Generating enhanced natural environments and terrain for interactive combat simulations (GENETICS) Virtual battlefields devoid of vegetation deprive soldiers of valuable training in the critical aspects of terrain tactics and terrain-based situational awareness. Creating believable landscapes by hand is notoriously expensive, requiring both proprietary tools and trained artists, which hampers rapid scenario generation and limits reuse. Our approach constructs large-scale natural environments at run-time using a procedural image-based algorithm without the need for artists or proprietary tools.This paper discusses the current state of the open source project GENETICS (Generating Enhanced Natural Environments and Terrain for Interactive Combat Simulations) and how simulationists can use GENETICS to quickly and cheaply build large-scale natural environments to improve training effectiveness. It will also briefly touch upon level-of-detail techniques and ecotype modeling. The terrains found within most military simulators (especially flight simulators) are bleak, desolate places that share a strong measure of commonality with desert environments. These relatively featureless environments hinder one's ability to grasp the scale of one's surroundings. It is nearly impossible to determine distances or speed in a world devoid of a single bush, tree, or surface detail necessary to establish depth cues [1] [2]. Likewise, successful navigation depends on ones' ability to visually recognize landmark features along a route to the final destination [3] . Large polygonal meshes draped with blurry satellite imagery (see Figure 1 ) are missing the visual cues and landmark features needed to aid the infantry soldier on the ground or the low flying helicopter or aircraft pilot. Similarly, within today's cockpit simulators it is too easy for a pilot to quickly "find, fix or track, and target anything that moves on the surface of the Earth.". 1 When the only object protruding from the terrain surface is an enemy tank with no cover to hide behind, the task of acquiring and destroying your enemy, while never easy, is greatly simplified. Adding vegetation to the synthetic environment provides needed ground clutter to a scene while enhancing realism. While previously it has taken teams of artists to crea te static, frozen-in-time landscapes, our ates large-scale, realistic terrains at is possible to have cy for small-scale, custom-tailored, approach automatically gener runtime using the latest source data. With a minimum amount of shared source data and parameters, terrains are easily synchronized between clients; thus guaranteeing the same terrain environment is created by all hosts within a heterogeneous networked simulation system (see Figure 13 ). Terrains can be reused or regenerated afresh with new parameters in response to the needs of the training audience. With the simple change of a random number seed, a new terrain database is generated without the need to manipulate a database (see Figure 14) . This capability allows trainers the flexibility to use the same terrain repeatedly or create a new one each time; forcing trainees to not depend upon the static nature of most simulation databases.. The immediate and practical benefit of this work is that training improves by giving players a more realistic environment in which to operate (see Figures 15a&b) . It Figure 13 Consistent environment between networked players players and simulated forces engage in a multi-spectrum conflict where the natural environment takes on an active role in the experience and is no longer simply a backdrop. It is only at this point, when the ground cover looks real and foliage hides your view of the enemy that terrain can truly work towards becoming a full-fledged entity within a distributed virtual environment. Our GENETICS NVIDIA 6800GT 256Mb) currently runs at 15 frames per second (1600x1200) with two million objects from five LCC types represented in the scene. From the start of execution with an empty cache to the first rendered frame takes approximately two minutes. Using cached data cuts this time in half.. Geometric object instancing [17] , still a graphics dependent feature, holds the promise of dramatically increasing our frame rate by storing an object's vertex and texture information on the card, thus requiring only the passing of position attitude transform information to the shader program. Additionally, we have chosen to use textures as a storage medium to facilitate the expanded use of shader programming to improve the performance of GENETICS.

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


Neuro-fuzzy approach to processing inputs with missing values in pattern recognition problems An approach to dealing with missing data, both during the design and normal operation of a neuro-fuzzy classifier is presented in this paper. Missing values are processed within a general fuzzy min-max neural network architecture utilising hyperbox fuzzy sets as input data cluster prototypes. An emphasis is put on ways of quantifying the uncertainty which missing data might have caused. This takes a form of classification procedure whose primary objective is the reduction of a number of viable alternatives rather than attempting to produce one winning class without supporting evidence. If required, the ways of selecting the most probable class among the viable alternatives found during the primary classification step, which are based on utilising the data frequency information, are also proposed. The reliability of the classification and the completeness of information is communicated by producing upper and lower classification membership values similar in essence to plausibility and belief measures to be found in the theory of evidence or possibility and necessity values to be found in the fuzzy sets theory. Similarities and differences between the proposed method and various fuzzy, neuro-fuzzy and probabilistic algorithms are also discussed. A number of simulation results for well-known data sets are provided in order to illustrate the properties and performance of the proposed approach. There are many real-world pattern recognition problems where input feature vectors are incomplete [1] [2] [3] 9, 13, 14, [16] [17] [18] 21, 23, 27, 28] . The reasons for missing data can be multifold ranging from sensor failures in engineering applications to deliberate withholding of some information in medical questioners. And though such problems are very interesting from the practical and theoretical point of view, there are very few pattern recognition techniques which can deal with missing values in a straightforward and efficient manner. It is in a sharp contrast to the very efficient way in which humans deal with unknown data and are able to perform various pattern recognition tasks given only a subset of input features.. There are two distinctive problems when dealing with incomplete data which can be stated in the following way: (a) How to use such data in the design stage of a pattern classifier, especially when one cannot afford to discard the inputs with missing values? and (b) How to perform the classification of an incomplete input vector during the normal operation of the classifier? Both learning on the basis of incomplete data and pattern classification using input vectors with missing features will be discussed in this paper. The proposed approach to handling incomplete data will be based on utilising hyperbox fuzzy sets within a general fuzzy min-max (GFMM) neural network architecture [5] [6] [7] [8] [9] [10] [11] [12] . The facts that inputs to the GFMM can be given in a form of upper and lower limits and that the hyperboxes are also represented by a pair of minimum and maximum coordinates for each dimension of the input space will be heavily used in the way the missing values are processed during the learning and recall stages. This specific representation of inputs and cluster prototypes allows for the missing values to be handled very efficiently. It is achieved by representing the missing features as real valued intervals spanning the whole range of possible values. It facilitates the classification of incomplete input patterns without any need for modifications of the neural network structure or substituting of missing features with estimated values. It is argued that since there is a certain level of uncertainty associated with unobserved features, this uncertainty has to be quantified in some way during the classification process. What distinguishes the proposed method from a vast majority of the algorithms for dealing with missing data is that the primary goal of classification is specified as the reduction of a number of viable alternatives on the basis of observed features. Additionally the upper and lower degrees of class membership are calculated in order to quantify the level of uncertainty associated both with the classifier model built on the basis of incomplete training data and incomplete input patterns to be classified. The use of upper and lower degree of class membership is very similar to the plausibility and belief measures used in the theory of evidence [22] or possibility and necessity values used in the fuzzy sets theory [19] . Despite