' Krug [15] . This is a key concept that expresses the complexity inherent in the diversity of users that could access a tool in the Web environment. This complexity becomes crucial in search engines as they are one of the main access point to information retrieval in the Web 1 and they are used by many different users.. Usability issues in search engines do not only concern designing interfaces but also what are the user needs, how he express them and how he reacts to the retrieved information. The diversity of user's profile makes very important to take care about how the user behaves in order to implement better information retrieval web tools. Questions about behaviour (How many searches does a user execute? How many words does he use to express his goal? How many results does he visit?) go beyond the interfaces and deserve answers from research community which has already addressed them.. Beyond analyzing the behavior of a hypothetical average user whose correspondence with actual users in all situation, there are some research initiatives that tries to group queries and user interactions according to some criteria so different access patterns can be identified and intentions can be inferred, This paper proposes a new criteria for grouping queries and user's interactions with search engines analyzing different measures traditionally extracted in previous research papers in order to check if this criteria allow us to separate different use cases of search engines with different characteristics.

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Control-flow checking by software signatures ABSTRACTThis paper presents a new signature monitoring technique called Control Transient or permanent faults introduced in a computer system during runtime can cause an incorrect sequence of instruction execution in the program and may cause control flow errors. If the system does not perform some run-time checking, the erroneous output may not be detected and serious damage may result. Therefore, it is important to monitor the program to detect any abnormality in the control flow or other error, and to take appropriate actions to avoid any incorrect output.. Signature monitoring and pure software methods have been proposed to check the control flow of a computer system. Signature monitoring is a signature analysis method in which a signature associated with a block of instructions (one or more nodes in the control flow graph) is calculated and saved somewhere during compile time; then, the same signature is generated during run time and compared with the saved one. Signatures are assigned arbitrarily (assigned signature) or derived from the binary code or the address of the instructions (derived signature). Structural Integrity Checking (SIC) [Lu 82 ] employs an assigned signature method while Path Signature Analysis (PSA) [Namjoo 82 ], Signatured Instruction Streams (SIS) [Shen 83 ], Asynchronous SIS (ASIS) [Eifert 84 ], Continuous Signature Monitoring (CSM) [Wilken 89 ], [Wilken 90 ], extended-precision checksum method [Saxena 90] , and On-line Signature Learning and Checking (OSLC) [Madeira 92 ] all use derived signatures.. In many signature monitoring techniques, dedicated hardware is used to calculate the run-time signatures and to compare them with the saved signatures. A watchdog processor is proposed for this purpose in [Lu 80] and [Mahmood 85 ]. On the other hand, when the hardware is fixed and cannot be changed, a software method that does not need any extra hardware has to be developed. Examples of software methods are assertions [Andrews 79], [Ersoz 85 ], watchdog task [Ersoz 85 ], Block Signature Self-Checking (BSSC) [Miremadi 92 Checking (VASC) has been proposed and evaluated in [Furtado 96] , in which the comparison of the ratio between control flow errors and data errors in RISC and CISC processors are reported. This paper presents the Control Flow Checking by Software Signatures (CFCSS), a new signature monitoring technique for control flow checking using a pure software approach.. CFCSS employs a run-time signature G and a run-time adjusting signature D to check the control flow of a program. Algorithm A detects the inter-block control flow errors by assigning a different signature to each node in the program graph. The XOR operation is used both to embed the signatures in the program and to check for them during run-time. In addition, the aliasing problem that may cause an undetectable illegal branch is discussed and the technique to avoid it is presented. Furthermore, Algorithm B is proposed to reduce the overhead by postponing the comparison of signatures.. The distinctive feature of the CFCSS over previous signature monitoring techniques is that the CFCSS needs no dedicated hardware such as a watchdog processor for control flow checking. Watchdog task in multi-tasking environment also needs no extra hardware but the advantage of the CFCSS over it is that CFCSS can be used even when the operating system does not support multi-tasking. If the hardware is fixed or a system is to be built using commercial-off-the-shelf components to keep the costs low, and control flow checking is needed, CFCSS can be used to enhance the reliability of the system.

