Local Dimensionality Reduction for Non-Parametric Regression Abstract Locally-weighted regression is a computationally-efficient technique for non-linear regression. However, for high-dimensional data, this technique becomes numerically brittle and computationally too expensive if many local models need to be maintained simultaneously. Thus, local linear dimensionality reduction combined with locally-weighted regression seems to be a promising solution. In this context, we review linear dimensionalityreduction methods, compare their performance on non-parametric locally-linear regression, and discuss their ability to extend to incremental learning. The considered methods belong to the following three groups: (1) reducing dimensionality only on the input data, (2) modeling the joint input-output data distribution, and (3) optimizing the correlation between projection directions and output data. Group 1 contains principal component regression (PCR); group 2 contains principal component analysis (PCA) in joint input and output space, factor analysis, and probabilistic PCA; and group 3 contains reduced rank regression (RRR) and partial least squares (PLS) regression. Among the tested methods, only group 3 managed to achieve robust performance even for a non-optimal number of components (factors or projection directions). In contrast, group 1 and 2 failed for fewer components since these methods rely on the correct estimate of the true intrinsic dimensionality. In group 3, PLS is the only method for which a computationally-efficient incremental implementation exists. A brief summary of the presented dimensionality-reduction methods is presented in Table 1 . This table further shows the computational complexity of each method, for training and testing, and, if available, for the incremental version of the algorithm. Only the dominant complexity term is shown, assuming n > d > k. For linear regression, dimensionality-reduction methods that maximize the correlation between projection directions and output data do better than methods that model only the variance of the data distribution. The drawback of these latter methods is that they need to assume that the number of factors or components is at least as high as the intrinsic dimensionality of the data-this makes incremental addition of components problematic. Ideally, one projection direction would be sufficient, namely the direction in input space that maximally correlates with the output. The computation of this direction, however, involves calculations which are at least as expensive as the ordinary weighted least squares on the full-dimensional data set.. Fortunately, a locally weighted, incremental reformulation of partial least squares (PLS) provides an ideal dimensionality-reduction technique: PLS does well with only a few projection directions (it works with only one projection in the special case of spherically distributed input data), and since the projection directions are orthogonal, additional relevant dimensions can be added without relearning existing projection directions, e.g., as in use with online regression algorithms like locally-weighted projection regression [48] . We compared the regression performance of non-parametric dimensionality-reduction methods on synthetic and real-world data. The synthetic data were constructed such that they were restricted to a lower q-dimensional subspace. Our basic finding was that if the number k of factors, components, or projection directions is smaller than q then methods that are based on maximizing the correlation between projection directions and output (PLS and RRR) do better than methods that model the distribution of data in joint space (FA, PPCA, and PCAJ), which in turn do better than methods that reduce the dimensionality without taking the regression target into account (PCR). This finding is significant in view of the practical requirements of incrementally adding projection directions in a real-world application without significant prior knowledge of the true intrinsic dimensionality.. The data generation mechanism fulfilled all assumptions of factor analysis except for the true intrinsic dimensionality q. Thus, the breakdown of FA for incorrect number of factors demonstrates its brittleness with respect to the match between k and q; making it inherently difficult to use for a constructive, incremental algorithm. In contrast, PLS shows promising results with graceful degradation for k < q.. As the comparison with RRR shows, the dimensionality could be even reduced to one by finding a direction whose projections maximally correlate with the output-this is nothing but the direction of the true regression vector β. However, as mentioned earlier, computationally, RRR is equivalent to doing ordinary least squares on the full data set, i.e., the full covariance matrix needs to be inverted, and there are no computationally-attractive incremental implementations. Thus, PLS appears as the best compromise: the method maximizes only the correlation in the special case of spherically-distributed input data, but it provides an efficient implementation, which can be also written in an incremental way.. In a further test, we showed that this advantage of PLS for k < q actually holds also in the incremental version of the algorithm. The lower performance of the incremental methods for k > 1 is probably due to simultaneous component estimation, which is known to accumulate error in incremental PCA [29] . For k ≥ q, PCAJ did best. In our incremental PCAJ implementation, inconsistencies between the incrementally updated variables are probably compensated by the Gram-Schmidt orthonormalization, which was computed every 200 steps (see Sect. 5.1.2). For the other methods, this compensation probably did not work as well, because additional variables have to be updated (β for PCR and the residual variance for PPCA). Moreoever, our PLS implementation did not use any such compensation step.. Apart from the breakdown for k < q, all of PCAJ, PCR, FA, and PPCA showed an additional specific weakness. PCAJ deteriorates for k > q; particularly, for only output noise, one principal component points into the direction of this noise, and regression cannot proceed because the principal subspace is perpendicular to the input space. PCR fails dramatically if k < q because part of the input that contributes to the output value is ignored. FA shows higher regression errors for k > q if input dimensions have zero noise, probably, because the model does not consider zero noise variance.. PPCA fails if we add irrelevant noise dimensions because the ellipsoid associated with the eigenvectors and eigenvalues includes this noise (see Sect. 4.4). Thus, to compensate for this additional noise, as many components as noise dimensions need to be added [21] . PCR and PCAJ do better with irrelevant noise because the data's variance equals d/q on the embedded subspace; the variance in the noise dimensions equals 1 (see Sect. 5.1.1). Here, FA was not affected, because the irrelevant noise dimensions are covered by the model assumptions. This feature may explain why FA did better than PCAJ, PCR, and PPCA on the real-world data.. Furthermore, we illustrated the influence of the noise-generation model on the regression results. Apart from PCAJ, all tested methods produce an optimal regression result for only output noise, but not for isotropic noise. The model assumptions of FA and PPCA are consistent with isotropic noise, and these methods do reproduce the underlying model correctly-the density p(y, x) matches the distribution of the data in joint space. However, maximizing p(y|x) for a given x results in the ordinary-least-squares solution, which is not optimal for noise-free test data. PCAJ also aligns its principal subspace with the data distribution for isotropic noise, but PCAJ uses this subspace directly as the result for regression, which is optimal for noise-free test data.. On non-linear data sets, with locally-linear regression, the prediction error depends on the width of the kernel function. An optimal width exists, which is a trade-off between finding a small enough value for a good linear approximation and a large enough value to avoid over-fitting (because of the noise). In incremental learning, when the number of components q changes, a robustness of the optimal width about q is desirable. This robustness was only found in PLS (see Fig. 6 ).. Using the real-world data, we could reproduce the main findings obtained from the synthetic data, particularly, the advantage of maximizing the correlation versus maximizing the variance. Thus, our synthetic data seems to have basic characteristics found in real applications: being lower-dimensional than the embedding space, including dimensions with irrelevant noise, being globally non-linear, and locally linear. In robotics, such characteristics are likely to occur in sensory data, for example, in visual and tactile input. For both data sets, the results show many of the characteristics as seen in Sect. 5.1.3 for the synthetic data (Fig. 8) . RRR provides a kind of base-line for the optimal performance. Among the remaining methods, PLS' regression error decreases the most quickly with increasing number of factors. FA, as in the case with irrelevant noise dimensions, converges more quickly than the remaining methods PCR, PCAJ, and PPCA. On the vision-robot data, the prediction error for FA drops sharply at the intrinsic dimensionality (k = 2). The error for PCR, PCAJ, and PPCA drops at a higher k-value (k = 4). When increasing k further, only PCAJ fails. The experiments are split in two parts. First, on a data set with known structure, we show how the dimensionality-reduction methods compare with each other for different factor numbers and kernel widths. Second, we demonstrate that key results observed in the synthetic data are also visible in two real-world data sets. Two data sets were used: vision-robot and census-house data. The vision-robot data were taken from a study in which a six-degrees-of-freedom robot arm learned to grasp an object (a small brick) presented visually on a table surface [23] -see Fig. 7 . Each data point combines the visual information of the object with the grasping arm posture of the robot. To collect a data point, the robot put the brick on the table, recorded the corresponding arm posture, removed the arm, and took a picture of the brick. The visual information consists of a 4 × 4-pixels grid providing a coarse-grained view of the table surface and a histogram showing the edge distribution over four orientations within the camera image [23] . The arm posture consists of six joint angles. The vision-robot data set is locally low-dimensional, because the brick on the table has only three degrees of freedom (two for position and one for orientation). Thus, there is a lot of redundancy in the sensor values (the coarse-grained image and the edge histogram). A disadvantage of this data set for our regression study is that several redundant arm postures exist for a given image. For a given end-effector position, the inverse kinematics of the robot arm has several solutions; i.e., several combinations of joint angles lead to the same endeffector position [25, 30] . However, in the collected data, the shoulder joint showed almost no redundancy (three redundant postures could be identified and were removed). Thus, we used the angle of the shoulder joint as a target value for the 20-dimensional sensory input. Since the orientation of the brick is irrelevant for the angle of the shoulder joint, the intrinsic dimensionality of the relevant data is only two. The data set consists of 3,368 patterns; the first 2,000 were used for training, the rest for testing.. The census-house data set [27] is part of the Delve repository. The data show median house prices and demographic compositions of several survey regions, as obtained from the 1990 US Census. Here, we used a subset called 'house-price-16L', that contains only 16 of the demographic attributes. These attributes form the input and the corresponding house price is the output. Also this data set contains redundancy, since some of the attributes correlate with each other, for example, the number of families and the number of households. However, the intrinsic dimensionality is unknown. The pattern set consists of 22,784 data points; thereof 10,000 were used for training and the rest for testing.. Both training data sets were processed such that each attribute had zero mean and unit variance. For regression, as for the sine-function before, we used only one local model centered at the origin. The weight-function was uniform and had the same metric parameter D as before. The optimal D-value was obtained by, first, computing the regression using RRR with k = 1 for various D-values (in steps of 0.1), and second, choosing the D-value resulting in the lowest error (D = 2.4 for the vision-robot and D = 0.3 for the house data). Using this optimal D, the six methods were evaluated with varying number of factors k.