using the upper and lower classification limits, it has to be stressed that if the observed features are discriminative enough a single class can be produced as a result of the classification. However, when a number of alternatives is found and one winning class must be specified, data frequency or hyperbox cardinality information can be used to select the most probable class. In this way the proposed method combines some ideas from the probabilistic methods and fuzzy sets in order to produce more informative classification results. Before concentrating on the description of the algorithms let us first review the existing approaches to handling missing data and highlight some similarities between them and our approach.. One of the most common ways of dealing with missing values is to substitute the missing features with their estimates [3] . These could include the mean value calculated over all examples (or k-nearest neighbours) for a particular feature. However, as it has been pointed out in many papers [1, 13, 17, 18, 27] such a ''repaired'' data set may no longer be a good representation of the problem at hand (as illustrated in Fig. 9 ) and quite often leads to the solutions that are far from optimal.. Another approach presented in [23] advocates the creation of a set of classifiers that would work on different subsets of input features. Unfortunately, this method of training classifiers for all possible combinations of input features very quickly explodes in complexity with an increasing number of features.. Within a probabilistic setting much better results have been obtained by estimating conditional probability distribution over all unknown features given the known features. The emphasis here has been put on the input data density estimation. Ghahramani and Jordan [13] used a mixture of gaussians as a basis for the density estimation and the Expectation-Maximisation algorithm for learning the parameters of this mixture model. A very similar approach has been presented in [27] where the input data density estimation has been carried out by employing Parzen windows and gaussians centered at training data points, though the training algorithm was based on the modified backpropagation method. Tresp et al. [27] also point out that their proposed normalised gaussian formula (very similar to the formula (18)) could be used with various approximations where the gaussians do not necessarily have to be centred at the training data points. It is especially important when one has a large number of training data points. In terms of data density estimation and using clustering methods as one of the alternatives for the above-mentioned approximation, the method proposed in this paper can be regarded as one such approximation with hyperbox fuzzy sets used instead of multidimensional gaussians. There are however some crucial differences which will be pointed out in the later sections. Yet another example of neural implementation of the approach based on building the input data density model can be found in [18] where a radial basis Bolzmann machine has been used for conditional probability distribution estimation and modified learning rules have been proposed to deal with missing values during the training. And although all the methods discussed above are another form of estimating the missing values they proved to be more accurate than the heuristic approaches used before.. Apart from the above-mentioned probabilistic approaches there have been a number of fuzzy and neuro-fuzzy methods for dealing with missing values [1, 17] . Fuzzy sets/fuzzy rule based classification systems naturally lend themselves to efficient processing of missing features. Both in [1] and [17] , which discuss the processing of missing data within fuzzy rule based classifiers, the classification is only based on the known features without making any assumptions about the values of missing features. In other words, the assumption that any value for missing features is possible means that such a feature should not influence the computation of a degree of fulfilment of any of the classification rules. In [1, 17] this was achieved by assigning 1.0 as a degree of membership to the missing features. The general idea of using only the known features for classification and ensuring that the missing features do not affect the generation of classification membership values is also used in this paper. The way of achieving this is however completely different though the final result is very similar. As mentioned earlier, the main goal of the classification is defined as a reduction of viable alternatives rather than producing one winning class without supporting evidence. It is argued here that one of the potential drawbacks of the methods based on the estimation of missing features is the fact that once the estimated value has been used (in implicit or explicit way) the output of the classifier does not differ in any way from the output generated for examples with all features present. An extreme example would be an attempt to classify an input pattern with all features missing which in case of probabilistic models discussed above would result in a guess based on the estimated data density model while for the fuzzy methods all classification rules would be fulfilled with a degree of 1.0 making all classes equally viable. One difference of the proposed method from [1, 17] is that in the case of classification producing a number of viable alternatives the hyperbox cardinality information can be used to find the most likely class. The resulting formulas are very similar to the normalised gaussian formula presented in [27] . Additionally, by taking advantage of the interval representation of the inputs, a way of using some limits (if known) for missing features, which could help in the class discrimination, is proposed.. While classification on the basis of patterns with missing features is the same, the learning of fuzzy rules in [1] and [17] is performed differently. In [1] the rule model that evolves during the training to predict the most possible value for each of the missing features is used. These predictions are then subsequently used to complete the input pattern. In [17] when a missing feature is present all combinations of fuzzy sets, which are possible for such pattern, are created. Depending on the granularity (the number of fuzzy sets) used to partition each variable this approach can initially lead to generation of a large number of rules, especially for high dimensional problems with relatively large proportion of missing values. However, this general approach of assuming that a missing feature can take any value will be utilised in the training stage of our method. But again the implementation of this idea will be based on a suitable representation of the missing feature as a real valued interval spanning the whole range of possible values.. Some other approaches to dealing with missing values within a context of classification decision trees and hyperbox based neural networks can be found in [20, 28] .. The remaining of this paper is organised as follows. Section 2 presents a summary of GFMM neural network with definitions of hyperbox fuzzy sets and associated fuzzy membership function. It also provides a short description of one of the learning algorithms which can be used to place and adjust hyperboxes in the input space. A way of dealing with missing features within the GFMM NN structure is described in Section 3. Both learning on the basis of incomplete data and GFMM operation when given an incomplete input vector are discussed. The simulation results for a number of well known data sets follow in Section 4. And finally the discussion and conclusions are presented in the last section.. 2. An overview of GFMM neural network GFMM neural network for clustering and classification [8, 12] is a generalisation of and extension to the fuzzy min-max neural networks introduced in [24, 25] . The main changes in GFMM constitute the combination of unsupervised and supervised learning, associated with problems of data clustering and classification respectively, within a single learning algorithm and extension of the input data from a single point in n-dimensional pattern space to input patterns given as lower and upper limits for each dimension, i.e. a hyperbox in n-dimensional pattern space.. The GFMM NN can be realised as a three layer feedforward neural network shown in Fig. 1 . It consists of 2 Ã n input layer nodes, m second layer nodes representing hyperbox fuzzy sets and p þ 1 output layer nodes representing classes.. The basic idea of fuzzy min-max neural networks is to represent groups of input patterns using hyperbox fuzzy sets. A hyperbox fuzzy set is a combination of a hyperbox covering a part of n-dimensional pattern space and associated with it membership function. A hyperbox is completely defined by its min point and its max point. A membership function acts as a distance measure with input patterns having a full membership if they are fully contained within a hyperbox and the degree of membership decreasing with the increase of distance from the hyperbox. Learning in the GFMM neural network for clustering and classification consists of creating and adjusting