Distributed Fault Detection in Sensor Networks using a Recurrent Neural Network In long-term deployments of sensor networks, monitoring the quality of gathered data is a critical issue. Over the time of deployment, sensors are exposed to harsh conditions, causing some of them to fail or to deliver less accurate data. If such a degradation remains undetected, the usefulness of a sensor network can be greatly reduced. We present an approach that learns spatio-temporal correlations between different sensors, and makes use of the learned model to detect anomalous sensors by using distributed computation and only local communication between nodes. We introduce SODESN, a distributed recurrent neural network architecture, and a learning method to train SODESN for fault detection in a distributed scenario. Our approach is evaluated using data from a real-world sensor-network deployment, and shows good results even with imperfect link qualities and a number of simultaneous faults. Wireless sensor networks (WSN) are increasingly being deployed over extended periods of time [5] , in particular for environmental monitoring applications. To facilitate long term deployments in remote areas, nodes in WSN are typically small computers with little memory, powered by solar energy and rechargeable batteries, with some way of communication between each other. In this work, we are interested in detecting problems that manifest in faulty sensor readings as a result of unavoidable degradation of some of the sensors using recurrent neural networks (RNN).. RNN have successfully been used in applications to model nonlinear time series, such as electricity load prediction [1] , to model finance data [21] , or even migration patterns O. Obst [20] . Training RNN comes, traditionally, with a high computational cost, using approaches like backpropagation through time [26] . These training methods can suffer from problems like slow convergence and vanishing gradients [2, 11] . A number of approaches have been developed to overcome these difficulties (see also [3, 8] ). Long short-term memory networks [12] , for example, solve the problem by using a more sophisticated architecture and training method. Other approaches are self-organizing maps for time series [7] , or Evolino [22, 23] .. Another recent and efficient approach is called echo state networks (ESN) [15] , where the lower computational complexity is result of using a fixed, randomly connected "reservoir" of neural units in the recurrent layer. Here, only connections to output units are changed during training (see Fig. 1 ). This approach will form the base for our work.. ESN have also been applied to anomaly detection in WSN [4, 19] . In these approaches, either one ESN on a central machine has been used, or one instance of an ESN on each node. The former requires to pass all sensor readings through the sensor network, whereas the latter may have only limited information available locally. While the limitation to only locally available information could be overcome by explicit communication of sensor readings by either multi-hop connections or a fully connected network, this is typically not an option due to energy limitations. Multi-hop connections suggest some form of transmission control in the network protocol, resulting in multiple attempts to transmit the same data in case of poor connections. Transmitting information in a WSN is expensive in terms of the energy used, for example, [6] state the cost for transmitting 1 bit corresponds to 800-1000 instructions on a particular WSN and processor. We will also discuss energy consumption later in the paper.. Here, we introduce anomaly detection using spatially organized distributed ESNs (SODESN, [18] ), an architecture that distributes a recurrent neural network over an entire sensor network, and, at the same time, imposes a structure (topology) on its connectivity matrix that uses only local communication in the WSN (Sect. 3). ESN with a one-dimensional topology have already been subject to some investigations in [17] . In SODESN, all information processing to predict and detect anomalies is distributed over the entire sensor network. To have computation decentralized, and, instead of using multi-hop communication, to use only local communication with neighbor nodes without a requirement for end-to-end transmission control is one of the advantages of the proposed approach. This is beneficial if not all gathered information needs to be forwarded to a central database, but could be used to make local decisions about whether a particular situation is anomalous or not.. In Sect. 4, we present a training method for fault detection using the SODESN architecture and a fault detection approach using SODESN in a WSN. This learning method needs to be executed once per node, and can also be run on the WSN. SODESN learn a model of normal behavior of sensor nodes based on information from other sensors. The fault detection, once deployed, monitors differences between the model and actual sensor readings in a distributed way. In a neural network distributed over a WSN propagation of activations may be unreliable, and we are able to show that our approach is able to deal with these dropouts. Another advantage of our approach is that it is resilient to multiple faults because we can use recursive prediction to keep the network stable for longer. We demonstrate the capabilities of our approach with actual WSN data (Sect. 5) and discuss our results in Sect. 6. In the following section, we give a brief introduction to ESN, for a comprehensive introduction, we refer to [14] . Excellent overviews on recent advances in ESN can be found in [8, 16] . We presented SODESN, a distributed recurrent neural network architecture for creating models of dynamical systems. We introduced an offline learning approach for SODESN that is closely related to training ESN and inherits the low computational complexity of the original approach. We then presented an approach to train SODESN for fault detection in WSN, where predictions of sensor values are made based on information from sensors on local neighbor nodes. Our evaluations on real-world data show that our approach can be used to build models of dynamic time series and help to detect sensor faults. We have shown that the approach is robust to WSN link failures through its distributed computation and local communication. SODESN perform well for anomaly detection even in presence of multiple faults, assuming realistic link qualities. SODESN also scale well with increasing number of WSN nodes in the network, because they use only local communication with nearest neighbors. Our first experiment was used to show that SODESN can perform a useable one-step-ahead prediction even in the presence of a faulty sensor. With perfect communication, the prediction quality of the distributed approach only imposes a specific connectivity onto the network, similar to a network with a permutation matrix. Consequently, performance is close to the non-distributed ESN versions. Some of the differences may be explained by the different approach to scaling the weight matrix, as a result of treating proxy connections essentially as input weights. In all variants, the quality improves slightly in presence of faulty input-this may be due to our training method using random data locally. When the input is normal, this may bias the output so that the error of a one-step-ahead prediction becomes larger. Results from our second experiment are about the detection accuracy of our approach. These are interesting for a number of reasons: first, we have shown that using the SODESN is useful for anomaly detection even in the presence of several faults, and imperfect communication. With only 20 reservoir units per sensor node, the reservoir size was fairly small for ESN. This is an important feature in the context of sensor networks, since the number of units in our approach directly correlates with the amount of communication introduced. Less communication is desirable, since an increase in communication corresponds to an increase in energy consumption. The effect of dropouts may be similar to a regularization, similar to the approach recently suggested in [10] , but overall was small in our application. On the other hand, the randomized half of each local input sequence used during training has the important effect that the local outputs are not just a variation of the local