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 B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B I I 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 Perception Correlated Comparison Method for Dynamic Meshes Abstract-There are multiple areas of computer graphics where triangular meshes are being altered in order to reduce their size or complexity, while attempting to preserve the original shape of the mesh as closely as possible. Recently, this area of research has been extended to cover even a dynamic case, i.e., surface animations which are compressed and simplified. However, to date very little effort has been made to develop methods for evaluating the results, namely the amount of distortion introduced by the processing. Even the most sophisticated compression methods use distortion evaluation by some kind of mean squared error while the actual relevance of such measure has not been verified so far. In this paper, we point out some serious drawbacks of the existing error measures. We present results of the subjective testing that we have performed, and we derive a new measure called Spatiotemporal edge difference (STED) which is shown to provide much better correlation with subjective opinions on mesh distortion. F OR about 20 years, we have seen attempts to reduce the memory requirements of triangular meshes based on sophisticated encoding or simplification of surface meshes. These efforts do not lose their value with increasing availability of high performance hardware, because it is always useful to be able to store and transmit more detailed models. On the other hand, it had been the increase of processing power of current hardware that recently enabled the extension of the original static mesh compression problem to the more computationally demanding dynamic case, i.e., the compression of surface animations.. Most of the techniques for both static and dynamic mesh compression presented so far are lossy, i.e., the decompressed data is slightly different from the original data. The goal of lossy compression is to achieve the best possible rate/distortion ratio, i.e., we want to obtain a mesh which can be stored in a low number of bits, and at the same time, we want to preserve some property of the mesh (low distortion). Quite surprisingly, there has been only very little effort invested into the evaluation techniques. Most of the papers simply use the mean squared error (MSE), or some of its derivatives. This might seem sufficient, because the MSE behaves intuitively in cases such as adding Gaussian error with growing deviation (the MSE grows as well) or adding sinusoidal error with growing amplitude (the MSE again grows).. However, as soon as we start adding different kinds of distortion, the MSE fails completely. It is unable to distinguish between smooth error (neighboring vertices shifted by similar amount and in similar direction) and random error (neighboring vertices shifted by a random amount in random directions), even though intuitively there is a big perceptual difference between these two. Such situation is actually very common in practical compression and processing tasks, where usually multiple parameters control the quality of the result (roughness of quantization, number of basis vectors used for PCA based schemes, etc.), however, each parameter may introduce an error of different character. These parameters are usually set by some kind of rate/distortion optimization, and it is therefore, essential to have an error measure which accurately measures the property of the processed which we want to preserve. MSE works well only as long as the property being preserved is "vertices located near their original location," however, we will show that "visual similarity with the original" is a different property and it should be therefore evaluated by a different measure.. In the next section, we will show the mesh comparison methods used so far, identifying the same problems for all of them. Motivated by this insufficiency, we had performed a series of subjective experiments which showed that the correlation of the MSE (and other measures) with subjective distortion perception is only very limited. Based on the results, we have designed a new measure, the STED error, which will be described and configured to fit the subjective testing results. Finally, we will show that the STED measure had predicted some counterintuitive phenomena which were confirmed by our final subjective experiment. We have proposed a novel method for measuring the discrepancy between two dynamic meshes. Our contributions are:. . We have shown that there is only a weak relation between the MSE and the perceived similarity of yield results with high correlation with subjective testing. . We have determined constants to be used in the equations. These constants may not be exact, however, the result does not critically depend on their value. Using the proposed STED measure, we are able to robustly determine an error introduced by a compression. We can detect certain kinds of spatial and temporal artifacts, namely the difference between regular and random vertex shifts. The measure we are proposing yields results significantly different from the ones provided by existing measures, which has a substantial influence on dynamic mesh compression algorithm configuration, or even dynamic mesh compression algorithm design.. Quite surprisingly, our experiments show that the influence of temporal error is relatively small. The contribution of the temporal error term in (20) is about 12 percent for the chicken sequence, and less than 2 percent for the dance and cloth sequences. Nevertheless, the inclusion of the temporal term improves the correlation and may be important in cases where very little spatial error is introduced into the animation.. The method has some drawbacks as well. In some cases, such as rigid translation, rotation, and scale of the animation, it is possible that the measure does not report any difference. We consider such eventuality highly unlikely and easy to detect and compensate. It is possible that in some cases of extremely smooth deformation, such as global taper or shear of the animation, the measured error might be smaller than expected.. We have also identified a class of artifacts which will not be detected by the measure at all. One can easily imagine that the deformation depicted by Fig. 10 is well visible, while all the edge lengths remain unchanged. This is not only a problem of meshes with borders and with one edge around which the mesh is rotated. Situation depicted by Fig. 11 is also not detected by the measure. Such case may occur in practise, however, a larger deformation that leaves edge lengths unchanged is again very unlikely.. Despite the drawbacks, we have received measures that correlate with the subjective testing with Pearson correlation coefficient constantly higher than 0.9 during the testing. In the case of the mocap dance sequence, we have reached to a value of 0.976. In our experiments, such high correlations are not achieved by any existing error measure technique. Moreover, the STED measure is fast to evaluate, especially compared to the Hausdorff distance based measures such as the one used by the Metro tool. In the future, we intend to perform a study of existing compression algorithms and their performance with respect to the proposed STED measure. We have shown that moving from the KG error to the STED error requires reconfiguration of the Coddyac coder in order to obtain the best possible result. Similarly, other compression algorithms have multiple configuration parameters (quantization constant, level of simplification, number of clusters, depth of used space subdivision structure, etc.) which need to be optimized to obtain an optimal result with respect to a given error measure. It is therefore not easy to perform a fair comparison, and thus, we have not attempted to do such study in this paper.. Currently, the measure only works for compression evaluation, i.e., for cases when original and distorted sequences share the connectivity. There is a possibility to extend the measure also to the case of mesh sequences of unequal connectivity. This could be done by fine resampling of both meshes, which will convert them into a shared topology, using some strain minimization criterion. There are some techniques used for this purpose, such as the Face2Face algorithm [27] , [28] , the wavelet decomposition [29] , or remeshing [30] . Extending the measure in this way is a part of the intended future work.. We also intend to use the spatial part of the STED measure to test the possibility of measuring the difference between static meshes in order to compare results of static mesh compression algorithms.. Detailed information about our research in the field of dynamic mesh compression can be found on our website http://graphics.zcu.cz/compression. Fig. 11 . The "hill/valley" artifact, not detected by the STED measure. Fig. 10 . The "newspaper" artifact, not detected by the STED measure.