hyperboxes in the pattern space. In previous publications an on-line [12] and a batch (agglomerative) [8] versions of the training algorithm had been proposed. One of the two agglomerative learning schemes described in detail in [8] will be used in this paper and is summarised later in this section.. Once the network is trained the input space is covered with hyperbox fuzzy sets. Individual hyperboxes representing the same class are aggregated to form a single fuzzy set class. The cores of hyperboxes belonging to the same class are allowed to overlap while the cores of hyperboxes belonging to different classes are not allowed to overlap therefore avoiding the ambiguity of an input having full membership in more than one class.. The following are the definitions of input data format, hyperbox fuzzy sets, hyperbox membership function and hyperbox aggregation formula that are used within GFMM.. The input data used during the training stage of GFMM neural network is specified as a set of N ordered pairs. where. h is the hth input pattern in a form of lower, X l h , and upper, X u h , limit vectors contained within the n-dimensional unit cube I n ; and d h 2 f0; 1; 2; . . . ; pg is the index of one of the p þ 1 classes, where d h ¼ 0 means that the input vector is unlabelled. The jth hyperbox fuzzy set, B j is defined as follows:. for all j ¼ 1; 2; . . . ; m, where V j ¼ ðv j1 ; v j2 ; . . . ; v jn Þ is the min point for the jth hyperbox, W j ¼ ðw j1 ; w j2 ; . . . ; w jn Þ is the max point for the jth hyperbox, and the membership function for the jth hyperbox is Hyperbox fuzzy sets from the second layer are aggregated using the aggregation formula (4) in order to generate an output c k which represents the degree to which the input pattern X h fits within the class k. The transfer function for each of the third layer nodes is defined as. where U is the binary matrix with values u jk equal to 1 if the jth hyperbox fuzzy set is a part of the kth class and 0 otherwise; and c k 2 ½0; 1; k ¼ 0::p, represent the degrees of membership of the input pattern in the kth class with c 0 representing all unlabelled hyperboxes. Though seemingly very different the two groups of algorithms with their roots in probabilistic and fuzzy settings discussed in Section 1 have one major common characteristic. They all use a mechanism for building an input data model using a combination of local basis functions. In probabilistic approaches the data density estimation is carried out based on a number of multidimensional gaussians while in fuzzy approaches the triangular or trapezoidal membership functions serve a similar purpose of covering the input space. In other words after learning has been completed the input data space is covered by ''patches'' which are then combined to give a final classification model. The approach discussed in this paper also provides a mechanism for building an input data model by covering the input space with hyperbox fuzzy sets.. It has been shown that the GFMM neural network, with its definition of the inputs given in a form of real number intervals and hyperbox fuzzy sets used to represent input data clusters, provides a natural and efficient way of (c) Fig. 8 (continued ) representing and processing of missing data. No assumptions about the missing features or substitutions with fixed values have been used either during the classification or training of the classifier based on incomplete input patterns. What is also distinctive about our method is that the process of placing and adjusting the hyperboxes in the input space can lead to creation of some hyperbox fuzzy sets for which certain dimensions are not set. Such a hyperbox would effectively represent a subspace of the original input space. This could occur, for instance, when all the training data for one class would have consistently one or more features missing. Such a ''not missing at random'' ex- ample could cause serious problems for the methods which learn the parameters of the classifier model by using random initialisation and assume that some values have to observed for each feature and all the classes. In contrast such an example would be efficiently processed and is catered for within the GFMM training algorithm which is virtually the same whether the missing values are present or not. The classifier model uncertainty resulting from training on incomplete data (like in the example above) can be quite easily quantified by generating upper and lower class membership values during the GFMM classifier operation.. It has been argued that the primary goal of the classification when missing data are involved should be a reduction of the viable class alternatives based on the evidence present in the observed features. This can lead to the selection of a single winning class but in cases when two or more viable alternative classes are identified we have also proposed formulas utilising the hyperbox cardinality values (the data frequency information) to find the most probable class. Such a procedure leads to a more informative classification and could be crucial in situations when important decisions have to be made on the basis of such classification.. One more advantage of the GFMM classifier, shared with the decision trees or rule based classifiers, is its transparency and ability to provide easily interpretable explanation of the suggested classification. The hyperbox min-max values can be easily converted into a set of rules understandable to a nontechnical user. In cases when the classification results in a number of possible alternatives this would also provide an opportunity to highlight and suggest the features which would be most useful in reducing the number of alternatives. An example could include a hypothetical loan approval application where on the basis of initial processing of incomplete data a customer could be asked to provide an additional information, suggested by the classification system, which would reduce the risk/ambiguity of the decision to be made. The use of the proposed techniques within the medical diagnosis and financial decision support systems are currently under investigations. The testing of the proposed approach has been carried out on four wellknown data sets, namely IRIS, Wine, Ionosphere, and SatImage obtained from the repository of machine learning databases (http://www.ics.uci.edu/$mlearn/ MLRepository.html). The data sets have been selected due to their different characteristics concerning the number of samples, features and classes. They also represent four different classification problems drawn from different Table 2 Input patterns and classification results for the example shown in Fig. 4 Input patterns Classification Train Test IRIS 4 3 150 75 75 Wine 13 3 178 90 88 Ionosphere 34 2 351 200 151 SatImage 36 6 6435 3219 3216 domains, i.e. classification of iris plants (IRIS data set), three types of wine (Wine data set), radar signals used to describe the state of ionosphere (Ionosphere data sets) and multi-spectral remotely sensed data of satellite images (SatImage). The sizes and splits into the testing and training data for all four data sets are shown in Table 3 . In the first group, the GFMM neural network was trained using training data sets without missing values. Subsequently, the testing was conducted on all testing data sets for a factor of missing features ranging from 10% to 80%. At each level, the average values over 100 testing runs, each for randomly chosen missing features have been calculated. The results are shown in Figs. 5(a), 6(a), 7(a) and 8(a).. In the second group of experiments the GFMM was trained on the basis of training data sets with a factor of missing features ranging from 10% to 80%. In the third group of experiments both training and testing sets contained the same level of missing features again between 10% and 80% of all features. The results are shown in Figs. 5(c), 6(c), 7(c) and 8(c).. The greyed area in all figures represents cases where 2 or more classes have been identified as viable alternatives (according to Eqs. (22)- (24)) with the correct class always present. The area below the greyed area refers to the unique, correct classification where only one, correct class has been chosen. The area above the greyed area represents misclassified cases. The solid line marked with '+' within the greyed area represents the classification rates obtained when hyperbox cardinality information is used according to (16) . The dashed line with '' shows the classification results using (18) and the dash-dot line with '*' shows the classification results for the nearest neighbour classifier.. The first comment which can be made on the basis of Figs. 5(a), 6(a), 7(a) and 8(a) is that the substituting of the mean value for the missing features and classifying using the nearest neighbour classifier produced much worse results than the proposed method. This is not surprising since it is just a confirmation of what has been found by other researchers investigating such substitutions. It was though very surprising that the nearest neighbour method performed very well in the second group of experiments where the training data was generated with various levels of missing features and there were no missing data in