inputs of the same unit. In this way, the local inputs still contribute to the overall network dynamics, and have a chance to be used to help detect anomalies on neighbor units, but are less influential on the local outputs.. The detection was more difficult for stuck-at-fault errors, possibly a result of the usually slow change of the solar voltage readings. These types of errors may be easier to detect if the variables measured have a faster dynamics.. Our offline learning approach performs a computation on the whole training time series. The method can be executed locally, but for larger data sets it may also be shifted to a machine outside the network that is computationally more capable than typical WSN nodes. The learning then computes sets of output weights for each sensor node. A way to deal with less correlated sensors may therefore be to successively improve the SODESN by re-training on increasingly larger data sets and exchanging the learned weights over time.. The ability to work with imperfect communication also means that SODESN communication does not have to be tightly synchronized with each other, at least in typical sensor networks where communication rates are around 1 Hz. In our approach, it would also be possible to bridge gaps in communication by collecting data locally, and to run the SODESN on larger blocks of data when communication becomes available again.. From a network connectivity point of view, it appears that there are three options to use a neural network over the WSN, which do have an impact on the energy consumption: (1) forward all sensor data to a base station and process it using one ESN there, (2) use a neural network on each node that receives sensor information from all other nodes, (3) distribute the processing, like in the SODESN approach presented here.. In WSN that only cover a small area, each node can reach the base station without using multiple hops over neighbor nodes. In larger deployments, this will typically not be the case. A number of publications are dedicated to energy-efficient communication in WSN, and the actual energy consumption of a WSN depends strongly on the network topology and radio hardware of the system, e.g., [6, 9, 25] . To use our network topology in Fig. 2 as an example, we have 32 nodes that are able to talk to their nearest neighbors. For an estimate of energy consumption, we make the simplifying assumptions that the base station is one of the nodes in the WSN-typically, the base station is outside the WSN area-, that packets are transmitted via a shortest path over the network, with no overhead for multi-hop transmissions (i.e., a 2-hop transmission is twice as expensive as a single hop transmission), and that we need to transmit one packet for each transmission independent of the setup. In other words, for a central ESN in our example network, the number of packets to be sent for all nodes to talk to the base is the sum of the manhattan distances between each node and the base. For an ESN on each node, the number of packets is the sum of the manhattan distances from each node to all other nodes. For SODESN, the number of packets to be transmitted for all nodes is the sum of nearest neighbors of all nodes, since they talk to all their neighbors with a distance of 1. Table 3 shows the number of packets transmitted for the different connectivities. Only in the case where the base station is in the center of the WSN, the number of packets is lower for an approach processing all data on the base. An approach with a central ESN on the base station would thus be preferred in small WSN, when the number of hops is low. For larger WSN with local communication, the SODESN approach uses less packets than an approach with an ESN on the base node. The column 'total' is the product of the number of packets and the number of hops per packet

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


Browsing on small displays by transforming Web pages into hierarchically structured subpages We propose a new Web page transformation method to facilitate Web browsing on handheld devices such as Personal Digital Assistants (PDAs). In our approach, an original Web page that does not fit on the screen is transformed into a set of subpages, each of which fits on the screen. This transformation is done through slicing the original page into page blocks iteratively, with several factors considered. These factors include the size of the screen, the size of each page block, the number of blocks in each transformed page, the depth of the tree hierarchy that the transformed pages form, as well as the semantic coherence between blocks. We call the tree hierarchy of the transformed pages an SP-tree. In an SP-tree, an internal node consists of a textually enhanced thumbnail image with hyperlinks, and a leaf node is a block extracted from a subpage of the original Web page. We adaptively adjust the fanout and the height of the SP-tree so that each thumbnail image is clear enough for users to read, while at the same time, the number of clicks needed to reach a leaf page is few. Through this transformation algorithm, we preserve the contextual information in the original Web page and reduce scrolling. We have implemented this transformation module on a proxy server and have conducted usability studies on its performance. Our system achieved a shorter task completion time compared with that of transformations from the Opera browser in nine of ten tasks. The average improvement on familiar pages was 44%. The average improvement on unfamiliar pages was 37%. Subjective responses were positive. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or permissions@acm.org. Internet-enabled Personal Digital Assistants (PDAs) have become more and more powerful and compact. For instance, the HP iPAQ hx4700 Pocket PC features a 624 MHz processor, 128MB ROM, 64MB RAM, and 4 (display, all contained in a 3 × 5 × 0.6 ) thin pad. As these devices gain increasing popularity, it is desirable to make Web browsing, one of the most common activities on desktops, also convenient on these devices. However, the majority of current Web sites are designed for desktop displays (see Figure 1 ). Viewing a Web page designed for desktops becomes a heavy burden on the smaller screen of a PDA. Jones et al. [1999] indicated that small display users spent much more effort and time than their large screen counterparts to perform the same navigation tasks on Web pages designed for large screens. In addition to the small screen, low bandwidth and limited input devices also make Web browsing on handheld devices difficult.. One solution to this problem is to create special Web pages for handheld devices manually, but manual reauthoring is tedious and not adaptive to all Web pages. Another way is to adapt the original Web page designed for desktops to fit automatically on the small screen. Compared with manual reauthoring, automatic and dynamic page reauthoring is cost effective. Therefore, automatic on-the-fly adaptation of Web pages has become popular.. Currently, only a handful of browsers on PDAs (e.g., Handweb [Smartcode Software 2008] , Plamscape [ILINX 2008], and ProxiWeb [ProxiNet 2008] ) support limited automatic Web page adaptation for small displays. The performance of these browsers is limited either because PDA users have to scroll constantly with the system or because the visual information in the transformed pages is insufficient for PDA users to pick detailed information. In this article, we explore a new and efficient approach to do automatic page transformation for small displays.. In our approach, an original Web page that does not fit on the screen is transformed into a set of subpages, each of which fits on the screen. The transformed subpages preserve the contextual information in the original Web page and reduce scrolling. Our approach is based on the following two hypotheses: First, when PDA users access a Web page that they previously accessed on desktops, they recognize the content and links of their interest in the Web page based on their previous browsing experience. In particular, page layout and visual context information are crucial for users to identify their interest in a page. Second, when PDA users access a Web page that they are not familiar with, the summary of the content of each block in the page will be helpful for them to identify their interest.. Therefore, for an original Web page that does not fit on a