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


An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria The most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean. In many problems however, the criteria considered interact, and a substitute to the weighted arithmetic mean has to be adopted. We show that, under rather natural conditions, the discrete Choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by the taking into consideration of the interaction among criteria. The axiomatic that supports the Choquet integral is presented and an intuitive approach is proposed as well. Let us consider a finite set of alternatives A = {a, b, c, . . .} and a finite set of criteria N = {1, . . . , n} in a multicriteria decision making problem. Each alternative a ∈ A is associated with a profile x a = (x a 1 , . . . , x a n ) ∈ IR n , where, for any i ∈ N , x a i represents the partial score of a related to criterion i. We assume that all the partial scores are defined according to the same interval scale, that is, they are defined up to the same positive linear transformation.. From the profile of any alternative a, one can compute a global score M (x a ) by means of an aggregation operator M : IR n → IR which takes into account the weights of importance of the criteria. Once the global scores are computed, they can be used to rank the alternatives or select an alternative that best satisfies the given criteria.. Until recently, the most often used aggregation operators were the weighted arithmetic means, that is, operators of the form. with i ω i = 1 and ω i ≥ 0 for all i ∈ N . However, since these operators are not able to model in any understandable way an interaction among criteria, they can be used only in the presence of independent criteria.. In order to have a flexible representation of complex interaction phenomena between criteria, it is useful to substitute to the weight vector ω a non-additive set function on N allowing to define a weight not only on each criterion, but also on each subset of criteria. For this purpose the concept of fuzzy measure [31] has been introduced.. A fuzzy measure (or Choquet capacity) on N is a monotonic set function v : 2 N → [0, 1], with v(∅) = 0 and v(N ) = 1. Monotonicity means that v(S) ≤ v(T ) whenever S ⊆ T . One thinks of v(S) as the weight related to the subset S of criteria. Throughout this paper, we will denote by F N the set of all fuzzy measures on N . Now, a suitable aggregation operator, which generalizes the weighted arithmetic mean, is the discrete Choquet integral, whose use was proposed by many authors (see e.g. [7] and the references there). Given v ∈ F N , the Choquet integral of x ∈ IR n with respect to v is defined by. where (·) indicates a permutation of N such that x (1) ≤ . . . ≤ x (n) . Also A (i) = {(i), . . . , (n)}, and A (n+1) = ∅. We see that the Choquet integral (2) is a linear expression, up to a reordering of the elements. Moreover, it identifies with the weighted arithmetic mean (discrete Lebesgue integral) as soon as the fuzzy measure v is additive, that is such that v(S ∪ T ) = v(S) + v(T ) whenever S ∩ T = ∅.. The main aim of this paper is to present the Choquet integral as an appropriate extension to the weighted arithmetic mean for the aggregation of criteria. This operator offers indeed a large flexibility while keeping in some sense a linear form. Although its definition is not very intuitive, we will show that the Choquet integral can be characterized axiomatically by means of rather natural properties.. The outline of this paper is as follows. In Section 2 we examine three types of dependence between criteria: correlation, substitutiveness/complementarity, and preferential dependence. In Section 3 we set the framework of our study by introducing some properties often required for aggregation. In Section 4 we present the Choquet integral in a rather intuitive way and we propose an axiomatic characterization. Finally, in Section 5 we introduce the im-portance and interaction indices which enable to interpret the behavior of aggregation.. In order to avoid a heavy notation we will often omit braces for singletons, e.g. writing a(i), N \ i instead of a({i}), N \ {i}. Also, for pairs, we will often write ij instead of {i, j}, as for example a(ij).. For any subset S ⊆ N , e S will denote the characteristic vector of S in {0, 1} n , i.e. the vector of {0, 1} n whose i-th component is 1 if and only if i ∈ S. We also introduce the notation. Finally, ∧ and ∨ will denote the minimum and maximum operations, respectively. We have shown that the Choquet integral is an appropriate substitute to the weighted arithmetic mean to aggregate dependent decision criteria. The motivation is based mainly on an axiomatic characterization of the class of Choquet integrals, but also on an intuitive interpretation of its expression. We hope that this will encourage people of the multicriteria decision making community to use this innovative technique of aggregation, which seems very promising.

