Using support vector machines to enhance the performance of elastic graph matching for frontal face authentication A novel method for enhancing the performance of elastic graph matching in frontal face authentication is proposed. The starting point i s t o w eigh the local similarity v alues at the nodes of an elastic graph according to their discriminatory power. Powerful and well-established optimization techniques are used to derive the weights of the linear combination. More speci cally, w e propose a novel approach that reformulates Fisher's discriminant ratio to a quadratic optimization problem subject to a set of inequality constraints by c o m bining statistical pattern recognition and Support Vector Machines. Both linear and nonlinear Support Vector Machines are then constructed to yield the optimal separating hyperplanes and the optimal polynomial decision surfaces, respectively. The method has been applied to frontal face authentication on the M2VTS database. Experimental results indicate that the performance of morphological elastic graph matching, is highly improved by using the proposed weighting technique. Many techniques for face recognition have beendeveloped whose principles span several disciplines, such as image processing, pattern recognition, computer vision and neural networks 1]. The increasing interest in face recognition is mainly driven by application demands, such as nonintrusive identi cation and veri cation for credit cards and automatic teller machine transactions, nonintrusive access-control to buildings, identi cation for law enforcement, etc. Machine recognition of faces yields problems that belong to the following categories whose objectives are brie y outlined:. Face recognition. Given a test face and a set of reference faces in a database nd the N most similar reference faces to the test face.. Face authentication. Given a test face and a reference one, decide if the test face is identical to the reference face.. Face recognition has been studied more extensively than face authentication. The two problems are conceptually di erent. On the one hand, a face recognition system usually assists a human expert to determine the identity of a test face by computing all similarity scores between the test face and each human face stored in the system database and by ranking them. On the other hand, a face authentication system should decide itself if a test face is assigned to a client (i.e., one who claims his/her own identity) or to an impostor (i.e., one who pretends to besomeone else). The evaluation criteria for face recognition and face authentication systems are di erent. The performance of face recognition systems is quanti ed in terms of the percentage of correctly identi ed faces within the N best matches 2]. The performance of face authentication systems is measured in terms of the false rejection rate (FRR) achieved at a xed false acceptance rate (FAR) or vice versa. By varying FAR, the receiver operating characteristic (ROC) curve is obtained. If a scalar gure of merit is used to judge the performance of an authentication algorithm, then we usually choose the operating point having FAR=FRR, the so called equal error rate (EER). A third di erence is in the requirements needed when face recognition/authentication systems are trained. Face authentication systems need more than one training images perperson while face recognition systems are usually trained on sets having one frontal image per person.. A well-known approach to face recognition/authentication is the so-called dynamic link architecture, a general object recognition technique, that represents an object by projecting its image onto a rectangular elastic grid where a Gabor wavelet bank response is measured at each node. A simpli ed implementation of dynamic link architecture, the so-called elastic graph matching (EGM), is often preferred for locating objects in a scene with a known reference 3, 4, 5] . It was found that elastic graph matching achieves a better performance than the eigenfaces 6, 7] , auto-association and classi cation neural networks 8, 9] due to its robustness to lighting, varying face position and facial expression variations. The research on elastic graph matching and its applications has been an active research topic since its invention. A di erent topology cost for a particular pair of nodes, that was based on the radius of the Apollonius sphere de ned by the Euclidean distances between the nodes beingmatched, was proposed in 10]. Three major extensions to elastic graph matching that allowed for handling larger galleries, tolerated larger variations in pose and increased its matching accuracy were introduced in 11]. Procedures that increase the robustness of elastic graph matching in translations, deformations and changes in background were proposed in 12] . Recently, a variant of elastic graph matching based on multiscale dilation-erosion, the so-called morphological elastic graph matching (MEGM) 1 , was proposed and tested for face authentication 13, 14] . This paper addresses the derivation of optimal coe cients that weigh the local similarity v alues determined by the elastic graph matching procedure at each grid node. The starting point is the observation that the distance measure between two persons in elastic graph matching is simply the sum of local similarity values. However, this observation contradicts the fact that some facial features (e.g., the eyes, the nose, etc.) are more distinctive for one person than other features. Therefore, it is reasonable to assume that the response of local image descriptors attempting to capture the image properties in the region of these features should be weighted heavier than the other responses. To alleviate the just