small screen, our browser first displays its textually enhanced thumbnail image, which fits on the small screen. When a block within the image is pen-tapped, the textually enhanced thumbnail image of this block is then displayed. Again, this thumbnail image fits on the small screen. Since we are zooming-in the page, users can observe additional information on the selected block. This tap-and-zoom-in process may continue for a few more times until the target of interest is found (Figures 2(a)-(d) ). During the entire browsing process, the original page layout and context information are preserved and scrolling is seldom needed.. We use textually enhanced thumbnails in our approach. Compared to plain thumbnails, which keep the original visual information of the Web pages, textually enhanced thumbnails draw additional textual summary of the Web pages on the images [Woodruff et al. 2001] . This additional feature can help users identify interest in content. Hence, a user familiar with the page can browse through the thumbnail images; otherwise, the textual summaries drawn on the images can facilitate user browsing. For example, Figure 3 shows a plain thumbnail image and its textually enhanced version produced by our system. Compared to the plain thumbnail image B of the original Web page A, textually enhanced thumbnail image C is enhanced by the textual summary of A, which consists of important pieces of text extracted from the Web page.. There are several interesting research questions to be answered in our approach. How do we divide a large page into several smaller ones so that each of 4:4 • X. Xiao et al. Fig. 2 . Transformed pages of the original Yahoo! homepage. After the user pen-taps a block in a subpage, which is indicated by the shaded box, the corresponding subpage of the block, indicated by the arrowed line, is presented to the user. The circled portions are the additional summaries that appear at the current subpage, but not at the upper-level pages.. the subpages fits on the screen? What parts of a page are "good" to put into one subpage? What is the right number of subpages in one page, since too many subpages in one page is hard for users to browse, and too few subpages may result in too many pen-taps during navigation? Moreover, what are the effects of user browsing experience with the transformed pages in comparison with the original pages, as scrolling is nearly eliminated but the number of pen-taps increases?. In our system, we first use a vision-based page segmentation algorithm to divide a large Web page into small blocks. Then, we propose an algorithm to merge small blocks, to form new leaf blocks, given the screen size limit, and organize these new leaf blocks into a tree, which we call an SP-tree (SP is the abbreviation of SubPage). In the SP-tree organization of the set of transformed pages for an original Web page, each internal node is a thumbnail of the original Web page (the root) or that of an intermediate subpage, and each leaf node is a leaf page (a sufficiently small block in the original Web page).. We show the SP-tree of an example Web page in Figure 4 . When constructing the SP-tree, we require the out-degree of the nodes in the tree to be no more than 8 and the height of the tree no more than 3, that is, F = 8 and H = 3. In the example reported in this figure, the size of the screen is 450 pixels × 300 pixels, that is, s w = 450 and s h = 300. On the left of Figure 4 , the leaf blocks are extracted using the SP-tree construction algorithm. They are bordered with thick lines. These leaf blocks correspond to the leaf nodes of the SP-tree. The SP-tree of the Web page is shown on the right of Figure 4 . We have implemented a transformation module, which we call SPTransform, on a proxy server and have conducted experimental evaluation of it on PDAs. Participants in our study found that this transformation method helped them in browsing Web pages on PDAs.. The remainder of this article is organized as follows. In Section 2, we discuss related work. We describe the interaction paradigm of our system in Section 3. We overview the architecture of our system in Section 4. We present the SPTransform algorithm in Section 5 and experimental results in Section 6. Finally, we conclude our work in Section 7. Schilit et al. [2002] classify into four categories the techniques of fitting desktop content into a small display: scaling, manual authoring, transducing, and transforming. Scaling can reduce scrolling, but it reduces Web page readability as well. Manual authoring is laborious since it requires professional Web designers to manually tailor Web pages to fit on particular devices. In contrast, automatic Web page re-authoring methods, transducing and transforming, release Web designers from heavy manual workload. Transducers such as AvantGo [Sybase 2008 ], translate HTML and images into some formats that are supported by mobile devices so that the display of the Web page matches device characteristics such as screen size and image resolution. Transforming goes further, modifying both content and structure of Web pages originally designed for desktop browsing to make them suitable for display on small screens. It may introduce a new organizational structure, such as a table of contents or other categories of overview pages. In this article, we have proposed an SP-tree-based page transformation method to facilitate Web browsing on small terminals. In our approach, a Web page is transformed into a set of screen-fitting thumbnail index pages and leaf pages. The thumbnail index pages and leaf pages form a multilevel tree structure with a bounded fanout. This transformation is done through slicing the original page into blocks iteratively with several factors considered, including the size of the screen, the size of each page block, the number of blocks in each transformed page, the depth of the result hierarchy, as well as semantic coherence within blocks. Our transformation retains the page layout of the original Web page, so that users can utilize previous browsing experience on desktops for familiar Web pages. Furthermore, we draw textual summaries on the thumbnail images. With these textually enhanced thumbnails, PDA users can also efficiently browse unfamiliar Web pages. We have implemented our algorithm in a proxy server. In our experiments, we compared our system with Opera. Our approach improved user browsing experience in terms of task completion time and input effort with comparable bandwidth consumption.. Our current page transformation system still has some limitations. One is that we must be able to obtain the HTML source of the original Web page, as it is an input of our page transformation module. We need the HTML source of the original Web page to build the HTML DOM tree and take this tree as input to construct the VIPS tree. We further process the VIPS tree in the leaf extraction and fanout-height balance subroutines. For Web pages in sites of banks and governments, the HTML source may be unavailable.. Another limitation of our page transformation approach is that it does not handle dynamic Web pages well. In the leaf page generation module, we copy the header section of the original Web page to all of the transformed leaf pages. If the script functions invoked in a section of the Web page are not declared in the header section, the browser will fail to find the declaration of the functions in the transformed leaf page. As a result, the leaf page cannot offer to the PDA users the functions that are provided by the original Web page. The existing transforming methods can be further classified according to the resulting transformed pages they provide to users: single column, fisheye, and overview + detail.. The single-column method adapts the multicolumn layout table of a Web page into a table with a single narrow column so that horizontal scrolling can be reduced. Opera SSR is an example system that zooms-out font size and image size, and transforms page layout to a single-column view for small displays. While this single-column view eliminates scrolling in the horizontal dimension, it greatly increases the amount of scrolling in the vertical dimension. Xie et al. [2005] also proposed a one-column view that displays blocks of a page in a list and places important blocks in the top of the list.. The fisheye visualization [Furnas 1986; Gutwin and Fedak 2004] shows a focus region at a readable scale while