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


Detecting Moving Shadows: Algorithms and Evaluation Abstract-Moving shadows need careful consideration in the development of robust dynamic scene analysis systems. Moving shadow detection is critical for accurate object detection in video streams since shadow points are often misclassified as object points, causing errors in segmentation and tracking. Many algorithms have been proposed in the literature that deal with shadows. However, a comparative evaluation of the existing approaches is still lacking. In this paper, we present a comprehensive survey of moving shadow detection approaches. We organize contributions reported in the literature in four classes two of them are statistical and two are deterministic. We also present a comparative empirical evaluation of representative algorithms selected from these four classes. Novel quantitative (detection and discrimination rate) and qualitative metrics (scene and object independence, flexibility to shadow situations, and robustness to noise) are proposed to evaluate these classes of algorithms on a benchmark suite of indoor and outdoor video sequences. These video sequences and associated "groundtruth" data are made available at http://cvrr.ucsd.edu/aton/shadow to allow for others in the community to experiment with new algorithms and metrics. DETECTION and tracking of moving objects is at the core of many applications dealing with image sequences. One of the main challenges in these applications is identifying shadows which objects cast and which move along with them in the scene. Shadows cause serious problems while segmenting and extracting moving objects due to the misclassification of shadow points as foreground. Shadows can cause object merging, object shape distortion, and even object losses (due to the shadow cast over another object). The difficulties associated with shadow detection arise since shadows and objects share two important visual features. First, shadow points are detectable as foreground points since they typically differ significantly from the background. Second, shadows have the same motion as the objects casting them. For this reason, the shadow identification is critical both for still images and for image sequences (video) and has become an active research area, especially in the recent past. It should be noted that, while the main concepts utilized for shadow analysis in still and video images are similar, typically, the purpose behind shadow extraction is somewhat different. In the case of still images, shadows are often analyzed and exploited to infer geometric properties of the objects causing the shadow ("shape from shadow" approaches) as well as to enhance object localization and measurements. Examples of this can be found in aerial image analysis for recognizing buildings [1] , [2] , for obtaining 3D reconstruction of the scene [3] , or even for detecting clouds and their shadows [4] . Another important application domain for shadow detection in still images is for the 3D analysis of objects to extract surface orientations [5] and light source direction [6] .. Shadow analysis, considered in the context of video data, is typically performed for enhancing the quality of segmentation results instead of deducing some imaging or object parameters. In the literature, shadow detection algorithms are normally associated with techniques for moving object segmentation. In this paper, we present a comprehensive survey of moving shadow detection approaches. We organize contributions reported in the literature in four classes and present a comparative empirical evaluation of representative algorithms selected from these four classes. This comparison takes into account both the advantages and the drawbacks of each proposal and provides a quantitative and qualitative evaluation of them. Novel quantitative (detection and discrimination rate) and qualitative metrics (scene and object independence, flexibility to shadow situations, and robustness to noise) are proposed to evaluate these classes of algorithms on a benchmark suite of indoor and outdoor video sequences. These video sequences and associated "ground-truth" data are made available at http://cvrr.ucsd.edu/aton/shadow to allow for others in the community to experiment with new algorithms and metrics. This availability follows the idea of data-sharing embodied in Call for Comparison, like the project of European COST 211 Group (see http://www.iva.cs.tut.fi/COST211/ for further details).. In the next section, we develop a two layer taxonomy for surveying various algorithms presented in the literature. Each approach class is detailed and discussed to emphasize its strengths and its limitations. In Section 3, we develop a set of evaluation metrics to compare the shadow detection algorithms. This is followed by Section 4, where we present a results of empirical evaluation of four selected algorithms on a set of five video sequences. The final section presents concluding remarks. Development of practical dynamic scene analysis systems for realworld applications needs careful consideration of the moving shadows. The research community has recognized this and serious, substantive efforts in this area are being reported. The main motivator for this paper is to provide a general framework to discuss such contributions in the field and also to provide a systematic empirical evaluation of a selected representative class of shadow detection algorithms. Papers dealing with shadows are classified in a two-layer taxonomy and four representative algorithms are described in detail. A set of novel quantitative and qualitative metrics has been adopted to evaluate the approaches.. The main conclusion of the empirical study can be described as follows: For a general-purpose shadow detection system with minimal assumptions, a pixel-based deterministic nonmodel-based approach (DNM1) assures best results. On the other hand, to detect shadows efficiently in one specific environment, more assumptions yield better results and the deterministic model-based approach should be applied. In this situation, if the object classes are numerous to allow modeling of every class, a complete deterministic approach, like the DNM2, should be selected. If the environment is indoor, the statistical approaches are the more reliable since the scene is constant and a statistical description is very effective. If there are different planes onto which the shadows can be cast, an approach like SNP is the best choice. If the shadows are scattered, narrow, or particularly "blended" to the environment, a region-based dynamic approach, typically deterministic, is the best choice (as DNM2 in the Highway II scene reported in this paper). Finally, if the scene is TABLE 4 Qualitative Evaluation noisy, a statistical approach or a deterministic approach with effective pre and postprocessing steps should be used. Finally, we want to remark that all the evaluated approaches exploit a large set of assumptions to limit complexity, and to avoid being unduly constrained to a specific scene model. This limits their shadow detection accuracies. This, in fact, points to the limitations of using only image-derived information in shadow detection. Further improvements would require feedback of specific task/scene domain knowledge.. A very interesting future direction has been suggested by an unknown reviewer. He/she suggested considering the physically important independent variables to evaluate the algorithms. If we can consider as parameters of the scene, for example, the type of illumination for indoor scene or the surface type upon which the shadows are cast in outdoor environments, we can build up a benchmark on which to test the different approaches. Results on accuracy on this benchmark would be more useful to future reserachers/developers of shadow detection (and motion detection) algorithms since they are more physically linked to the considered scene.

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


Storage coding for wear leveling in flash memories Abstract-NAND flash memories are currently the most widely used flash memories. In a NAND flash memory, although a cell block consists of many pages, to rewrite one page, the whole block needs to be erased and reprogrammed. Block erasures determine the longevity and efficiency of flash memories. So when data is frequently reorganized, which can be characterized as a data movement process, how to minimize block erasures becomes an important challenge. In this paper, we show that coding can significantly reduce block erasures for data movement, and present several optimal or nearly optimal algorithms. While the sorting-based non-coding schemes require O( n log n) erasures to move data among n blocks, coding-based schemes use only O( n) erasures and also optimize the utilization of storage space.