the testing set. The reason for this can be explained on the basis of the two dimensional example shown in Fig. 9 . The data are the normal mixtures data introduced by Ripley [20] and used in many other publications [5] [6] [7] [8] 15] . The training data consists of two classes with 125 points in each class. Each of the two classes has bimodal distribution and the classes were chosen in such a way as to allow the best-possible error rate of about 8%. An independent testing set of 1000 samples has been drawn from the same distribution. In the example shown in Fig. 9 the training data with 80% of missing features have been utilised. It means that out of the total number of 500 feature values only 100 have been used. As we can see from Fig. 9(a) substituting the missing values with mean values completely changes the distribution of the data which makes such ''repaired'' data completely inappropriate to use for training of other classifiers though the performance of the 1-nn classifier on the testing set does not decrease dramatically. On the other hand, the hyperboxes shown in Fig 9(b) generated on the basis of the same training data with 80% of missing features cover the original training data very well and the classification performance is also very good.. Another interesting observation based on Figs. 5(a), 6(a), 7(a) and 8(a) is that the level of misclassified cases remains roughly the same for a whole range of missing features. The increasing level of missing features is reflected in a higher percentage of cases for which classification resulted in producing more than one viable alternative but almost always including the correct class. This can be particularly observed for IRIS and Wine data sets. Please note that 80% of missing features for IRIS data set means that a significant proportion of the testing cases will have all features missing.. However, as it can be seen for Ionosphere and SatImage (up to 60% of missing features) data sets even for a high level of missing features, one winning class is produced for a vast majority of testing cases (i.e. there is a very small grey area). This is an example of how the classification with missing features can result in a one class being selected if only the known features are discriminative enough to do so.. Another effect characteristic for the method used is illustrated for the Ionosphere data set in Fig. 7(b) and (c). It is an example for which a large number of hyperboxes have been retained in the final classifier model many of which did not have some of the dimensions set during the training. The upper and lower degrees of class membership overlap for both classes for a large number of testing cases which according to the test defined in Section 3.2 means that the two classes are initially treated as viable alternatives (large greyed areas). The use of cardinality information and Eqs. (16) and (18) result in a very good classification performance even for high ratios of missing features.. When all the hyperbox fuzzy sets are used for classification as in (18) it is very important that the parameter c is chosen carefully as it decides how wide an influence of a hyperbox fuzzy set around the core hyperbox will be. It is essentially the same problem when one needs to select the Parzen window width or more specifically the variance of a gaussian when multidimensional gaussians are used for the data density approximation as in [27] . While there are procedures for finding a good estimate for c based on cross-validation it does not change the fact that the results of applying (18) are very sensitive to the changes in c. The illustration of this phenomena is given for SatImage data set in Fig. 8(a) and Wine data set in Fig. 6(b)-(c) . The value of parameter c ¼ 1 was clearly too small (the hyperbox membership values did not decrease (c) Fig. 7 (continued ) quickly enough) since the performance of (18) was significantly worse than (16) . By increasing c to 2, therefore making the hyperbox fuzzy sets more local, a significant improvement could be noticed as shown in Fig. 8(b)-(c) for Satlmage data set and in Fig. 6(a) for the Wine data set. It is worth noting that the performance based on (16) was not affected by changes in c. It is because the hyperboxes to be used in (16) are selected as the best locally performing ones in the first place. The performance of the GFMM classifier utilising formula (16) on IRIS data set is similar to the results reported in [13, 18] and for the SatImage data set to the results reported in [1] though it has to be said that the emphasis in the above-mentioned publications had been on the learning with missing values while the testing was carried out on testing sets without missing features which in our case refers to results in Figs. 5(b) and 8(b) respectively.

O O O O O O O O O B I O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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Semantically enabled exploratory video search With the exponential growth of video data on the World Wide Web comes the challenge of efficient methods in video content management, content-based video search, filtering and browsing. But, video data often lacks sufficient metadata to open up the video content and to enable pinpoint content-based search. With the advent of the 'web of data' as an extension of the current WWW new data sources can be exploited by semantically interconnecting video metadata with the web of data. Thus, enabling better access to video repositories by deploying semantic search technologies and improving the user's search experience by supporting exploratory search strategies. We have developed the prototype semantic video search engine 'yovisto' that demonstrates the advantages of semantically enhanced exploratory video search and enables investigative navigation and browsing in large video repositories. The popularity of video and multimedia data has become a driving force for growth in the World Wide Web (WWW). Portal sites such as, e. g., YouTube. 1 and other content aggregators provide billions of video resources. While content management systems already are able to maintain large video repositories, search and pinpoint access of video content is left over to the user. Textual metadata enable contentbased retrieval of video resources. But, for the most part, WWW2010, April 26-30, 2010 , Raleigh, North Carolina.. . textual metadata originate from manual annotation carried out by authors or expert users. Automated video analysis techniques mostly provide low-level descriptions with sufficient accuracy, but the subsequent aggregation of different low-level features to derive high-level features concerning the contents is error-prone and algorithmic expensive. With the advent of Web 2.0 also user generated textual metadata such as, e. g., user tags or comments complement the authoritative manual annotation. Nevertheless, insufficient textual metadata is responsible for poor search performance.. Today, Google 2 has become synonymous for web search. The user enters a query string that might consist out of one or several keywords and Google's web search engine delivers (text) documents containing these keywords or multimedia documents annotated with metadata containing these keywords. In the majority of cases this approach is absolutely sufficient. But, users might have different information needs and different ways to search for information. If the search task is getting more complex, i. e. if a single document is not the answer to the user's search problem, different search strategies have to be applied. Moreover, if the user tries to get an overview of available information about a certain topic, today's web search engines often are flooding search results by millions. Thus, giving the user no chance to review but the first few resulting pages.. Sometimes, users are looking for a specific set of documents that do contain almost all the keywords of the query string (navigational searches), while in many other cases the user tries to gather information about a specific subject with no particular document in mind (research searches) [12] . In complex search tasks, the user first has to retrieve some facts (i. e. documents containing those facts), which are required to enable further search queries to solve the overall search problem. Often, the user is not familiar with the topic she is searching for, and sometimes, the user isn't sure about her search goal in the first place. This kind of search is known as 'exploratory search' [17] . In this paper, we address the problem of how to deploy exploratory search for video data by using semantic search technology.. In the Semantic Web as being an evolving extension of the WWW the meaning of information and services is well defined. Linked Data has become one of the most popular topics among the emerging Semantic Web [2] . By Linked Data we refer to a method of exposing, sharing, and connecting data via dereferenceable URI on the WWW. The Linking Open Data (LOD) project aims at making semantic data freely available to everyone and provides starting points to extract relationships among information resources. By mapping available video metadata to Linked Data, we show how to utilize cross-connections and