spatially compressing page content outside the focus region, as its name suggests. This is achieved by showing more remote regions in successively less detail. Fishnet ] is an example of a browser that shows a fisheye view.. The overview + detail method splits a Web page into multiple sections and provides an overview page with links to these sections. The overview page can be either a thumbnail image, as in our work, or a text summary of the Web page. The objects in an overview + detail system form a multilevel hierarchy.. The overview + detail kind of transformation methods can be further categorized into two types. One provides a textual overview and the other a graphical overview, namely the thumbnail overview.. Both the Document Segmentation and Presentation System (DSPS) [Lee et al. 2003 ] and the Stanford Power Browser [Buyukkokten et al. 2002 [Buyukkokten et al. , 2000a [Buyukkokten et al. , 2000b [Buyukkokten et al. , 2001a [Buyukkokten et al. , 2001b Kaljuvee et al. 2001] provide textual summaries in the overview page and reveal the detailed Web page content progressively. One advantage of using textual summaries is their simplicity in the display, but the downside is that the visual context (e.g., styles, images) in the original Web page is lost. Consequently, users' browsing experience on desktop PCs cannot be utilized on PDAs to help discover the target information.. In contrast to textual overviews, thumbnail overviews [Baluja 2006; Björk et al. 1999; L.-Q. Chen et al. 2003; Y. Chen et al. 2005 Milic-Frayling and Sommerer 2002; Rodden et al. 2003; Wobbrock et al. 2002; MacKay et al. 2004; Xie et al. 2005 ] preserve the visual context information of the original Web pages. Baluja [2006] first displays a Web page as a thumbnail image which is divided into 9 regions. By pressing the corresponding number keys 1-9 on a phone, users can select a region to zoom-in. In addition to zooming into detail, collapse-to-zoom ] allows users to collapse areas. WEST, proposed by Björk et al. [2000 Björk et al. [ , 1999 provides three modes of viewing blocks of. a page: a thumbnail view, a keyword view, and a link view. However, the view rendered by the original HTML source is not available. DRESS, proposed by L.-Q. , adapts the layout of a page to generate an overview page under various constraints on display size. It allows users to enlarge the display size of a desired block on the overview page. To achieve this, a binary slicingtree-based document representation was proposed. Y. Chen et al. [2005 use heuristic methods to split a Web page into blocks. In their transformation method, an index page is generated which contains a thumbnail image of the original page and hyperlinks to subpages corresponding to blocks. In addition to a thumbnail overview and detailed view of blocks, SmartView [Milic-Frayling and Sommerer 2002] displays the number of hits on the overview page and highlights query terms within a detailed view of a block. SearchMobil [Rodden et al. 2003 ] extends SmartView with the aim of providing support for searching. It displays the locations of search terms on an overview of a page if it is retrieved by a search engine. WebThumb [Wobbrock et al. 2002] provides a function to pick up and zoom into an item from the thumbnail view. Gateway [MacKay et al. 2004 ] allows users to select blocks on a thumbnail overview of a page. These selected blocks are then expanded and superimposed on the overview. Xie et al. [2005] proposed a block importance model and three views of a page based on the model: a 2D thumbnail view, a one-column view, and a main content view that only displays important blocks. The 2D thumbnail view is of the thumbnail overview + detail type.. Although the thumbnail overview + detail transformation method has been studied in the literature, previous work is limited to a two-level hierarchy, that is, a single thumbnail overview page linked to a set of subpages for detailed information. When an original Web page is large, the thumbnail overview is either too large to fit on the small screen or too crowded for users to identify the blocks of interest. As a result, scrolling and pen-taps may not be effectively reduced for large pages.. Our work falls into the thumbnail overview + detail category of transformation. Instead of the two-level hierarchy of overview + detail transformed pages, we allow transformed pages to form a multilevel hierarchy. Each transformed page fits on the screen and consists of a small number of blocks. This generalization reduces input effort for users by providing sufficient but not overwhelming visual context information. Additionally, our work uses textually enhanced thumbnails instead of traditional plain thumbnails.. People have studied how thumbnails help users to recall their memories on previously visited Web pages and how summaries help users to make correct relevance decisions [Czerwinski et al. 1999; Woodruff et al. 2001; Kaasten et al. 2002; Sweeney and Crestani 2006] . Woodruff et al. [2001] compared the efficiency of textually enhanced thumbnails, plain thumbnails, and textual summaries in navigating Web pages to find different types of information. Their studies show that, for all categories of questions, the performance of enhanced thumbnails is comparable to textual summaries or plain thumbnails, whichever is better. Lam and Baudisch [2005] also provided text-readable thumbnail images by enlarging the font size of text in thumbnails. When zoomed-in, the abbreviated text is replaced by the complete text. However, in their system, the hyperlinks and forms on the original Web pages are no longer accessible on the transformed images. Sweeney and Crestani [2006] have studied the relationship between summary size and screen size. Their study indicates that for all screen sizes, shorter summaries are more effective. Their results also show that since the screen size of a PDA is small, a short and precise summary of a page is especially helpful for users to identify their interest in the page.. Previous study shows that a Web page on small displays should present its summary information before detailed content [Jones et al. 1999] . This requirement is the driving force for page summarization. Document summarization has been studied in the literature [Luhn 1999; Mani and Maybury 1999] . Text summaries can consist of important words, sentences, or their combination. Stanford's Power Browser project [Buyukkokten et al. 2001b ] presents five methods for summarizing the textual units of a Web page: incremental, all, keywords, single-sentence summary, and keyword/single-sentence summary. Their results shows that the combination of keywords and single-sentence summary (the last mentioned method) is the best method among the five in terms of task completion time. The Power Browser uses the well-known TF/IDF technique [Salton and McGill 1986 ] to find keywords and summary sentences. TF/IDF works well on full-text resources. However, Web pages, especially those browsed on PDAs, contain much less text than do traditional environments. In addition, there is rich formatting information in Web pages which is useful in identifing important sentences/words. Therefore, we chose to mainly use formatting information to determine summary words.. Our previous work [Xiao et al. 2005 ] provides a brief overview of our page transformation system. DRESS has a problem common to two-level overview + detail transformation methods. Specifically, when the layout of a Web page is complex, the blocks in the overview page are sometimes too crowded to be distinguishable. Additionally, the tree representation of a Web page proposed in DRESS is always binary, which is different from our work. We do not use the block importance model proposed by Xie et al. [2005] because importance of blocks is sensitive to user purposes and tasks. For example, on the homepage of CNN, the "Top Stories" block is important for news readers; the "Market Indexes" block is important for investors; the navigation bar block is the target of users who intend to link to a subsite of CNN through the homepage. Thus, their three proposed view modes based on the block importance model do not work for users with various purposes or interest.