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Comparative study of analytical and numerical algorithms for designing reinforced concrete sections under biaxial bending This paper presents a comparative study of different integration methods of stresses (both analytical and numerical) for concrete sections subjected to axial loads and biaxial bending. Such methods are applied to circular and rectangular sections. The constitutive equation used is a parabola-rectangle from the Eurocode-2. The comparison was performed with regard to the accuracy and the computational speed of each method. The objective of the paper is to determine which of the integration methods compared is more efficient in computing the interaction surfaces for rectangular and circular sections. Computer programs for designing reinforced concrete structures need to compute the internal forces through the integration of stresses over the concrete cross-section in order to obtain the interaction surface (N u , M uy , M uz ). This action is performed many times, and thus its optimisation represents an important reduction in computing time. Romero et al. [3] demonstrated that the runtime to obtain the internal forces of the section is mainly consumed during the evaluation of the internal equilibrium forces of concrete sections (N c , M cy , M cz ) and not in the steel bars.. Generally, the columns of such structures are subjected to axial loads and biaxial bending moments as a result of their geometry, the shape of the cross-section or the type of external forces exerted. In edification many examples follow this behaviour such as the columns in the corners of building and elements affected by seismic and wind forces. For this type of structures the sections are typically rectangular or circular. The classical method of integrating the stresses [4, 5] , etc.) in the concrete is usually performed by dividing the section into layers or fibers (also called cells). This technique is not numerically efficient due to the huge amount of information that is required to characterise the section and the large number of numerical operations needed to reach an acceptable level of error, Miguel et al. [6] . Moreover, this integration method could produce convergence problems for non-linear structural analysis programs, since the variation of the internal forces is not continuous for different locations of the neutral axis and depends on the fiber mesh size or the layer height. There are a lot of alternative methods to integrate the concrete stresses over the cross-section, some of the most interesting of which are those by Basu [7] , Brö ndum [8, 9] and Yen [10] . These authors study the integration of stresses if the constitutive equation is the equivalent rectangular stress block. Rodríguez and Aristizabal [11] also proposed computing such integrals by decomposing the section into many trapezes. In this case the constitutive equation is a parabola-rectangle diagram for the ascending branch and linear for the descending branch. They obtain the solution to the stresses integral analytically. Recently Barros et al. [1] obtained the stress integral for the rectangular section analytically using the Heaviside functions. This method is valid for ultimate loads, axial load and uniaxial bending and a parabola-rectangle diagram. Previously, the same authors, Barros et al. [12] , used the constitutive equation of Model Code 90 [14] for a non-linear structural analysis under axial load and biaxial bending.. Fafitis [13] developed a method for computing the internal forces of the concrete using Green's theorem. Use of this method allows the stress integral of the compressed area to be transformed into a path integral over the perimeter. This integral is solved numerically using the GaussLegendre quadrature. Bonet et al. [2] also proposed two methods based on the Gauss quadrature. The first method was valid for a non-cylindrical stress field and the authors presented an automatic algorithm which subdivided the non-cracked concrete area with any polygonal shape (including holes) into a small number of quadrilateral areas. The second method is suitable only for polygonal sections (including holes) in which the stress field is uniform over one direction (cylindrical stress fields) and it was decomposed the integration area into thick (or wide) layers parallel to the neutral axis. The integral of each layer was transformed into a path integral along the perimeter. This last method was termed the ''Modified Thick Layer Integration'' (MTLI) method. In this former paper, Bonet et al [2] compared these two new methods based on the Gauss quadrature, the fiber method and the method proposed by Fafitis [13] in terms of accuracy and speed (efficiency). In this study, the constitutive equation respected was the Model Code 90 [14] for a non linear analysis. These authors concluded the method most efficient is the MTLI. This paper presents a comparative study of different analytical and numerical methods in order to determine which is the most efficient algorithm to compute the concrete interaction surface for rectangular and circular sections.. Two new methods, one analytical and the other numerical, have been proposed for the circular sections. The analytical method proposed in this paper obtains the stress integral by using Heaviside functions to define the failure of the section. Moreover, the numerical method proposed here decomposes the integration area into thick layers parallel to the neutral axis. The definition of such layers depends on the concrete constitutive equation used. The internal forces (N c , M cy , M cz ) are computed as a sum of the numerical integrals using a Gauss-Legendre quadrature for each layer the compressed concrete zone is decomposed into.. In general, it was observed that the level of accuracy of the fiber method or layers depends on the mesh density and how it adapts to the compressed concrete zone.. For the rectangular section, the most efficient integration method is the ''Modified Thick Layer Integration'' (MTLI) method with regard to the analytical method and the fiber method. If the parabola-rectangle diagram is selected, three Gauss points are suggested for each side of the perimeter for each layer the compressed zones are broken down into.. For the circular section, both the analytical method and the numerical method are very efficient in comparison with the layer integration method. If an error below 0.04% is accepted, the numerical method with 15 Gauss points for each layer is recommended.. Consequently, the integration methods based on the Gauss-Legendre quadrature that decompose the compressed are into thick layers parallel to the neutral axis become very effective when it comes to computing the interaction diagram of the concrete section. The optimisation of this procedure gives rise to a notable decrease in the computation times in the design programs that require an iterative process to obtain the reinforcement bars of the section. The most important results for both types of sections are analysed in this section. For each position of the neutral axis and for each integration method, the error and computing time were calculated.. It was noticed that the computing times are extremely small for both the analytical and numerical cases, and were in fact lower than what could be measured accurately by the computer. For this reason, the times presented here are the average value of five series of executions. Each of the series consisted in repeating the stress integral 1000 times.. The errors obtained for each method of computing the internal forces of the concrete (N c , M cy , M cz ) are presented in relative form with respect to the analytical solution, that is:. where F i is the value obtained for the ith-case for an internal force F (axial load or bending moment) and F exac is the exact value of the internal force F (axial load or bending moment) obtained for the analytical solution. Table 2 shows the computing time consumed for the rectangular section, as well as the maximum errors for any of the internal forces (axial load or bending moment) Table 1 Cases analysed in the comparative study Position of the neutral axis. and any inclination of the neutral axis (b), in terms of the number of Gauss points and of the relative depth of the neutral axis (a). In general, the maximum errors are reached for the maximum inclination of the neutral axis. Nevertheless, it can be observed that with only three Gauss points, the error is almost zero and the computing time is very small. Table 3 presents the results of the circular section for different number of Gauss points and relative depths of the neutral axis (a). In comparison with the results for the rectangular section (Table 2) , it was observed that for the same number of Gauss points for each layer and for the same a, the error for the circular section is higher. This behaviour is due to the shape of the section. Furthermore, if 15 Gauss points are applied to each layer, an admissible degree of accuracy (lower than 0.1%) is obtained with a reasonable computing time in comparison to other integration methods (Tables 5 and 7) . Tables 4 and 5 show the results from the fiber and layer integration methods respectively. With respect to the fiber method, for the rectangular section it can be inferred that for a higher relative depth of the neutral axis a the error decreases because the mesh of fibers adapts better to the concrete stress field. It can also be seen that for very small values of a the error is not admissible, because the size of the fiber is quite a lot higher than the depth of the neutral axis. In other words, the error obtained is mesh-dependent. Moreover, in comparison with the results obtained in Table 2 , the execution time is quite a lot higher for the same error order.. On the other hand, with respect to the layer integration method for the circular section it can be observed that the errors increase as the relative depth of the neutral axis (a) decreases. In addition, for very small values of a the errors are not admissible. It can be seen that the error is also mesh-dependent. Finally, on comparing with the results from Table 3 , the computing times are much higher for a similar error.. The times from the analytical integral for the rectangular section are shown in Table 6 . If the integral is performed with a unique function, the execution time is seven times slower than for the numerical integral with three Gauss points (Table 2) . However, if the analytical integral is performed by separated components, the computing time decreases considerably, the numerical integral being between 0.8 and 2.7 times faster. Table 7 presents the time for the analytical integral and the circular section. It can be shown that the time of the numerical integral is slightly lower than the analytical one with 15 Gauss points per layer.. In order to compare the three integration methods applied to the circular section, Fig. 9 displays the error for the axial load (Eq. (18)) on the y-axis, and the computation time on the x-axis for the average value of the five series. The results from the numerical and analytical methods are included in this graph for the following relative depths of the neutral axis (a): 0.01; 0.10; 0.20 and 0.80. The average computing integration time corresponding to the cases performed with the analytical method is also represented with a continuous vertical line (1.76 sg).. As is shown in Fig. 9 , if the integration is performed using the layer method, the time increases as the relative depth increases because there is a higher number of layers in the compressed area. Moreover, if the proposed numerical method is compared with the layer method, the former is found to be more efficient.. Finally, it can be inferred that for a level of errors that could be considered to be admissible in a non-linear program (between 0.01% and 0.001%) the computing time of the numerical method is similar to that of the analytical method.