data interdependencies to enable new ways of search space visualization and navigation. One of the key interlinking hubs of the LOD cloud is DBpedia 3 , the semantic counterpart of the online encyclopedia Wikipedia [1] . But, DBpedia often provides too many facts and details about entities. Thus, sophisticated heuristics have to be applied to achieve a ranking to identify the most important, i. e. interesting facts for a given entity [28] .. Contrariwise to facetted search approaches, which aim to further refine an original search query by clustering the search results according to common properties, exploratory search broadens the scope of the search query by suggesting associated terms, concepts, and resources. These exploratory search suggestions can be used to navigate among the entire search space and to explore the repository content by user guided browsing [27] . Thus, enabling the possibility to achieve search results that the user wasn't looking for in the first place by serendipity.. The major contributions of this paper are the following: We have developed a frequency-based heuristics for ranking entity properties and relationships of LOD resources including efficient offline indexing for semantic video search. The most relevant properties are applied for the generation of contentually related search suggestions, and the quality of the overall exploratory search approach is shown with appropriate evaluation methods.. The paper is organized as follows. Section 2 presents related work and introduces to exploratory search, the yovisto semantic video search, and the general problem of evaluating exploratory search systems. Section 3 details, how Linked Data facilitates exploratory semantic search. In Section 4, our evaluation method for exploratory video search is presented together with first qualitative evaluation results. The last section concludes the paper with a brief outlook on future work. In this paper, we addressed the problem of how to deploy exploratory search for video data by using semantic search technology and demonstrated an improved exploratory search for the yovisto video search engine along with an evaluation of the exploratory search process. Exploratory search is supported by heuristics based on semantic data from the LOD resources, which are used to augment direct search result with navigational information that might be also relevant for the user. We presented a new frequency-based heuristic for determining related information resources and evaluated the exploratory search feature by using a comparative evaluation technique. Thus, we were able to objectively point out the positive aspects of the exploratory search approach such as a higher task accomplishment, higher motivation and satisfaction rates.. Exploratory search is at it's early stages as a research area. Currently, there does not exist an overall accepted bestpractice neither on how to realize nor on how to evaluate exploratory search. We aimed to overcome this shortcoming by strongly relating our accomplishments and methods to existing research in this area.. Since the heuristics for the determination of 'interesting' relationships among data entities play a major role in the exploratory search approach, we are developing objective evaluation procedures for the optimization of these heuristics. In addition, improvements of the graphical user interface explicitly supporting the investigative and navigational aspect of our approach will be considered in future work.. For better support in data space navigation, future work is focussed on the combination of facetted and explorative search features to satisfy the searchers curiosity and to foster serendipitous discovery. This section covers research related to the work presented in this paper. This includes various approaches of semantic and exploratory search emphasizing the focus on video search in general and the yovisto academic video search platform in particular, including evaluation.. According to Guha et al. 'Semantic Search is the application of the Semantic Web to search' [12] . Correspondingly, Semantic Web resources and technologies are applied to augment the traditional search scenario. In the Semantic Web as being an extension of the current WWW information is complemented by formal knowledge representations with well defined meaning [3] . Linked Open Data has become one of the most popular topics among the emerging Semantic Web [4] . It supports the Semantic Web vision by leveraging free access to interconnected semantic data. In this paper we present a semantically enhanced exploratory search based on LOD resources, in particular on DBpedia, which contains structured information from the popular online encyclopedia Wikipedia [1] . A comprehensive survey on different approaches to Semantic Search is given in [16] , while in [26] a formal model of ontology-based Information Retrieval in general is presented.. There are two main challenges for Semantic Search [12] : (1) the query input has to be mapped to concepts and entities [14, 18] and (2) the search domain has to be augmented with semantic content [9] . We address both challenges in this paper.. Alternatively, Semantic Search is often referred to as retrieval of data from the Semantic Web, as being represented by semantic search engines such as Sindice [20] . The work in this paper does not relate to this interpretation of Semantic Search.. Exploratory search assists in exploring the data space by providing data visualization, related content, a wide range of sophisticated selection options, and various possibilities for analysis. In order to allow a multi-facetted, multi-tactical, opportunistic, and open ended search, exploratory search is targeting learning and investigation tasks [17, 29] . To enable this complex and investigative retrieval option, underlying metadata including implicit and formerly unknown relationships among resources need to be fully made accessible. We extend and complement previous work on this topic by applying improved exploratory search to video data on the WWW and by framing suitable evaluation procedures for exploratory search [27, 28] . A motivation to use exploratory search over traditional search methods is given in [17] .. State of the art video retrieval systems use a combination of visual and textual feature extraction for search index generation and combine these techniques with machine learning procedures [25, 11, 8] . However, the chosen feature extraction depends on domain and task characteristics and determines the quality of the retrieval system.. Yovisto is a video search engine specialized on video recordings of academic lectures and conference talks [22, 13, 27] . To enable pinpoint search access within the video recordings a combination of automated video analysis and user generated collaborative annotation is deployed in connection with fine granular, time-dependent metadata. We extend the search capabilities of yovisto by adding an exploratory search feature that enables the user to browse the content of the underlying video repository in a multi-facetted way. In difference to popular recommender systems, our approach is neither based on logfile analysis and statistical usage analysis of content popularity, nor on similarity-based methods such as query by example [15, 10, 5] .. Evaluation of traditional information retrieval systems is based on rather quantitative than qualitative measurements of the achieved retrieval results. The retrieval results are compared to a ground truth resulting in an objective assessment of the achieved quality, i. e. by statistical classifications such as recall and precision. Evaluation strategies from the well known TrecVid benchmarks fall into this category. Their focus lies on pure system evaluation. Evaluation based on direct user involvement, referred to as 'User evaluation' is explicitly mentioned as out of scope for these benchmarks [24] .. To demonstrate the added value of newly implemented retrieval features it is suitable to execute the same evaluation task with and without application of the specific retrieval feature. The differences between the resulting measurements mark the effect of the new retrieval feature. Singh et al. apply this evaluation strategy in [23] . We adopt this approach for our evaluation to demonstrate the usefulness of exploratory search in yovisto. The motivation to use this strategy lies in the subjective and investigative nature of exploratory search [17] , which makes it difficult to determine an objective ground truth for reference. Therefore, we apply additional qualitative evaluation measures by monitoring user satisfaction throughout the work task, as proposed in [21] .. In [7] a framework for evaluation of interactive information retrieval systems is presented where the user task is formulated in a cover story leading to the work task and finally to the actual search task. Two evaluation strategies are compared, which distinguish multiple types of relevance, inter alia, so-called 'Situational Relevance', which reflects the dynamic nature of relevance [6] . Situational relevance also applies to exploratory search scenarios, where the user's relevance scale may be influenced by the receipt of new information.