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A Framework for Realistic and Systematic Multicast Performance Evaluation Previous multicast research often makes commonly accepted but unverified assumptions on network topologies and group member distribution in simulation studies. In this paper, we propose a framework to systematically evaluate multicast performance for different protocols. We identify a series of metrics, and carry out extensive simulation studies on these metrics with different topological models and group member distributions for three case studies. Our simulation results indicate that realistic topology and group membership models are crucial to accurate multicast performance evaluation. These results can provide guidance for multicast researchers to perform realistic simulations, and facilitate the design and development of multicast protocols. In this paper, we propose a realistic and systematic performance evaluation framework. Our work can be summarized as follows:. • We identify a wide variety of multicast performance metrics.. • We investigate a variety of real and generated network topologies at both AS-level and router-level (especially those produced by power-law topology generators), and examine their impact on the performance of various multicast protocols.. • We propose a generic group participation probability model (truncated normal distribution), which contains the commonly accepted uniform membership model, evaluate multicast performance with this membership model and explore how different parameters affect various multicast metrics.. • We conduct three case studies, namely, multicast routing protocols CBT vs.. SBT, multicast state scalability, and application layer multicast, for which we focus on the topology sensitivity and group distribution sensitivity of multicast performance metrics.. Through thorough analysis and extensive simulation studies, we draw the following conclusions:. • Unrealistic topology models can significantly affect multicast performance of various protocols (including application layer multicast).. • We identify the GLP topology generator (for AS-level topology) and the Transit-Stub topology generator (for router-level topology) to be realistic in terms of multicast performance metrics that we investigated.. • Many multicast performance metrics, such as delay ratio, multicast efficiency, and state scalability are sensitive to group size.. • For protocols/approaches exploiting multiple group properties (such as intergroup aggregation), their performance can be significantly affected by the group membership distribution. • Therefore, simulation studies on multicast should appropriately gear simulation environments with realistic topology models and group models that best match their target applications.. Future Work While we have presented a realistic and systematic multicast evaluation framework and investigated many typical metrics and representative case studies, there are still many issues to be addressed in future work. First, we would incorporate other spatial group properties (such as clustering and group participation correlation) and temporal group properties into our framework. Second, we would like to investigate the impact of small-world property of topology graphs on multicast performance. We also want to use a more detailed network topology model to take link asymmetry, link capacity and delay, and background traffic into consideration. Third, more case studies, such as QoS multicast and reliable multicast, obviously deserve further exploration. In this section, we discuss related work in areas of topology generators, group member models, and multicast performance evaluation. These results have greatly influenced our work.

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Multigrid preconditioned conjugate-gradient solver for mixed finite-element method The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugategradient algorithm is used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the hydraulic conductivity tensor. Lowest-order Raviart-Thomas mixed finite-element approximations to second-order elliptic PDEs with variable hydraulic conductivity coefficients are recognized as important methods for modeling subsurface flow in heterogeneous media [12, 14, 16] . In this paper, we consider the three-dimensional second-order elliptic problem given by. where is a bounded domain in 3 , p is a pressure or hydraulic head potential, n denotes the unit outward normal vector, f is a sink/source term, and q and g are given functions. K is a bounded, symmetric, and uniformly positive-definite matrix-valued function related to the intrinsic permeability of the porous media. The flux v in subsurface flow models, given by the Darcy relation. is often of primary interest. The use of a mixed method, which approximates both the hydraulic head and flux simultaneously, is desirable in the presence of rough coefficients since traditional methods, using post-processing of primal hydraulic head solutions, generally lead to loss of accuracy when estimating the flux vector. Mixed methods also have the advantage of conserving mass locally, which can be critical in modeling subsurface flow. Using the Darcy relation given by Eq. 2, problem 1 can be rewritten as a system of first-order equations. When discretized using the Raviart-Thomas finiteelement spaces [25] on triangular or hexahedral elements, an indefinite system of equations results with the form given by. The matrix M is real symmetric