described weakness of the elastic graph matching, we propose to weigh the local similarity v alues at the grid nodes by a n o vel approach that combines statistical pattern recognition (i.e., discriminant analysis) 15, 16] and Support Vector Machines 17, 18, 19] .. Our approach reformulates Fisher's discriminant ratio to a quadratic optimization problem subject to a set of inequality constraints. Both linear and nonlinear Support Vector Machines are then constructed to yield the optimal separating hyperplanes and the optimal polynomial decision surfaces, respectively. The proposed method has been applied to frontal face authentication on the M2VTS database 20]. Experimental results indicate that the performance of morphological elastic graph matching, a variant of elastic graph matching, is highly improved by using the proposed weighting technique reaching an EER of 2.4 %. It is worth noting that the objective of our approach di ers signi cantly from that of:. Linear Discriminant Analysis (LDA) to face r ecognition 21, 22, 23] , because there Fisher Discriminant Analysis is applied to the entire face image as a linear projection algorithm, i.e., a parametric projection pursuit, or LDA for feature s e l e ction 24, 13] , because there Fisher Discriminant Analysis is applied to feature vectors for selecting the most discriminating features, that is, to reduce the dimensionality o f feature vectors by throwing away the local image descriptor responses that are not signi cant t o t h e authentication task.. On the contrary, our objective is closely related to the Bayesian approach that yields the most reliable nodes for gender identi cation, beard and glass detection in bunch graphs 4, 11] or the algorithm for learning the weights in discrimination functions using a priori constraints 25]. The latter algorithm uses the Simplex method to nd the local minima of a multidimensional function in order to optimize the weights on the local similarity values.. The outline of the paper is as follows. A brief description of elastic graph matching and the problem treated throughout the paper is given in Section II. Local discriminatory power coe cients are derived in Section IIIthrough the formulation of the problem as a constrained Least Squares optimization problem. The extension of constrained Least Squares formulation to the construction of a class of Support Vector Machines (both linear and nonlinear ones) is described in Section IV. The weights derived by the approaches described in Sections III and IV are incorporated to the morphological elastic graph matching and the combined scheme is tested for frontal face authentication. The performance of the weighted morphological elastic graph matching is assessed in Section V and conclusions are drawn in Section VI.

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


Towards Semantic-driven, Flexible and Scalable Framework for Peering and Querying e-Catalog Communities Given that e-catalogs are often autonomous and heterogeneous, effectively integrating and querying them is a delicate and time-consuming task. More importantly, the number of e-catalogs to be integrated and queried may be large and continuously changing. Conventional approaches where the development of an integrated e-catalog requires the understanding of each of the underlying catalog are inappropriate. In this paper, we use the concept of e-catalog communities and peer relationships among them to facilitate the querying of a potentially large number of dynamic e-catalogs. E-catalog communities are essentially containers of related e-catalogs. We propose a flexible and user-centric query matching algorithm that exploits both community descriptions and peer relationships to find e-catalogs that best match a user query. The user query is formulated using a description of a given community. Web portals, such as Expedia.com and Amazon.com, are becoming more and more prominent feature of the World Wide Web. They offer tremendous opportunities for empowering users and organisations in various application domains including electronic commerce, travel, intelligence information gathering and analysis, digital government. For example, citizens and social workers will be able to get timely information and services by accessing state and federal government department Web portals [1] . Travellers will get timely information and services from relevant sources such as route information, travel operations schedule, and entertainment Web portals.. However, the technology to organize, search, integrate, and evolve these portals has not kept pace with the rapid growth of the available information space. Enjoying the above benefits using current Web technologies is a time-consuming, frustrating, and a very complex process for end users. Currently, users need to access a large number of heterogeneous and distributed data sources in order to become self-supported. For example, in digital government example, users need access to several Web portals of different government offices separately, manually filter and organise the search results to get the information about the benefits they are entitled to. This ad-hoc process of searching and combining required information sources has largely hampered a faster pace in allowing users to automate their tasks. Clearly, to effectively realize the potential of Web-based information access, there is a need for facilitating the integrated access to relevant information sources.. At present, the mainstream approach for locating web resources is through keyword-based search engines relying on information retrieval techniques. Although search engines are certainly useful, they present fundamental drawbacks such as lack of information-space organisation [2] . To