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


Cost Considerations in Network on Chip Abstract-Systems on Chip (SoCs) require efficient inter-module interconnection providing for the required communications at a low cost. We analyze the generic cost in area and power of Networks on Chip (NoCs) and alternative interconnect architectures: a shared bus, a segmented bus and a point-to-point interconnect. For each architecture we derive analytical expressions for area, power dissipation and operating frequency as well as asymptotic limits of these functions. The analysis quantifies the intuitive NoC scalability advantages.Next we turn to NoC cost optimization. We explore cost tradeoffs between the number of buffers and the link speed. We use a reference architecture, termed QNoC (Quality-of-Service NoC), which is based on a grid of wormhole switches, shortest path routing and multiple QoS classes. Two traffic scenarios are considered, one dominated by short packets sensitive to queuing delays and the other dominated by large block-transfers. Our simulations show that network cost can be minimized while maintaining quality of service, by trading off buffers with links in the first scenario but not in the second.Index Terms-network on chip, scalable interconnect, wormhole buffering, cost minimization.Networks on Chip (NoCs) can help solve major design challenges of integrated Systems on Chip (SoCs) [1]-[12] , including modularity and reuse, design productivity, global wire speed/power optimization, synchronization, and communication error recovery. However, since VLSI is extremely cost-sensitive, the required communication Quality-of-Service (QoS) must be provided at a minimal cost [9] , [11] , [26] , [27] . QoS is associated primarily with latency and throughput. Cost is measured by chip-area and power dissipation. The purpose of this paper is to make a quantitative comparison between the generic cost of a NoC and the cost of other interconnection schemes, and to explore cost minimization options within a specific NoC architecture.Traditional [10] . However, no quantitative cost analysis has been conducted so far. This paper analyzes and quantifies the cost and performance advantages of a network based interconnection scheme over other interconnection alternatives for future SoCs. In particular, we analyze the area and power cost of a packet-switched NoC in comparison with non-segmented (shared) system bus (NS-Bus), segmented system bus (S-Bus) and point-to-point (PTP) interconnect. Assuming a given set of Quality-ofService requirements we derive analytical expressions for the wire area, power and operating frequency of each interconnection scheme. With an increasing number of system modules, simple asymptotic limits of these expressions are derived. The results clearly quantify the scalability advantage of NoC over the traditional alternatives.Switched networks and techniques for their design have been developed for computer networks and for multiprocessor systems, for example [16]-[22]. However, a unique set of resource constraints and design considerations exists for an on-chip environment. [23] . Thus, the area of a generic NoC can be approximated by the wiring area used for the NoC links. Shortest-path routing guarantees minimal wire length and power dissipation in the links.Some studies investigated optimum wormhole buffering for increased router performance in general computer networks [28] , [29] . Performance-power cost tradeoff was explored by selecting appropriate packet size in [26] . Unlike computer networks which are built for on-going expansion, future growth and standards compatibility, on-chip networks can be designed and customized for an a-priori known set of computing resources and pre-characterized traffic patterns among them. These imply that various design parameters of the network architecture such as buffer size and link bandwidth allocation can be designed for Haifa 32000, Israel 2 specific implementations in order to provide a required QoS for known traffic patterns. Moreover, one can apply a tradeoff between these parameters to achieve a more cost-effective NoC implementation at a given QoS specification. Based on the above considerations, we present a NoC cost minimization process by exploring the influence of increasing the number of wormhole buffers versus decreasing link bandwidth (by reducing the number of wires). For this tradeoff study we use a specific NoC architecture termed QNoC (Quality-of-Service NoC) [1], which is based on a planar grid of switches that route the traffic according to a fixed shortest path (X-Y based) discipline. It uses input buffering scheme and employs multi-class wormhole forwarding to support multiple service priority classes. The optimization process attempts to reduce the cost while supporting the different QoS classes and the QoS requirements for each class. We study two different system traffic scenarios. The first scenario is dominated by short packets that are sensitive to queuing delays. The second scenario is dominated by long block-transfers consisting of long packets. We show by simulations that in the first case NoC area cost minimization is achieved by adding wormhole buffers and decreasing link bandwidth up to an optimal value. However, this is not true in the blocktransfer dominated traffic scenario where there is no cost advantage in increasing the number of buffers above the minimum. The total area cost is estimated by calculating total area occupied by wires, and adding to it the estimated area occupied by the packet switch logic (buffers, tables etc.). The power cost is based on summation of the traffic that traverses each wire length and is received by input stages.The rest of this paper is organized as follows: Section 2 describes the example QNoC architecture, Section 3 presents an analytical comparison between a generic NoC and alternative architectures, Section 4 presents QNoC cost minimization process and provides cost minimization examples for several system traffic scenarios along with simulation results, and finally Section 5 concludes.The QNoC architecture and design process were presented in [1] . In this section we first present a brief overview and then develop additional architecture details (subsection 2.4). The QNoC architecture is based on a grid topology and wormhole packet routing. Links are assumed reliable 1 and backpressure is applied between stages resulting in a loss-less network. Packets traverse the network along the shortest route, thus minimizing power dissipation and maximizing network resource utilization.QNoC comprises routers interconnected by point-to-point links. Network topology can vary depending on system needs and module sizes and placement. Each system module is connected to a router ( Figure 1 ) via a standard interface, whose bandwidth is adapted to the communication needs of that module. The bandwidth of each inter-router link is similarly adjusted to accommodate the expected traffic and fulfill QoS requirements at the specific link. Link and interface bandwidth are adjustable by changing either the number of wires or the data frequency, or both. In addition, a module may be connected to the network through more than one interface. Routing is performed over fixed shortest paths, employing a symmetric X-Y discipline whereby each packet is routed first in an "X" direction and then along the perpendicular dimension or vice versa 2 . Network traffic is thus distributed non-uniformly over the mesh links, but each link's bandwidth is adjusted to its expected load, achieving an approximately equal level of link utilization across the chip.We identify four different types of communication requirements and define appropriate service levels (SL) to support them:1 Or made reliable using error correction. 