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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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Numerical simulation of incompressible fluid flow using locally solenoidal elements In this paper we present a numerical study of the performance of a discontinuous Galerkin formulation for the Navier-Stokes equations. This method is characterized by the fact that the velocity field is approximated using piecewise polynomial functions that are totally discontinuous across interelement boundaries and which are pointwise divergence-free on each element (locally solenoidal). In particular, numerical results are presented for two well known benchmark problems. We consider the stationary Navier-Stokes equations for viscous incompressible flow as given in the primitive variable formulation. Here u = (u 1 , · · · , u N ) : Ω → R N represents the velocity field, and p : Ω → R, the pressure; the function f = (f 1 , · · · , f N ) : Ω → R N denotes the prescribed external body forces, g = (g 1 , · · · , g n ) : ∂Ω → R N , the admitted flux across the boundary ∂Ω and ν > 0 is a constant measuring viscosity. Note that g must satisfy the compatibility condition ∂Ω g · n dσ = 0.. One of the attributes of the numerical method presented herein is the use of totally discontinuous piecewise polynomial vector functions to approximate the velocity field u. These functions satisfy the incompressibility condition ((2)) pointwise on each element of a partition of Ω. A special weak formulation is designed to account for the interelement jumps that ensue. The weak form includes penalty jump terms designed to enforce continuity in a weak sense, [1] , [2] . This is common practice in discontinuous Galerkin methods, [3] , [4] , [5] . Recently, Oden and Baumann introduced a method which dispenses with such terms and achieves stability by reversing the sign of one of the terms in the bilinear form, cf. [6] and the references therein. For the pressure, standard continuous piecewise polynomial functions are used. For the method that we shall describe below, a range of theoretical issues including the stability and convergence of the approximations at the optimal rates were presented in [2] . These results extended those obtained in [1] for the Stokes problem. Also, in [7] , the application of implicit Runge-Kutta methods to the corresponding time dependent problem were analyzed.. Continuing our earlier work, [8] , we present a much more expanded set of numerical experiments designed to gauge the performance of our method and to offer a study of its characteristics. For the sake of completeness, we include in Section 2 needed expository material borrowed from [2] and [8] . This includes a description of the energy spaces, the discontinuous Galerkin formulations and enumeration of some relevant analytical results. Section 3 contains the new material in this paper. It describes some implementational aspects of the method and includes data confirming the convergence rates established in [1] and [2] . The two benchmark problems of the driven cavity and backward facing step provide an important test for our approach and allow a comparison with other available methods. First a triangulation of Ω is generated using the Modulef library [15] . The triangles are enumerated and other appropriate data structures are established. The various arrays and vectors are constructed using suitable basis functions. Those for the velocity are as described in section 2.2. For the pressure, we use standard Lagrangian nodal basis functions expressed locally in terms of barycentric coordinates, [10] . . where. To solve the linear system ( (17)), we adopted an algorithm devised by Bramley [16] . Indeed it turned out to be the most efficient among several we tested, [7] . We describe briefly its derivation. For simplicity we assume thatḡ = 0, ν = 1. We solve ( (17)) for b in the least squares sense :. † is the Moore-Penrose pseudoinverse of B and substituting b with this in( (17)), we get P Sa = Pf where P = I − BB † is the orthogonal projection onto ker(B T ). On the other hand B T a = 0 if and only if a = P a so P SP a = Pf. The method now applies the standard conjugate gradient method to P SP a = Pf . Assuming that B has full rank then. The matrix B T B is symmetric positive definite and its dimension is small compared to that of S. We notice also that for the time-dependent problem, B does not depend on time so B T B can be factored once at the beginning. In this method, if the projection P is computed exactly, then the residual of B T a = 0 is zero and the method tries to bring the residual of Sa + Bb = f to zero iteratively. Finally a preconditioned conjugate gradient method, with preconditioner P S −1 P can be used to solve P SP a = Pf . The Galerkin formulation ( (14)) for the stationary Navier-Stokes problem is, on the other hand, nonlinear. We solve it using Newton's method, which takes the form. h ] given. We found it efficient to take as [u. h ] the solution of the corresponding Stokes problem. We then iterate until u. Indeed, in view of the asymptotic quadratic convergence of Newton's method, this will ensure that u k − u k ≤ T OL, where T OL is a prescribed tolerance parameter.

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


Normal Helly circular-arc graphs and its subclasses A Helly circular-arc model M = (C, A) is a circle C together with a Helly family A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs. A graph is an interval graph if its vertices can be arranged into a one-to-one mapping with a family of intervals of the real line in such a way that two vertices of the graph are adjacent if and only if their corresponding intervals have nonempty intersection. The family of intervals is a called an interval model or interval representation of the graph. Similarly, a graph is a circular-arc graph if its vertices can be arranged into a one-to-one mapping with a family of arcs of some circle in such a way that two vertices of the graph are adjacent if and only if their corresponding arcs have nonempty intersection. The circle together with the family of arcs is a called a circular-arc model or circular-arc representation of the graph.. Circular-arc graphs are a natural generalization of interval graphs; the intersection of segments of a line (i.e. intervals) is replaced with the intersection of segments of a circle (i.e. arcs). Despite the replacement of a line with a circle can seem like a petty modification, circular-arc graphs have a much more complex structure than interval graphs. One of the main reasons behind this complexity is that arcs can intersect in different ways as intervals do. In an interval model, each family I of pairwise intersecting intervals is associated with a unique nonempty segment (s, t) of the real line such that (s, t) is contained in every interval of I, and (s, t) is inclusion-maximal. The situation for circular-arc models is quite different, since for some families of pairwise intersecting arcs there could be zero or more than one of such segments. As a result of this generality, many properties that hold for interval graphs cannot be naturally generalized to circular-arc graphs. Let C be the class of circular-arc models such that every family A of pairwise intersecting arcs is associated with a unique nonempty segment (s, t) of the circle such that (s, t) is contained in every arc of A, and (s, t) is inclusion-maximal. It is not hard to see that a circular-arc model M belongs to C if and only if M has no two nor three arcs that together cover its circle.. The circular-arc models in C are related with two important subclasses of circular-arc models, namely normal and Helly circular-arc models. A circular-arc model is normal when no two arcs cover the circle, while it is Helly when every family of pairwise intersecting arcs share a common point of the circle. Lin and Szwarcfiter (2006) proved that if a circulararc model is not Helly, then it contains two or three arcs that together cover the circle. Furthermore, if a normal circular-arc model has three arcs that cover the circle, then the model is not Helly. Consequently, C is precisely the class of circular-arc models that are simultaneously normal and Helly. We define the normal Helly circular-arc graphs as those circular-arc graphs that admit a model in C.. The purpose of this paper is to study the class of normal Helly circular-arc graphs, as well as its subclasses. In Section 2 we give the basic definitions used throughout the paper, and we introduce the families of circular-arc models and graphs. The reasons for studying these subclasses of