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


Relevance-Zone-Oriented Proof Search for Connect6 Index Terms-Board games, Connect6, -in-a-row games, lambda search, threat-based proof search, threat-space search. A generalized family of -in-a-row games, named Connect [30] , [31] , was introduced and presented by Wu et al. Two players, named Black and White, alternately place stones on empty squares 1 of an board in each turn. Black plays first and places stones initially. The player who first gets consecutive stones of his own horizontally, vertically, and diagonally wins. Both players tie the game when the board is filled up with neither player winning. Games in this family are also called Connect games 2 in this paper. For example, Tic-tac-toe is Connect (3, 3, 3, 1, 1) , Go-Moku in the free style (a traditional five-in-a-row game) is Connect (15, 15, 5, 1, 1) , and Connect6 played on the traditional Go board is Connect (19, 19, 6, 2, 1) . For simplicity, let Connect denote the game Connect , played on infinite boards. For example, when played on infinite boards, Go-Moku becomes Connect (5, 1, 1) and Connect6 becomes Connect (6, 2, 1) .. Among these Connect games, Connect6 attracted much attention due to three merits: fairness, simplicity of rules, and high game complexity as described in [30] and [31] . Since Connect6 The authors are with the Department of Computer Science, National Chiao Tung University, Hsinchu 30050, Taiwan (e-mail: icwu@csie.nctu.edu.tw; bhlin@csie.nctu.edu.tw).. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.. Digital Object Identifier 10.1109/TCIAIG.2010.2060262 1 Practically, stones are placed on empty intersections of Renju or Go boards. In this paper, by squares, we mean intersections. 2 The term of connect games defined in [10] covers the games such as Hex, Connect Four, etc. In this paper, Connect are capitalized to indicate all the games in the family of Connect(m; n;k;p;q).. was introduced, hundreds of thousands of Connect6 games have been played on web sites, such as littlegolem.net [14] and cycgame.com [21] . Since 2006 , several Connect6 open tournaments [20] for human players have been held, such as NCTU Open, ThinkNewIdea Open, Russian Open, and World Open. Connect6 has also been included as one of the computer game tournaments at the Computer Olympiad [24] and Chinese Computer Games Contest [9] , since 2006 and 2007, respectively.. For Connect6, researchers in [30] and [31] mentioned a simple threat-based proof search method for solving Connect (6, 2, 3) . Section II shows that many more winning positions cannot be solved by such a method. This paper proposes a new threat-based proof search method, named relevance-zone-oriented proof (RZOP) search, developed from the lambda search proposed by Thomsen [22] . Section IV presents this novel, general, and elegant method of constructing and promoting relevance zones for Connect6. The proposed method is also generalized to all Connect games in the Appendix. Together with a proof number search [3] , [28] , it solved effectively and successfully many new Connect6 game positions, including several Connect6 openings, especially the Mickey Mouse opening, as described in Section V. This opening used to be one of the popular openings before we solved it. All definitions and notations used in this paper are given in Section III. Concluding remarks are made in Section VI. Theorem 2 concludes that the verifier in all cases satisfies Property RZV. Therefore, if returns value 1, the constructed is in , and the attacker wins in from Corollary 2.. Theorem 2: The verifier satisfies Property RZV in all cases.. Proof: By induction, the verifier satisfies Property RZV in all cases from Lemma 4 to Lemma 10. This paper investigates a new threat-based proof search for Connect games. The contribution of this paper is mainly the new search method, named RZOP search that uses relevance zones to help solve many positions in Connect6 as well as Connect games. In theory, this method can be applied to Connect games with infinite boards. Practically, this paper demonstrates the method by solving two typical winning positions in Figs. 3(a) and 4(a) on 19 19 boards, as well as many Connect6 positions and openings in Section V. In addition, the method can also be easily incorporated into Connect6 program, such as NCTU6.. This paper also leaves some open problems. 1) Investigate more winning positions in Connect6 that require -strategies, such as the one in Fig. 4(a) . 2) Investigate whether there exists a -strategy in Connect6.. 3) Optimize the proof search tree by pruning more branches efficiently [29] . 4) Apply the new method (in the Appendix) to solving some real positions in general Connect games. 5) Investigate whether dual lambda search [19] is useful for Connect6 or Connect games.. Using the JL-PN search together with our RZOP search, we successfully solved up to 65 positions with -strategy. The 65 positions include 12 three-move openings; in particular, Mickey Mouse opening, which used to be one of the popular openings before we solved it. One might ask whether or when Connect6 on 19 19 boards will be solved. So far, we still could not solve tens of the common openings, many of which experts believed to be well balanced for both players. Hence, the answer to this question is still unknown.