positive definite and, using the Raviart-Thomas finite element space, the discretization satisfies the inf-sup condition [10] so that B T has full column rank. In this paper, a lowest-order Raviart-Thomas finite element discretization on hexahedral grids is assumed.. Problem 3 can be transformed symbolically into coupled symmetric positive-definite matrix equations, or a Schur complement problem, using block Gauss elimination:. where S M = BM −1 B T . A solution of this system of equations is possible but difficult; S M must be inverted to solve for p, and subsequently, M must be inverted to solve for v. The system involving S M can be solved using the preconditioned conjugate gradient (PCG) method [17] , but requires the evaluation of M −1 for each PCG iteration and an effective preconditioner. These difficulties are overcome by approximating M with a positive-definite matrix Q M , which is relatively easy to invert, and approximating the inverse of the resulting Schur complement S Q M = BQ −1

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Efficient algorithms for minimizing tree pattern queries We consider the problem of minimizing tree pattern queries (TPQ) that arise in XML and in LDAP-style network directories. In [Minimization of Tree Pattern Queries, Proc. ACM SIGMOD Intl. Conf. Management of Data, 2001, pp. 497-508], Amer-Yahia, Cho, Lakshmanan and Srivastava presented an O(n 4) algorithm for minimizing TPQs in the absence of integrity constraints (Case 1); n is the number of nodes in the query. Then they considered the problem of minimizing TPQs in the presence of three kinds of integrity constraints: required-child, required-descendant and subtype (Case 2). They presented an O(n 6) algorithm for minimizing TPQs in the presence of only required-child and required-descendant constraints (i.e., no subtypes allowed; Case 3). We present O(n2), O(n 4) and O(n 2) algorithms for minimizing TPQs in these three cases, respectively, based on the concept of graph simulation. We believe that our O(n 2) algorithms for Cases 1 and 3 axe runtime optimal. In XML and in LDAP-style network directories, data is represented as a tree; associated with each node of the tree is an element type from a finite alphabet E. In XML (see [1] ), each node corresponds to an XML element; the children of a node are ordered from left to right, and represent the content (i.e., list of subelements) of that element. In LDAP directories (see [13] ), the children of a node axe not ordered; the paxent-child edges represent hierarchical information.. XML queries in languages such as XPath [24] , XQuery [25] , XML-QL [8] and Quilt [6] use tree patterns to extract relevant portions from the input database. LDAP directory queries [14] use tree patterns to specify certain structural relationships (child, descendant, etc.) between the desired entries. A tree pattern query that we consider in this paper, 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 flail 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. In any directed acyclic graph (dag), a node v is said to be a descendant of a node u if there exists a path (sequence of edges) from u to v. In the case of a TPQ, this path could consist of any sequence of c-edges and/or d-edges.. An embedding of a TPQ Q into a tree database db is a mapping ~ : Q ~ db, from the nodes of Q to the nodes of db, that satisfies the following conditions:. 1. Preserve node types: For each node u E Q, u and fl (u) are of the same type. Note that an embedding could map several nodes of the query (of the same type) to the same node of the database.

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Inference and Risk Measurement with the Pari-Mutuel Model We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assessing an upper probability from a precise one. We discuss a naive extension of the PMM considered in insurance, compare the PMM with a related model, the Total Variation Model, and generalize the natural extension of the PMM introduced by P. Walley and other pertained formulae. The results are subsequently given a risk measurement interpretation: in particular it is shown that a known risk measure, Tail Value at Risk (TVaR), is derived from the PMM, and a coherent risk measure more general than TVaR from its imprecise version. We analyze further the conditions for coherence of a related risk measure, Conditional Tail Expectation. Conditioning with the PMM is investigated too, computing its natural extension, characterising its dilation and studying the weaker concept of imprecision increase. The pari-mutuel model (PMM) formalizes a very intuitive and therefore widely used method of assigning an upper probability starting from a precise probability. To introduce it, consider, following [3] , a probability P for event A as a fair price for a bet which returns 1 unit to the bettor if A is true, 0 units if A is false, i.e. returns the indicator I A of A. The bettor's gain is G = I A −P (A), while that of his opponent, House, is −G = G H = P (A)−I A .. In most real-world betting schemes House is unwilling to accept such a game, which is fair to both counterparts (the expectations E(G H ), E(G) are 0), but asks for a positive gain expectation. It is so when House is a bookmaker, an insurance company, the organizer of a lottery, and so on. A way to achieve this goal is to raise the bettor's price without altering his reward, and a naive method multiplies P by a constant greater than 1, say 1+δ, where δ > 0 is a loading constant. The bettor pays