circular-arc graphs are given in Section 3, where we compile several properties of interval graphs that hold for normal Helly circular-arc graphs but do not hold for the general classes of normal circular-arc and Helly circular-arc graphs. Of course, there is also a completely theoretical motivation for studying normal Helly circular-arc graphs, namely to study all the circular-arc subclasses that are obtained by the intersection of the most well known subclasses of circular-arc models. Next, in Section 4, we describe the forbidden induced subgraph characterizations of the normal Helly subclasses, while in Section 5 we deal with the corresponding recognition problems. Finally, in Section 6 we provide some additional results on normal Helly circular-arc graphs that further relate this class of graphs with the class of interval graphs, and in Section 7 we discuss some possibilities for future research. In this paper we studied the NHCA, PHCA, and UHCA classes of graphs. In Section 3 we showed that these classes arise naturally when some properties are generalized from a family of intervals on the real line to a family of arcs of a circle. NHCA, PHCA, and UHCA also appear when studying some problems on circular-arc graphs that are to hard for the general case but easy for interval graphs. Given the close relationship between interval and NHCA graphs, we could expect to find good results for such problems, restricted to NHCA or PHCA graphs. The coloring problem on circular-arc graphs is perhaps the most well known example in which NHCA and PHCA play a major role. Let χ and ω be the chromatic number and clique number of a circular-arc graph G, and r be the maximum number of arcs that share a common point of a circular-arc model M of G. Tucker (1975) proved that χ ≤ 3/2ω when G is NHCA, while χ ≤ 3/2r ≤ 3/2ω when M is a PHCA model. Given the close relationship between interval and NHCA graphs, we expect that many properties and algorithms on (proper) interval graphs can be easily generalized for (PHCA) NHCA graphs. In Section 4 we described characterizations for the classes of NHCA, PHCA, and UHCA graphs. These characterizations imply a complete family of forbidden induced subgraphs for the PHCA and UHCA classes. That is, adding W 4 and the 3-sun to Tucker's forbidden subgraphs for PCA graphs, we obtain the complete list of forbidden subgraphs for PHCA graphs, while adding W 4 to Tucker's forbidden family for UCA graphs, we obtain all the subgraphs forbidden for UHCA graphs. On the other hand, the characterization for NHCA graphs is partial; we know which circular-arc graphs are not NHCA, but the complete family of forbidden subgraphs for NHCA is still unknown.. On the algorithmic side, Nussbaum (2008) showed how to transform a circular-arc model into an equivalent PCA model in O(n) time. Combining this results with those in Section 5, we obtain O(n) time algorithms for transforming a circular-arc model into an equivalent PHCA or UHCA model. For NHCA graphs we did not find such an algorithm. Furthermore, the O(n + m) time algorithm for the recognition of NHCA graphs is not based in transformations on the input circular-arc model. The problem is that did not find how to transform an HCA model into an interval model in O(n) time.. The recognition problem for NHCA, PHCA, and UHCA graphs is well solved when the input is given by the adjacency lists of the graph. However, the algorithm for the recognition of NHCA graphs given in Section 5 is hard, because it requires the build of a circular-arc model of the input graph. So, it would be nice to find a direct algorithm for the recognition of NHCA graphs. The local interval property could play an important part in such an algorithm. In fact, many ideas that were successful for other recognition algorithms can perhaps be used for the recognition of NHCA graphs. For instance, we could test whether G is an interval graph. If so, output the interval model of G. Otherwise, try to split G into several interval graphs, resembling the process done by Deng et al. (1996) . Then, recognize each part as an interval graph and "rebuild" the model. Other possibility is to find the cliques of each part so as to build a P C-tree latter (see Hsu and McConnell, 2003) , as Booth and Lueker (1976) do. The major inconvenient of this approach is that finding the cliques of an NHCA graphs directly seems difficult. The third possibility is to extend the incremental data structure used by Korte and Möhring (1989) in their interval graph recognition algorithm. Since the insertion of a vertex depends only on its neighborhood, then it is perhaps possible to exploit the local interval property. Also, it could be interesting to generalize the incremental algorithm by Crespelle (2010) , so as to insert each vertex in O(n) time.

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


A Decision Tree-Based Attribute Weighting Filter for Naive Bayes The naive Bayes classifier continues to be a popular learning algorithm for data mining applications due to its simplicity and linear run-time. Many enhancements to the basic algorithm have been proposed to help mitigate its primary weakness-the assumption that attributes are independent given the class. All of them improve the performance of naive Bayes at the expense (to a greater or lesser degree) of execution time and/or simplicity of the final model. In this paper we present a simple filter method for setting attribute weights for use with naive Bayes. Experimental results show that naive Bayes with attribute weights rarely degrades the quality of the model compared to standard naive Bayes and, in many cases, improves it dramatically. The main advantages of this method compared to other approaches for improving naive Bayes is its run-time complexity and the fact that it maintains the simplicity of the final model. All practical learning algorithms based on Bayes' theorem make some independence assumptions. The naive Bayes method takes this to the extreme by assuming that the attributes are statistically independent given the class. This leads to a simple algorithm where training time is linear in both the number of instances and attributes. Although the independence assumption is grossly violated in practice, naive Bayes performs surprisingly well on many classification problems [7] . However, because of this assumption, the posterior probabilities estimated by naive Bayes are typically poor. For example, in an extreme case where a single redundant attribute (i.e., an attribute that is perfectly correlated with another) is present in the data, that attribute effectively has twice as much influence as the other attributes.. Many techniques have been developed to reduce the 'naiveity' of the naive Bayes algorithm. Zheng and Webb [28] provide a comprehensive overview of work in this area. One simple approach that often works well is to combine naive Bayes with a preprocessing step that attempts to remove redundant attributes from the training data. Various methods from the attribute selection community have been applied to naive Bayes for just this purpose. However, one related area that has received little attention with regards to naive Bayes is the use of attribute weights 1 . This paper presents a filter method that sets attribute weights for use with naive Bayes. The assumption made is that the weight assigned to a predictive attribute should be inversely related to the degree of dependency it has on other attributes. Our method estimates the degree of attribute dependency by constructing unpruned decision trees and looking at the depth at which attributes are tested in the tree. A bagging procedure is used to stabilize the estimates. Attributes that do not appear in the decision trees receive a weight of zero. Our experimental results show that using attribute weights with naive Bayes improves the quality of the model compared to standard naive Bayes in terms of probability estimation and area under the ROC curve.. This paper is structured as follows. In Section 2 we present our approach for enhancing naive Bayes by learning attribute weights. Section 3 contains experimental results for a collection of benchmark data sets and shows that the performance of naive Bayes can be improved by using attribute weights. Section 4 discusses related work on enhancing the performance of naive Bayes. Section 5 summarizes the contributions made in this paper. This paper has investigated a decision