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


Predicting remote versus collocated group interactions using nonverbal cues This paper addresses two problems: Firstly, the problem of classifying remote and collocated small-group working meetings, and secondly, the problem of identifying the remote participant, using in both cases nonverbal behavioral cues. Such classifiers can be used to improve the design of remote collaboration technologies to make remote interactions as effective as possible to collocated interactions. We hypothesize that the difference in the dynamics between collocated and remote meetings is significant and measurable using speech activity based nonverbal cues. Our results on a publicly available dataset -the Augmented Multi-Party Interaction with Distance Access (AMIDA) corpus -show that such an approach is promising, although more controlled settings and more data are needed to explore the addressed problems further. Increasingly, teams in the workplace are expected to collaborate across different physical locations. The challenges involved in designing effective remote collaboration systems are many-communication infrastructure, human-computer interfaces, improving the awareness of the participants, etc. Often, the goal of such designs is (implicitly or explicitly) to make remote meetings as close as possible to face-to-face interactions [11] .. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. The difference in the dynamics between collocated and remote meetings is significant and it can lead to poorer performance in distributed teams [9] . Some of the difficulties of the remote participants and the group as a whole in remote meetings include the inability to conduct side conversations, the challenge of occupying the floor because of the lack of eye contact, the inability to utilize posture shifts, and the phenomenon of in-room attendees forgetting about the remote people [7, 13] . In this work, we attempted to characterize the differences in group dynamics between collocated and remote meetings on a very recent data that is publicly available (the AMIDA corpus), by computing speech activity based nonverbal cues. We evaluated the effectiveness of these cues in predicting the remote versus collocated meetings, as well as the remote participant. Based on the results we noticed that collocated meetings have less turns and the remote participant talks less. Clearly, such observations are limited given the size of our dataset. Furthermore, it is noteworthy that the AMIDA corpus was neither designed nor collected with the task of classifying remote and collocated meetings in mind. Therefore the scenarios in these meetings make our tasks challenging. In the future, we would like to expand our dataset to study the generality of our findings, and to extract more relational and visual features. Task 1: For the first task, we learnt a Gaussian mixture model using the Expectation Maximization (EM) algorithm [1] for each of the group features. Figure 4 shows cross-validation performance of this task with one and two Gaussians. For this task, the Group Speaking Turns (GST) with two Gaussians had the best performance (70%) which is slightly above random performance (66%), but the difference is not statistically significant (given that the size of the data set is small). Figure 5 suggests that while learning a global threshold to classify collocated and remote meetings might be difficult (due to inter-group variations), making a local decision (when looking at only the meetings with the same participants) would still be possible. The results of task 2 verify that this is true. Figure 4 : Performance of group features on predicting the collocated and remote meeting (Task 1).. Task 2: For the second task, we have only 3 meetings with the same participants (9 such sets). Therefore, we use a simple unsupervised approach to predict the collocated meetings -hypothesizing that collocated meetings have either the minimum or the maximum value of the group nonverbal cue. Figure 6 shows average performance for the various features. For this task the Group Speaking Turns (GST) again performed the best (81% accuracy), showing that the collocated meetings have lesser number of turns as compared to the remote meetings. One possible reason might be that because of the presence of a remote participant the group needs more turns per unit time to achieve their desired objective. Also, this result is statistically significant (p = 0.01) compared with random performance (33%). From Figure 5 we can observe that collocated meetings are the ones with less speaking turns. Figure 6 : Performance of group features on predicting the collocated meeting (Task 2).. Task 3: For the task of predicting the remote participant in a meeting, we hypothesized that the remote participant has either the minimum or the maximum value of the individual features. Figure 8 shows the results. The minimum value of speaking length (ISL) is the individual feature that better predicts the remote participant (50% accuracy) higher but not statistically significantly, rather than random performance (25% accuracy). Figure 7 shows the average speaking length per role on remote meetings. We can see that the role with less speaking length is the user interface designer(UD) followed closely by the industrial designer(ID). Dataset: The Augmented Multi-Party Interaction with Distance Access (AMIDA) corpus [6] consists of 10 hours of recorded, transcribed, and annotated four-person meetings recorded at the University of Edinburgh, see Figure 2 . Recordings were gathered using 24 microphones (twocircular, eight-microphone arrays, four headset mics and four lapel mics), six cameras (four close-ups, two views of the room-center of the table and corner), and the output from a slide projector. There are three four-person meetings (for a total of 27-meetings) of which two have a remote participant (18-meetings). Figure 3 shows the scenarios of the AMIDA corpus. Meeting C -Detailed Design: During this meeting participants present their individual work, and present and evaluate the clay prototype again via videoconferencing.. Every meeting has a participant with one of the following roles: project manager (PM), industrial designer (ID), marketing expert (ME) and a user interface designer (UD). In all the remote meetings the user interface designer is the remote participant (as shown in Figure 3 ). We use these meetings to study the group interactions in remote versus collocated settings. The average duration for collocated (also called A-meetings) is 18.6 minutes; for remote meetings (called B and C meetings) the average duration is 37.7 minutes. While the collocated meeting was a pure discussion type meeting, the remote meeting had presentations followed by discussions. In order to have a fair comparison, we consider only the last five minutes of B and C meetings (which mostly were discussions) for our subsequent analysis.. Tasks: In order to model the difference between collocated and remote meetings, we define three tasks.