P (A) = (1+δ)P (A), while the gain for House is now G H = (1 + δ)P (A) − I A . Alternatively, House might ask the same price to pay a reduced reward (1 − τ )I A , where 0 < τ < 1 is interpreted as a commission, or also a taxation. This originates a gain G * H = P (A) − (1 − τ )I A = (1 − τ )(. . Thus, up to a scaling factor, the two methods are equivalent if τ = δ 1+δ. ; the latter is formally more adherent to common betting systems, called in fact pari-mutuel systems. In the theory of imprecise probabilities, P is an upper probability, but a slight adjustment to P is necessary to achieve the customarily adopted consistency notion of coherence, recalled in Section 2. In fact, Walley [16] terms pari-mutuel model the upper probability P (A) = min{(1 + δ)P (A), 1}.. (. Intuitively, the correction is needed: when P (A) > 1 1+δ. , it is G H > 0 in the naive method, i.e. a bettor suffers from a sure loss no matter whether A is true or false. This paper investigates further the pari-mutuel model, extending the analyses in [13, 16] . Preliminary issues are recalled in Section 2, very concisely in general, more extensively as for 2-monotone and 2-alternating previsions, since the upper probability P in (1) is 2-alternating. In Section 3 we discuss extensions of the PMM and compare it with a similar model. First, we consider alternative expressions for the natural extension E(X) of P , defined on a field A, to any A-measurable gamble X. These expressions were stated in [16] , but we make a more detailed analysis of the conditions ensur-ing that E(X) is equal to a certain conditional prevision (P (X|X > x τ )), which has a risk measurement interpretation. In Section 3.1 we restrict to non-negative gambles and compare the natural extension E with the naive extension P N (X) = min{(1 + δ)P (X), sup X}, showing that quite often P N is not coherent. The motivation for this work is that P N is a premium in insurance, although with different premises: the starting point is not the PMM but a set of non-negative gambles. In Section 3.2 we compare the PMM with the Total Variation Model (TVM) [8] and compute the natural extension of the TVM to A-measurable gambles. In Section 3.3 we generalize Walley's approach, obtaining a formula for E(X) when the PMM is given on a lattice of events and X is not necessarily measurable: this makes it possible to evaluate E(X) for all X defined on a suitable reference partition while assessing a probability on a relatively small set of events (this and other concepts are illustrated in Example 2).. These results have an interesting and, to the best of our knowledge, so far not considered interpretation in the realm of risk measurement. This is the main topic of Section 4, where the natural extension of the PMM defined on a field is shown to correspond to a coherent risk measure, called Tail Valueat-Risk or TVaR (in [5] ; other authors may use a different terminology). When the PMM is defined on a lattice, we obtain a generalization of TVaR (not discussed in the risk literature), which replaces precise with imprecise uncertainty measures; we name it ITVaR. Thus the PMM supplies a motivation for introducing 'imprecise' risk measures: one of them, ITVaR, is the natural extension of a PMM assigned on a lattice. Conditioning the PMM defined on a field is discussed in Section 5. We specialize general formulae for the natural extension of 2-alternating and 2-monotone probabilities to the case of the PMM and explore the effect on them of dilation and of a weaker phenomenon, imprecision increase. We characterise dilation and give sufficient conditions for dilation or imprecision increase in Section 5.1. Further, operational conditions for the most relevant cases (when the commission τ is 'low' or event A is 'rare') are given in Section 5.2. Ideas about varying the extent of dilation or imprecision increase are outlined in Section 5.3. Section 6 concludes the paper. The proofs of the main results are given in the Appendix. The pari-mutuel model represents a simple and natural way of eliciting upper and lower probabilities, and can be extended in more directions, thanks to the availability of standard procedures for 2-monotone and 2-alternating previsions. We computed explicitly its natural extension E starting from a PMM assignment on a lattice of events, generalizing the approach in [16] , which is anyway discussed, focusing on comparing the different formulae available for E. While a naive extension, considered in insurance premium pricing, does not seem to be a valuable alternative to the natural extension, being generally not coherent, the various formulae for the natural extension have a notable meaning in risk measurement. In fact, they correspond to known measures of risk or generalize them. We discussed also how to use the natural extension when conditioning, characterising dilation and investigating imprecision increase for the PMM.. A tempting new direction would, in a sense, merge our analysis in the conditional and unconditional framework, studying the natural extension to conditional gambles. Here a difficulty arises: available generalizations of equations (21), studied in [12] , are lower/upper bounds for the natural extension and might not be attained, even when P is 2-alternating. In other words, the available procedures seem to give weaker results.. This and the considerations in Section 5.3 on how to limit dilation or imprecision increase might motivate investigating coherent extensions of the PMM alternative to the natural extension.