tree-based filter method for setting attribute weights for use with naive Bayes. Empirically, our attribute weighting method for naive Bayes outperforms both standard naive Bayes and weighting methods based on information gain and the ReliefF algorithm. Furthermore, it has performance that is comparable with a more computationally intensive wrapper-based feature subset selection for naive Bayes.. In terms of computational complexity, our weighting method increases naive Bayes' runtime from linear in the number of attributes and examples to linear in the number of attributes and log-linear in the number of instances. This compares favourably with other enhanced versions of naive Bayes that maintain a single interpretable model such as naive Bayes trees [15] and selective Bayes, both of which are quadratic in the number of attributes. A fair amount of work has been done in investigating attribute weighting schemes in the context of nearest neighbor learning. The primary goal of these methods is to mitigate the "curse of dimensionality", where the number of training cases needed to maintain a given error rate grows rapidly with the number of attributes. Methods can be roughly divided into two groups: wrapper approaches, i.e. those that use performance feedback from the nearest neighbor method to adjust the values of the weights, and filter methods, i.e. those that incorporate another model's fixed bias in a preprocessing step to set the weights. Examples of the former include Salzberg's EACH system [23] , Aha's IB4 [1] , The Relief system and its extensions [14, 22] and the DIET scheme [16] (which, unlike the other methods, restricts the weight space to a small user-selectable set of discrete values). Examples of the latter include using information theoreticbased measures such as mutual information and gain ratio to assign feature weights [25] , setting weights based on class conditional probabilities [5] and Stanfill and Waltz's value-difference metric [24] . Weighting schemes for nearest neighbor can also be separated into those methods that find one globally applicable set of weights (all the previously mentioned methods fall into this category) and those that find locally applicable weights-either local to training instances or specifically tailored to the test instance. Examples of local weighting schemes include those by Cardie and Howe [4, 11] and the RC algorithm of Domingos [6] . For a good survey of attribute weighting methods for nearest neighbor algorithms see Wettschereck et al. [25] .. There is comparatively less work on using attribute weights in conjunction with naive Bayes. Zhang and Sheng [27] investigate a gain ratio-based weighting scheme and several wrapper-based methods for finding attribute weights in order to improve AUC performance for naive Bayes. In a text classification setting Kim et al. [13] explore information gain and chi-square statistics to set attribute weights for a Poisson naive Bayes model. Ferreira et al. [9] discretize numeric attributes and then compute a weight for each attribute that is proportional to how predictive of the class it is. Classification accuracy for a weighted version of naive Bayes is compared to standard naive Bayes and C4.5 on a small selection of UCI data sets. Unfortunately, comparison based on accuracy is unfair if calibration or threshold selection is not used (which would be necessary to maximize the accuracy of each classifier).. Using information captured in decision trees to improve the performance of other learning algorithms was first explored by Cardie [3] and Kubat et al. [17] . Cardie used only those features appearing in a C4.5 [20] tree as input to a nearest neighbor learner, while Kubat et al. did the same for naive Bayes. In an approach very similar to these two, Ratanamahatana and Gunopulos' [21] Selective Bayesian classifier (SBC) uses attributes that appear in only the top three levels of a decision tree to improve the performance of naive Bayes. Similar to our method, SBC uses a bagging procedure to generate multiple trees; unlike our method bagging is used primarily to speed up the tree growing process and so only a small percentage (10%) of the training data is sampled in each iteration. In an approach related to the one presented in this paper, Cardie [4] used an information gain metric based on the position of an attribute in a decision tree to derive feature weights for a nearest neighbor algorithm. This method differs from ours in that it is a local weighting scheme (i.e. weights are derived for each test instance according to the path it takes through the tree), it derives weights directly from information gain scores, it uses a single pruned decision tree and it is aimed at improving the prediction of minority classes.. The literature on feature selection in machine learning is too extensive to review in detail here, so we will mention just a few articles relevant to the investigation reported in this paper. John et al. [12] are credited with coining the terms "wrapper" and "filter" to describe those methods that use performance feedback from a learning algorithm to guide the search for good features versus those that incorporate another model's fixed bias to select features. An example of the wrapper approach specifically tailored to naive Bayes is Langley and Sage's selective Bayesian classifier [18] . This method uses the accuracy of naive Bayes on the training data to evaluate feature subsets and a conservative forward selection search that continues to add attributes as long as the predictive performance does not decrease. CFS (correlation-based feature selection) [10] is an example of a filter approach that is well suited to naive Bayes. CFS uses a heuristic that is biased towards subsets of features that are highly correlated with the class attribute and have low levels of redundancy.

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 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 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 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 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


Discrete approximations to real-valued leaf sequencing problems in radiation therapy For a given m × n nonnegative real matrix A, a segmentation with 1-norm relative error e is a set of pairs (α, S) = {(α 1 , S 1 ), (α 2 , S 2 ), . . . , (α k , S k )}, where each α i is a positive number and S i is an m × n binary matrix, and e = |A − k i=1 α i S i | 1 /|A| 1 , where |A| 1 is the 1-norm of a vector which consists of all the entries of the matrix A. In certain radiation therapy applications, given A and positive scalars γ , δ, we consider the optimization problem of finding a segmentation (α, S) that minimizes z = k i=1 α i + γ k + δe subject to certain constraints on S i . This problem poses a major challenge in preparing a clinically acceptable treatment plan for Intensity Modulated Radiation Therapy (IMRT) and is known to be NP-hard. Known discrete IMRT algorithms use alternative objectives for this problem and an L-level entrywise approximationĀ (i.e. each entry in A is approximated by the closest entry in a set of L equally-spaced integers), and produce a segmentation that satisfies A = k i=1ᾱ i S i . In this paper we present two algorithms that focus on the original nondiscretized intensity matrix and consider measures of delivery quality and complexity ( α i + γ k) as well as approximation error e. The first algorithm uses a set partitioning approach to approximate A by a matrixĀ that leads to segmentations with smaller k for a given e. The second algorithm uses a constrained least square approach to post-process a segmentation {(ᾱ i , S i )} ofĀ to replaceᾱ i with real-valued α i in order to reduce k and e. Intensity Modulated Radiation Therapy (IMRT) is being successfully used to treat various types of cancers (i.e., prostate, head and neck, pancreas etc.). The radiation dose is designed to conform to the three-dimensional (3-D) shape of the tumor by modulating or controlling the intensity of the radiation beam to focus a higher radiation dose on the tumor while minimizing radiation exposure to surrounding normal tissue. At an intermediate stage of IMRT treatment planning, nonnegative matrices known as intensity maps are generated to represent the grid of beamlet intensities for each angle (relative to the patient) from which radiation is delivered. The radiation delivery process approximates