2 Simple "around the block" modification is employed where needed.Signaling covers urgent messages and very short packets that are given the highest priority in the network to assure shortest latency. This service level represents interrupts and control signals and alleviates the need for dedicated wires.Real-Time service level guarantees bandwidth and latency to real-time applications, such as streamed audio and video processing. This service is packet based; a maximal level of guaranteed bandwidth is allocated to each real-time link and should not be violated.Read/Write (RD/WR) service level provides bus semantics and is designed to support short memory and register accesses.Block-Transfer service level is used for the transfer of long messages and blocks of data, such as cache refill and DMA transfers.We establish a priority ranking, where Signaling is given the highest priority and Block-Transfer the lowest. QNoC employs preemptive communication scheduling where data of a higher priority packet is always transmitted before that of a lower service level (a round-robin is employed within service levels). Additional service levels may be defined if desired. For instance, the RD/WR service level may be split into normal and urgent RD/WR sub-levels.Packets carry routing information, command and payload. The command field identifies the payload, specifying the type of operation. The packet is divided into multiple flits following [22] . Flit transfer over the inter-router link is controlled by handshake. Networks on Chip (NoCs) can help solve major design challenges of integrated Systems on Chip (SoCs) [1] - [12] , including modularity and reuse, design productivity, global wire speed/power optimization, synchronization, and communication error recovery. However, since VLSI is extremely cost-sensitive, the required communication Quality-of-Service (QoS) must be provided at a minimal cost [9] , [11] , [26] , [27] . QoS is associated primarily with latency and throughput. Cost is measured by chip-area and power dissipation. The purpose of this paper is to make a quantitative comparison between the generic cost of a NoC and the cost of other interconnection schemes, and to explore cost minimization options within a specific NoC architecture.. Traditional solutions for on-chip global communication include various shared-bus structures [13] - [15] and ad-hoc point-topoint interconnections. The lack of scalability of these approaches was qualitatively discussed in [2] , [7] , [10] . Advantages of spatial-reuse packet/wormhole switched networks were reported and explored in comparison with buses in [1] , [2] , [4] , [6] , [9] , [10] . However, no quantitative cost analysis has been conducted so far. This paper analyzes and quantifies the cost and performance advantages of a network based interconnection scheme over other interconnection alternatives for future SoCs. In particular, we analyze the area and power cost of a packet-switched NoC in comparison with non-segmented (shared) system bus (NS-Bus), segmented system bus (S-Bus) and point-to-point (PTP) interconnect. Assuming a given set of Quality-ofService requirements we derive analytical expressions for the wire area, power and operating frequency of each interconnection scheme. With an increasing number of system modules, simple asymptotic limits of these expressions are derived. The results clearly quantify the scalability advantage of NoC over the traditional alternatives.. Switched networks and techniques for their design have been developed for computer networks and for multiprocessor systems, for example [16] - [22] . However, a unique set of resource constraints and design considerations exists for an on-chip environment. As described in [2] , [10] , memory and computing resources are relatively more expensive on-chip, while relatively more wires are available. As a result, many NoC architectures are based on wormhole packet routing [1] , [2] , [7] , [8] , since wormhole routing reduces latency and buffer requirements in the routers [2] , [22] , [23] . Thus, the area of a generic NoC can be approximated by the wiring area used for the NoC links. Shortest-path routing guarantees minimal wire length and power dissipation in the links.. Some studies investigated optimum wormhole buffering for increased router performance in general computer networks [28] , [29] . Performance-power cost tradeoff was explored by selecting appropriate packet size in [26] . Unlike computer networks which are built for on-going expansion, future growth and standards compatibility, on-chip networks can be designed and customized for an a-priori known set of computing resources and pre-characterized traffic patterns among them. These imply that various design parameters of the network architecture such as buffer size and link bandwidth allocation can be designed for Haifa 32000, Israel specific implementations in order to provide a required QoS for known traffic patterns. Moreover, one can apply a tradeoff between these parameters to achieve a more cost-effective NoC implementation at a given QoS specification. Based on the above considerations, we present a NoC cost minimization process by exploring the influence of increasing the number of wormhole buffers versus decreasing link bandwidth (by reducing the number of wires). For this tradeoff study we use a specific NoC architecture termed QNoC (Quality-of-Service NoC) [1] , which is based on a planar grid of switches that route the traffic according to a fixed shortest path (X-Y based) discipline. It uses input buffering scheme and employs multi-class wormhole forwarding to support multiple service priority classes. The optimization process attempts to reduce the cost while supporting the different QoS classes and the QoS requirements for each class. We study two different system traffic scenarios. The first scenario is dominated by short packets that are sensitive to queuing delays. The second scenario is dominated by long block-transfers consisting of long packets. We show by simulations that in the first case NoC area cost minimization is achieved by adding wormhole buffers and decreasing link bandwidth up to an optimal value. However, this is not true in the blocktransfer dominated traffic scenario where there is no cost advantage in increasing the number of buffers above the minimum. The total area cost is estimated by calculating total area occupied by wires, and adding to it the estimated area occupied by the packet switch logic (buffers, tables etc.). The power cost is based on summation of the traffic that traverses each wire length and is received by input stages.. The rest of this paper is organized as follows: Section 2 describes the example QNoC architecture, Section 3 presents an analytical comparison between a generic NoC and alternative architectures, Section 4 presents QNoC cost minimization process and provides cost minimization examples for several system traffic scenarios along with simulation results, and finally Section 5 concludes. In this paper we quantified analytically the scalability of NoC as a communication architecture for future SoCs and its generic advantage over alternatives in terms of wiring area and power cost for future SoCs, when the number of communicating modules on a chip rapidly increases. We also presented a NoC cost minimization process in which we explored the influence of increasing wormhole buffers versus reducing network links bandwidth, while preserving the required QoS for all classes of service, in order to further minimize the area cost of the NoC. We defined criteria for the characteristics of system communication traffic allowing such cost minimization. We showed several QNoC cost optimization examples in which different levels of area reduction were achieved according to the nature of system traffic among the modules. In a RD/WR dominated, highly utilized network, an example of 10% reduction from original QNoC area was demonstrated. No area cost minimization could be achieved in Block-Transfer dominated traffic scenario. The results clearly show the advantage of the wormhole routing technique in networks on chip, since in many cases adding network buffers beyond the minimum value is very expensive and provides a diminishing return.