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Initial observations on economics, pricing, and penetration of the internet of things market One expectation about the future Internet is the participation of billions of sensor nodes, integrating the physical with the digital world. This Internet of Things can offer new and enhanced services and applications based on knowledge about the environment and the entities within. Millions of micro-providers could come into existence, forming a highly fragmented market place with new business opportunities to offer commercial services. In the related field of Internet and Telecommunication services, the design of markets and pricing schemes has been a vital research area in itself. We discuss how these findings can be transferred to the Internet of Things. Both the appropriate market structure and corresponding pricing schemes need to be well understood to enable a commercial success of sensor-based services. We show some steps that an evolutionary establishment of this market might have to take. Wireless sensor network (WSN) islands for the civilian sector have reached a stage in which they can actually be deployed outside research labs and provide useful information (see, e.g., [6] ). Current sensor networks are typically used by their operators for their own purposes only, but a change is envisioned: End-users shall be enabled to request environmental data from various hot-spots all over the world. This is based on the vision that millions of micro-providers will deploy and manage sensor network islands locally and, in a next step, open them to the public. A trading service in the Internet-operated by a search engine provider or a service provider purely dedicated to the composition and trading of sensor services-will then ensure that requests for sensor information will be sent to the right WSN island. This trading service can also take care of payments, thus forming the basis for a new market. We will refer to this envisioned market as the Internet of Things (IoT) Market. A research project actually considering such a vision is the European project SENSEI. 1 . While technical challenges are currently being addressed in research, the economic and business aspects of establishing the IoT market leave some more uncertainties:

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Preserving quality of service guarantees in spite of flow aggregation Abstract-We investigate how quality of service may be guaranteed to a flow of packets in the presence of flow aggregation. For efficiency, multiple flows, known as the constituent flows, are merged together resulting in a single aggregate flow. After the network node where the aggregation occurs, packet schedulers are aware of the aggregate flow, but are unaware of its constituent flows. In spite of this, we show that quality of service may be guaranteed to the constituent flows, provided the aggregation is performed fairly. When the delay bound of a flow is de-coupled from the flow's reserved rate, flow aggregation preserves the delay bound. When the delay bound of a flow is coupled to the flow's reserved rate, flow aggregation preserves, and in some cases improves, the delay bound. Consider the problem of transferring packets from a realtime application across a computer network. This problem has been studied extensively, resulting in the development of guaranteed-rate schedulers. That is, scheduling protocols which guarantee a minimum bandwidth and a maximum packet delay to each application. Examples of these protocols may be found in [29] [30] .. In guaranteed-rate schedulers, resources, such as bandwidth and buffer space, are reserved for each application along the entire path to its destination. If not enough resources are available, the application's request for a network connection is denied. Due to the reservation of resources, the network can provide service guarantees to each application, provided the rate of the application does not exceed the agreed-upon rate. These service guarantees are necessary for real-time applications, such as interactive audio and video [14] .. Let us denote by a flow the sequence of packets generated by a single application. In this paper, we investigate the effects of aggregating multiple flows, known as the constituent flows, into a single aggregate flow. Once the aggregation is done, the remaining schedulers along the path have no knowledge of the constituent flows of the aggregate flow. They simply treat the aggregate flow as a single flow whose rate is the sum of the reserved rates of the constituent flows.. One of many possible implementations of flow aggregation is a virtual path in virtual circuit networks [26] . Mul- tiple virtual circuits may be combined into a single virtual path. Schedulers are aware of the virtual path, but are unaware of the virtual circuits constituting the virtual path. Thus, a virtual circuit may be viewed as a constituent flow, and a virtual path may be viewed as an aggregate flow.. The purpose of flow aggregation is to improve the efficiency of the schedulers and to simplify the management of flows. For example, buffer management is simplified, because only one queue per aggregate flow is required, rather than one queue per constituent flow. Scheduling efficiency is improved, because the number of flows is reduced. Finally, rerouting an existing aggregate flow in the event of a failed channel is much more efficient than rerouting each of the individual constituent flows.. In this paper, we examine if it is possible to provide quality of service guarantees to the constituent flows. In particular, we consider end-to-end delay guarantees. We show that, if the aggregation of flows is performed fairly, then an upper bound on end-to-end delay is guaranteed to the constituent flows, even though schedulers are unaware of these flows. If the per-hop delay is de-coupled from the reserved rate, we show that the end-to-end delay with flow aggregation is similar to the end-to-end delay without flow aggregation. In addition, if the per-hop delay is coupled with the reserved rate, we show that the end-to-end delay with flow aggregation may be lower than the end-to-end delay without flow aggregation.. The paper is organized as follows. In Section 2, we present the definition of packet deadlines and the type of packet schedulers we will be using. In Section 3, we present our definition of flow aggregation and flow aggregators. In Section 4, we present the properties necessary for a flow aggregator to preserve the quality of service to its input flows, and in Section 5, we present two possible implementations of aggregators. In Section 6, we show how flow aggregation may be applied to a core network. Finally, in Section 7, we present related work and concluding remarks.

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