each intensity map A by a collection of intensity apertures pairs that are delivered using a Multi Leaf Collimator (MLC) that employs a set of tungsten leaves to shape the beam aperture (these shapes correspond to a subset of a rectangular collection of beamlets and are modeled mathematically via the binary matrices S i ). Details of IMRT and MLC can be found in [2] [3] [4] 7, 9, 10, 12, 14] .. In practice, IMRT can be delivered in two ways, namely, fixed gantry and rotational. Fixed-gantry IMRT (i.e., traditional IMRT) is achieved by delivering overlapping fields from fixed beam directions. The field shapes either remain constant during the delivery of radiation or can dynamically change during radiation delivery. Rotational IMRT is achieved by dynamically moving the leaves of the MLC and the gantry during radiation delivery. Intensity-modulated arc therapy (IMAT) was first proposed by Yu in 1995 [15] as an alternative rotational IMRT delivery technique to tomotherapy [11, 5] . Details of IMAT can be found in [6] .. Intensity map segmentation requires solving the following optimization problem, which is known to be NP-hard [1] . For a given m×n nonnegative real matrix A which represents the ''intensity map'' of radiation beamlet intensities, a segmentation with 1-norm relative error e is a set of pairs (α, S) = {(α 1 , S 1 ), (α 2 , S 2 ), . . . , (α k , S k )}, where each α i is a positive number and S i is an m × n binary matrix, and e = |A − k i=1 α i S i | 1 /|A| 1 , where |A| 1 is the 1-norm of a vector which consists of all the entries of the matrix A. In the segmentation problem, we are given A and positive scalars γ , δ, and the goal is to find a segmentation (α, S) that minimizes z = k i=1 α i + γ k + δe subject to machine-specific constraints on the S i . In IMRT, A, S i and α i thus represent an intensity matrix, an aperture (or shape) and a beam-on time for that aperture, respectively. The penalty parameters γ and δ are selected to obtain desired values for k and e.. Reducing the aperture count k subject to achieving an acceptable value of e, is very desirable in both traditional IMRT and Intensity Modulated Arc Therapy (IMAT). Current segmentation algorithms use a variety of objectives for this problem, and approximate the intensity matrix A by an integer matrixĀ, and ultimately produce shapes S i and corresponding aperture weightsᾱ i . To obtain small values of k, these algorithms use a simple approximationĀ, which results in segmentations with large values of e, reducing the quality of the treatment plan.. In this paper, we present an effective heuristic algorithm for the above problem with an additional constraint k ≤ K , where K is a given positive integer. Our algorithm can be used to find a clinically deliverable solution with an appropriate tradeoff between k and e, which behave as two competing metrics. We achieve our task by developing two new algorithms, one of which acts as a preprocessor and the other as a postprocessor for a previously developed and tested segmentation metaheuristic [8] . The preprocessing algorithm uses a set partitioning approach to approximate A byĀ so that a segmentation algorithm applied toĀ will produce segmentations with small k for a given e. The postprocessing algorithm uses a constrained least square approach to replaceᾱ i in a given segmentation (ᾱ, S) with real-valued α i in order to further reduce k while optimizing e. With our set partitioning algorithm, we are able to generate many L-level approximations (i.e., using various values of the smoothing parameter β) in a preprocessing stage of the modified metaheuristic. We then employ our constrained least square algorithm as a postprocessing stage to control the number of shapes and error in the final segmentation. These extensions to the metaheuristic allow us to handle alternative objectives effectively. As demonstrated by our results, our modified metaheuristic presents an alternative to direct aperture optimization. We implemented a dynamic programming method for the Set Partitioning Approximation in C/Unix and used Matlab/Unix for the Constrained Least Square Algorithm. In Tables 1 and 2 , we show properties of treatment plans for a pancreas case with and without the smoothing preprocessor. (The error entries of the last row of the tables are average errors). The pancreas case consists of intensity maps generated by the Corvus 4 commercial treatment planning system for 7 different angles. The largest value of entries in a map was scaled to 100 to calculate BT . To see the effects of smoothing alone, we have not used the postprocessing algorithm in the modified metaheuristic in this case. It can be seen that if we are willing to tolerate an error of about 11%, we can use use either L = 4 or L = 5. Table 1 shows that if we use the level L = 5 with smoothing β = 1, we get a much better treatment plan than the entrywise approximation in terms of the number of segments and beam-on time at the expense of slightly greater error. Table 2 shows the effect of smoothing for a larger number of levels (L = 8). Again, smoothing improves the treatment complexity, while slightly increasing the error.. Next we show the effect of optimizing beam weights at the postprocessing stage. In Table 3 , we compare the number of shapes (k), percentage relative error (E 1 = E 1,rel (A,Ā)), and total beam-on time (BT = k i=1 α i ) produced by Metaheuristic (H) and Modified Metaheuristic (MMH) algorithms for three clinical prostate maps. MMH was run with β = 0 (i.e. entrywise approximation) for a L-level approximation. The largest value of the entries in a map was scaled to 100 to calculate BT . Table 1 Count (C ), number of shapes (k), beam-on time (BT ), and percentage relative error (E 1 = E 1,rel (A,Ā)) (average error in last row) results for Modified Metaheuristic (MMH) (without postprocessing) for a set of pancreas intensity maps Table 2 Count (C ), number of shapes (k), beam-on time (BT ), and percentage relative error (E 1 = E 1,rel (A,Ā)) results for Modified Metaheuristic (MMH) (without postprocessing) for a pancreas case Table 3 The number of shapes (k), percentage relative error (E 1 = E 1,rel (A,Ā)), and beam-on time (BT ) produced by Metaheuristic (H) and Modified Metaheuristic (MMH) (with β = 0 and L = 5, 10) for three prostate maps Table 4 Given a % error bound E, a level L, and a smoothing parameter β, the values of the smallest possible k with corresponding E 1 ≤ E (using MMH) for the prostate map, MAP 1. 1. 4 8 171 10 0 12 203 9 204 10 0.5 11 192 8 192 40 0 32 309 26 323 18 293 11 285 40 1.0 27 229 21 225 17 225 9 209 60 0 38 303 35 305 28 310 19 320 14 345 80 0 41 248 32 249 25 247 18 237 10 233 We now focus on the prostate map MAP 1 of Table 3 (see Fig. 5 for this map). Given L, we can vary β to get L-level approximationsĀ with different amounts of smoothness. Figs. 6 and 7 show the graphs of the error measure E 1 and the complexity measure Count(Diff(Ā)) versus β. These graphs show the tradeoff between E 1 and the number of segments k, and facilitate the selection of a pair (L, β) to achieve a targeted tradeoff.. We now run MMH with alternative objectives to demonstrate tradeoffs between values of E and k. Tables 4 and 5 present results for two versions of MMH. In Table 4 , given a percentage error bound E, a level L, and a smoothing parameter β, we use MMH to calculate the smallest possible k with corresponding E 1 ≤ E.. For treatment plans, it is often desirable to reduce the number of segments k to below 10 for each beam angle. Table 5 shows E 1 and BT for various pairs of (L, β) when we apply MMH to select k = 5, 6, 7, 8, 9.

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


The tree width of auxiliary storage We propose a generalization of results on the decidability of emptiness for several restricted classes of sequential and distributed automata with auxiliary storage (stacks, queues) that have recently been proved. Our generalization relies on reducing emptiness of these automata to finite-state graph automata (without storage) restricted to monadic second-order (MSO) definable graphs of bounded tree-width, where the graph structure encodes the mechanism provided by the auxiliary storage. Our results outline a uniform mechanism to derive emptiness algorithms for automata, explaining and simplifying several existing results, as well as proving new decidability results.

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