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


Spatial-temporal understanding of urban scenes through large camera network Outdoor surveillance cameras have become prevalent as part of the urban infrastructure, and provided a good data source for studying urban dynamics. In this work, we provide a spatial-temporal analysis of 8 weeks of video data collected from the large outdoor camera network at UCSB campus, which consists of 27 cameras. We first apply simple vision algorithm to extract the crowdedness information in the scene. Then we further explore the relationship between the traffic pattern observed from the cameras with activities in the nearby area using additional knowledge such as campus class schedule. Finally we investigate the potential of discovering aggregated human movement pattern by assuming a simple probabilistic model. Experiment has shown promising results using the proposed method. There is a growing interest in studying and analyzing urban dynamics for purposes such as traffic forecasting, city planning and facility management. However, obtaining and modeling large, real world observational data is a challenging and costly task. In the past, researchers have used different digital traces of city-wide urban infrastructure as medium to study urban dynamics. For example, González et al. used cellular network data to study city dynamics and human mobility [5] . McNamara et al. [10] and Liu et al. [8] used data collected from RFID-enabled metro systems to monitor and predict co-location patterns among mass transit users. Froehlich et al. analyzed the digital footprint of bicycle usage from shared bicycling system to uncover patterns of human behavior and infer cultural and geographic aspects of the city [4] . All these methods utilize implicit data source to study urban dynamics.. On the other hand, outdoor cameras have become prevalent as part of the urban infrastructure. They provide an excellent data source to directly observe and understand urban dynamics. Previously, researchers have attempted to study videos from outdoor cameras, but they mainly focused on a few cameras covering small area and did not consider any relationship between visual data and external knowledge of the scene [6] . Here we demonstrate the potential of urban dynamic understanding through multiple cameras covering a large area. In particular, we collected videos from a network consisting of 27 cameras at the University of California at Santa Barbara (UCSB) [7] spanning a period of eight weeks. We show that better scene understanding can be achieved by analyzing the videos with simple vision techniques along with additional information of the area, such as campus class schedule. The main contributions of this paper are:. • Gaining insights of the urban scene through analyzing videos in a large camera network. To the best of our knowledge, this is the first attempt to understand the dynamic scenes in large area through a camera network over an extended period.. • Reducing uncertainty in visual observation through information fusion. By incorporating campus class schedules, we can better explore the relationship between the traffics observed from the cameras with activities in the nearby area.. • Demonstrating the potential of discovering aggregated human movement pattern. Assuming a simple probability model, we can obtain a fine-scale spatial-temporal estimation of the pattern. The rest of the paper is organized as follows. Section 2 explains the camera network dataset for our experiment. Section 3 shows the details of our approach to analyze the videos to obtain spatial-temporal understanding of the scenes. Interesting findings from our reasoning will be shown in this section. Finally we will have our conclusion and future work at section 4. In this paper, we have demonstrated the potential of spatialtemporal scene understanding through large camera network. In particular, we used simple vision algorithm to extract the crowdedness of the scene, and explored its relationship with campus activities based on class schedules. Similar analysis can be applied to other urban area with more diverse sources of information, e.g., theater schedule, mall/restaurant open time, office hours etc. In the future, we will deploy other sensors such as GPS to gain more quantitative ground-truths of human movement.

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


Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors We study a new method for proving lower Arithmetic circuits are the standard model for computing polynomials. Proving super-polynomial lower bounds for the size of arithmetic circuits and formulas is an outstanding open problem. Here we study a new method for proving lower bounds for several subclasses of circuits. Roughly speaking, this method is based on bounding the correlation between the coefficients' vector of a polynomial and the coefficients' vector of any product of two polynomials with disjoint sets of variables.. We prove tight exponential size lower bounds for two previously studied models of arithmetic circuits: monotone circuits (that are circuits that use only positive real numbers), and orthogonal multilinear formulas (that are formulas that are only allowed to sum two polynomials whose coefficients' vectors are orthogonal). We also prove n Ω(1) lower bounds for the depth of two new models of multilinear formulas: non-cancelling formulas, and noise-resistant formulas. Here are rough definitions of these two models: Non-cancelling formulas are formulas that are not allowed to sum two polynomials that almost cancel each other -the non-cancelling model is a generalization of both the monotone model and the orthogonal model. Noise-resistant formulas are formulas that compute well even when some small noise occurred during the computation.. One important ingredient of our proof is an explicit map f : {0, 1}. n → {0, 1} that has exponentially small maximalpartition discrepancy (see Section 3.1 for a formal definition). This notion is also related to extractors construction and to communication complexity, as we will describe below. For the rest of this section, n = 12sp is an integer, where p ∈ N is prime and s ∈ N is a large enough constant (given in Theorem 5.1), and f is the multilinear polynomial over the set of variables X = {x 1 , . . . , x n } with coefficients in {1, −1} defined below in Section 5.2. We note that f is in VNP, Valiant's algebraic analog of NP.. The polynomial f has negative coefficients, and so it can not be computed by a monotone circuit. However, we can use f to define a new polynomial F ∈ C[X] with coefficients in {0, 1}, for which we will also be able to prove lower bounds. The polynomial F is defined as follows: for a monic monomial m in the variables X, the coefficient of. The following theorem gives a tight lower bound for the size of monotone arithmetic circuits for F . Theorem 2.3. Let Φ be a monotone arithmetic circuit over the field R and over the set of variables X computing the polynomial F defined above. Then, |Φ| = 2 Ω(n) .. The following theorem gives a tradeoff between the depth and τ (the "amount of non-cancelling") for a syntactically multilinear arithmetic formula computing f . For example, a 2 − √ n -non-cancelling syntactically multilinear arithmetic formula that is at least 2 − √ n correlated with f is of depth Ω( √ n). Note that the smaller τ is, the less restricted the formula is. So, for proving a lower bound, the smaller τ is, the stronger the lower bound is. . where f is the polynomial defined in Section 5.2, and we think of Φ and f as vectors of coefficients. Then,. In particular, if τ < 2 and c ≥ 1/2, d = Ω n log(2/τ ) , and if τ ≥ 1 and c ≥ 1/2, |Φ| = 2 Ω(n) .. Since we do not know how to balance arithmetic formulas in the non-cancelling model, Theorem 2.4 does not imply an exponential lower bound for the size (for small τ ). However, since every orthogonal arithmetic formula is 1-non-cancelling, we have the following exponential lower bound for the size of orthogonal syntactically multilinear arithmetic formulas computing f . A similar trade-off holds for a noise-resistant computation of f . For example, a syntactically multilinear arithmetic formula that is 2 − √ n -noise-resistant to computing g is of depth Ω( √ n).. Theorem 2.6. Let 0 < ε < 1, and let Φ be a syntactically multilinear arithmetic formula of depth d ∈ N over the field C and over the set of variables X that is ε-noise-resistant to computing g, where g is defined in (3.1) .. The proof of Theorem 2.3 is in Section 4. Due to space limitations, the proofs of all the other theorems are not given here. However, the proof of Theorem 2.3 already contains most of the main ideas needed for the proofs of the other theorems. The proofs of the theorems above consist of two main steps. To describe the main ideas, we focus on the proof for monotone circuits (the proofs for multilinear formulas are more technical).. The first step is showing that polynomials that are computed by monotone circuits have a 'special' structure -they can be represented as a sum of what we call product polynomials (see Section 3.1 below for the definition of product polynomials). More specifically, a multilinear polynomial g that is computed by a monotone circuit of size roughly s can be written as g = s i=1 g i for product polynomials g 1 , . . . , g s . This representation of monotone circuits follows by an inductive argument (can be thought of as 'balancing' the circuit).. The second step of the proof employs the fact that f (defined above) has 'small' correlation with any product polynomial to conclude the lower bound. Here is a rough sketch of this argument. Let F be the monotone polynomial defined above using f . Assume that F has more 1's than 0's; that is, cor(f, F ) is large. Also assume that F can be computed by a monotone circuit of size s. Using the first step, F = s i=1 g i for product polynomials g 1 , . . . , g s . Thus, cor(f, g i ) ≥ cor(f, F )/s for some i. Since f has small correlation with any product polynomial, and in particular with g i , it follows that s must be large. We note that there are some subtleties here that we did not address.

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