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


The optimal sequenced route query Abstract Real-world road-planning applications often result in the formulation of new variations of the nearest neighbor (NN) problem requiring new solutions. In this paper, we study an unexplored form of NN queries named optimal sequenced route (OSR) query in both vector and metric spaces. OSR strives to find a route of minimum length starting from a given source location and passing through a number of typed locations in a particular order imposed on the types of the locations. We first transform the OSR problem into a shortest path problem on a large planar graph. We show that a classic shortest path algorithm such as Dijkstra's is impractical for most real-world scenarios. Therefore, we propose LORD, a light threshold-based iterative algorithm, which utilizes various thresholds to prune the locations that cannot belong to the optimal route. Then we propose R-LORD, an extension of LORD which uses R-tree to examine the threshold values more efficiently. Finally, for applications that cannot tolerate the Euclidean distance as estimation and require exact distance measures in metric spaces (e.g., road networks) we propose PNE that progressively issues NN queries on different point types to construct the optimal route for the OSR query. Our extensive experiments on both real-world and synthetic datasets verify that our algorithms significantly outperform a disk-based variation The objective of the nearest neighbor query is to find the object(s) with the shortest distance(s) to a given query point. Although this type of query is useful, more often what the user is really after is to plan a trip to several (and possibly different types of) locations in some sequence, and is interested in finding the optimal route that minimizes her total traveling distance (or time). This type of query is important to both commercial applications such as in-car navigation systems or for online map services as well as non-commercial applications such as in crisis management, emergency response and defense/intelligence systems. Similar types of "planning queries" have also been studied by the database community in other application domains such as in air traffic flow and supply chain management [5, 10, 14] . In this paper, we formally introduce and address this specific type of trip planning query in the context of spatial database systems. We studied the new problem of optimal sequenced route query in both vector and metric spaces. To tackle the problem, we first proposed a Dijkstra-based approach and showed that it is not efficient for large point sets and/or routes with large number of points. Hence, we proposed a novel threshold-based algorithm, LORD, for vector spaces. Subsequently, we proposed R-LORD which transforms the LORD's thresholds into a range query and then utilizes an R-tree index structure to support OSR queries more efficiently. We conducted an extensive set of experiments to evaluate R-LORD and our main observations are -R-LORD is light in terms of required workspace; as compared to the Dijkstra-based approach, it always reduces the required workspace by a factor of (on average) 55-90%. The maximum of this space reduction reaches 99.6% for some instances of our experiments. -R-LORDisefficient in terms of query response time;. it answers an OSR query in a time which increases with almost a linear rate as the sequence size |M| increases. Moreover, R-LORD substantially outperforms the Dijkstra's classic algorithm. -R-LORD is I/O efficient; it accesses at most 10.5% of the R-tree nodes.. We also proposed PNE, a progressive algorithm for metric spaces that generates the optimal route from the starting to the ending point. We showed that PNE performs efficiently, particularly when the distributions of the datasets of M are either similar, or increasing from U M 1 to U M i . Moreover, we showed that the overhead of PNE is always negligible as compared to the overhead of the nearest neighbor approach that it employs.. This paper is our first attempt to tackle the core problem of OSR and evaluate its base solutions. As part of our future work, we intend to study other variations of the OSR query as well as improving our base solutions.. In particular, we plan to extend our definition of OSR query to include more general precedence constraints on the points of the optimal route. Suppose we are planning a Saturday trip around the town as follows: first we intend to visit a shopping center in the afternoon to check the season's new arrivals, figure) with the length of 12 units is the optimum answer to our query. Note that s 1 is not the closest shopping center to p and r 1 is actually the farthest restaurant to s 1 . Hence, the optimum route for an OSR query can be significantly different from the one found by the greedy approach. In this section, we first review the related work in the area of graph theory. We then provide an overview of the related studies on variations of the nearest neighbor queries in spatial databases.. The only similarity between traveling salesman problem (TSP) and OSR is that both search for a route of minimum cost in a graph. The general form of TSP was first studied in the 1930s by Karl Menger in Harvard [2] . The most similar instance of TSP to OSR is sequential ordering problem (SOP) which sets some precedence constraints on the route [1] . Each constraint requires that a node of the graph be visited before some other node. Although the number of nodes is small in SOP and general TSP, the unknown traveling sequence makes them NP-hard. However, OSR dictates a given strict sequence order of point types where each point must be selected from a large set per type.. The OSR problem is also related to the problem of finding shortest path (SP) in directed weighted graphs. Two classic algorithms for solving SP in main memory are Dijkstra's and Bellman-Ford algorithms [4] . However, for addressing SP on the huge graph G of Sect. 3.1, an external memory algorithm is required. Hutchinson et al. [7] propose a tree data structure for answering SP queries on a planar graph stored in external memory. Chan at al. [3] describe a disk-based algorithm to find SP on large network systems. They partition the original large graph and search for the shortest path by locally searching in its smaller pieces. While these approaches eliminate the overheads of loading the huge graph in main memory, they are not applicable for OSR queries. The reason is that OSR graph's topology is dependent on the user's query Q(p, M). Since point p and sequence M are not known in advance, this graph must be built on demand as described in Sect. 3.1. Therefore, if we intend to use an external memory SP approach, we need to store the graph on disk blocks before processing it. This makes the approach expensive and therefore impractical.. Numerous algorithms for k-nearest neighbor queries in spatial databases have been proposed. A majority of these algorithms are based on utilizing spatial index structures such as R-tree and usually perform in two filter and refinement steps. Roussopoulos et al. [12] present a branch-and-bound R-tree traversal algorithm that uses two mindist minmaxdist metrics. Hjaltason et al. [6] propose an incremental nearest neighbor algorithm that is based on utilizing an index structure and a priority queue. Jensen et al. in [8] discuss data models and graph representations for NN queries in road networks and provide alternative solutions for it. Papadias et al. [11] propose a solution for NN queries in network databases by generating and expanding a search region around a query point. In our previous work [9] , we proposed a solution for NN queries in road network databases that is based on utilizing network Voronoi diagrams. Other variations of k nearest neighbor queries have also been studied and their solutions are usually motivated by the solutions of their regular k nearest neighbor queries.. In an independent research, Li et al. [10] studied trip planning queries (TPQ), a class of queries similar to our OSR query. With a TPQ, the user specifies a subset (not a sequence) of location types R and asks for the optimal route from her starting location to a specified destination which passes through at least one database point of each type in R. In particular, TPQ eliminates the sequence order of OSR to define a new query. Consequently, finding accurate solutions to TPQ becomes NP-hard as the size of candidate space significantly grows. The paper proposes several approximation methods to provide near-optimal answers to TPQs. However, our proposed solutions to OSR efficiently compute the exact optimal route. In theory, OSR algorithms are able to address address TPQ queries. This can be done by running any OSR algorithm |R|! times, each time using a permutation of point types in R as the sequence M and returning the optimal route among all the resulted routes. This exponentially complex solution is inefficient in practice. The approximation algorithms of [10] are designed to handle the exponential growth of the problem's search space.. In parallel with our study, Terrovitis et al. [14] addressed k-stops shortest path problem for spatial databases. The problem seeks for the optimal path from a starting location to a given destination which passes through exactly k intermediate points of the location database. The k-stops problem is a specialized case of OSR queries. To be specific, OSR query Q(p, M) reduces to a k-stops problem where p is the starting location, M = (1, . . . , 1), |M| = k, and U 1 is the single given point set representing the location database. The destination is considered by the variation directed-OSR of the above OSR query. A major advantage of our OSR approach over k-stops is that we address solutions to OSR queries in both metric and vector spaces while [14] only considers Euclidean space. Our PNE approach can answer K best optimal routes of OSR in a metric space.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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


Strategical languages of infinite words Abstract: We deal in this paper with strategical languages of infinite words, that is those generated by a nondeterministic strategy in the sense of game theory. We first show the existence of a minimal strategy for such languages, for which we give an explicit expression. Then we characterize the family of strategical languages as that of closed ones, in the topological space of infinite words. Finally, we give a definition of a Nash equilibrium for such languages, that we illustrate with a famous example. Game theory [6] is usually defined as a mathematical tool used to analyze strategical interaction, the game, between individuals which are called players. The games studied in this paper are supposed simultaneous, noncooperative, infinitely repeated and with a perfect knowledge of the previous moves. We will elucidate these ideas through a famous example. In game theory, the distinction between the cooperative and noncoperative game is crucial. The Prisoner's Dilemma [3] is an interesting example to explain these notions. It is a game involving two players where each one has two possible actions : cooperate (c) or defect (d). The game consists of simultaneous actions (called moves) of both players. It can be represented using the matrix : where each entry e ij is an ordered pair of real numbers. The two players are referred to as the row player and the column player respectively. The actions of the first player are identified with the rows of the matrix and those of the second one with the columns. If the row player chooses action i and the second action j, the components of the ordered pair e ij are the payoff received by the first and the second player respectively. It is clear that if they could play cooperatively and make a binding agreement, they would both play c. If the game is noncooperative, the best action for each player is d. Suppose now that we consider infinite repetitions of a noncooperative base game. This game is just as noncooperative as the base one, but it allows a certain form of interaction. Suppose that each player has a perfect knowledge of the previous moves of all the others. Then his strategy may depend on these previous moves and he may coordinate it with that of his opponents. For instance, if the base game is the Prisoner's Dilemma, grim-trigger is the strategy of cooperating in the first move and until your adversary defects, then of always defecting after the first defection of your opponent. Tit-for-tat is the strategy of playing at each step the action played by your adversary at the previous one ; the initial move is free.. In this paper, we make use of infinite words to analyze the kind of games we want to model. A match of such a game is represented as an infinite word on the alphabet A of moves. In this context, a strategy for player i can be viewed as a relation from the set of finite words on A to that of the actions of this player. The whole strategy of the game is defined as the vector composed by using the strategies of all players. We can associate to each strategy vector a language L of infinite words on A, defined as the set of all matches that the players may make if everyone follows the strategy he decided to apply.. Nash equilibrium is one of the most important notions in game theory. The whole strategy of the game is defined as the vector composed by using the strategies of all players. Intuitively, a strategy vector is a Nash equilibrium if one player's departure from it while the others remain faithful to it results in punishment. The idea is that once the players start playing according to such a strategy vector, then they all have a good reason to stay with it.. More precisely, our study will be organized as follows. In Sections 1 and 2, we introduce some basic notions on words and game theory. In Section 3, we show that the same language can be generated by several strategies. We call "strategical" a language which is given by at least one strategy. Section 4 is devoted to the proof of the existence of a minimal strategy for a strategical language, for which we give an explicit description. The characterization of the family of strategical languages as that of closed sets in the topological space of infinite words is given in Section 5. Finally, in Section 6, we give a definition of a Nash equilibrium for strategical languages.

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