effectively exploit the Web's expanding information sources, the emerging semantic web efforts employ machine-understandable abstractions for the representation of resource semantics. In particular, the semantic web promotes the use of ontologies as a tool for reconciling semantic heterogeneity between web resources. Efforts in this area include the development of ontology languages such as RDF, DAML, OWL 1 .. Despite these early efforts, many of the semantic web objectives, including the technology to organise, search, integrate, and evolve web-accessible resources, remain difficult to achieve. Ontologies provide essential support for semantics rich queries (e.g., data source vocabulary and structure aware queries). However, several critical issues must be addressed to realize the potential of the semantic web vision [3] . In particular, the effective use of a potentially large and dynamic collection of Web resources requires more scalable and flexible information sharing and querying techniques, which leverages and seamlessly extends current mainstream technologies and practices.. In this paper, without loss of generality, we use e-catalog portals (also called e-catalogs or e-commerce portals) as representative Web information sources to illustrate the infrastructure we propose to support information sharing and querying over the Web. We present the design and implementation of the WS-CatalogNet system: a Web services based information sharing middleware infrastructure whose aims is to enhance the potential of e-catalogs by focusing on scalability and flexible aspects of their sharing and access. Our research in the WS-CatalogNet builds upon lessons learned in our previous work on Web information integration [2, 4] , in order to provide a peer-to-peer and semantic-based e-catalogs sharing middleware infrastructure through which: (i) e-catalogs can be categorised and described using domain ontologies, (ii) heterogeneous service ontologies can be linked via peer relationships, (iii) ecatalogs selection caters for flexible and user-centric matching between user queries and e-catalogs descriptions. In a nutshell, the salient features of WSCatalogNet are:. • Ontology-based Indexing of e-Catalogs: At present the main stream approach for providing unified access to multiple e-catalogs is the ad-hoc development of an integrated e-catalog by locating the e-catalogs to be integrated, understanding their interfaces, and statically linking them to the global or integrated e-catalog. This static approach is clearly tedious and hardly scalable because of the volatility and size of the Web. We use the concept of catalog community as a way of organizing and integrating a potentially large number of dynamic e-catalogs. A catalog community is a container of alternative e-catalogs (i.e., catalogs offering products of a common domain such as community of Laptopsproviders or community of Accommodationproviders). It provides an ontological description of desired products (e.g., product categories, product attributes) without referring to any actual provider (e.g., Dell Computers, The Hilton Hotel). Actual providers can register with any community of interest to offer the desired products. The e-catalog communities allow for meaningful organization and division of the information space.. • Peering e-Catalog Communities: Existing e-catalog organization techniques usually use centralized categorization and indexing schemes, whereas the participating e-catalogs are distributed and autonomous [5, 6] . A centralized categorization and indexing model has several drawbacks including scalability, flexibility, and availability [7, 8] . In addition, it is not realistic to assume that a global ontology that contains concepts used in all possible e-catalogs could exist. It is not realistic either to assume that there will be on agreement on the vocabularies used by different ontologies. It is important to provide mediation facilities between heterogeneous ontologies. Given the highly dynamic and distributed nature of e-catalogs, we believe that novel techniques involving peer-to-peer centric categorization and indexing schemes will become increasingly attractive. Our approach features the use of peer relationships among e-catalog communities to allow decentralized sharing of e-catalogs information [9] . Query routing between communities is based on query forwarding policies. Query forwarding policies are high-level directives that define interaction modes among communities based on different types of peer relationships (e.g., mappings among community descriptions are fully defined, partially described, not defined). The objective is to achieve scalable information sharing and access through the collaborative data sharing and query processing among peer communities.. • Flexible and User-centric Selection of Relevant e-Catalogs: Because of the variety of e-catalogs offering similar information and the large number of available e-catalogs, it is important to provide appropriate support to select those e-catalogs that are relevant to a specific user query before actually querying them. In addition, e-catalog selection techniques should support flexible matching since it is unrealistic to expect queries and e-catalog descriptions to match exactly. In our approach, a user query is expressed using descriptions from a community ontology. We formalize relevant e-catalog selection as a new instance of the query rewriting problem [10, 11] , where a user query is reformulated in terms of: · a local query (i.e., the part of the user query that can be answered by some e-catalogs registered with the actual community), · the remaining part of the user query that cannot be answered by the actual community (i.e, cannot be answered by e-catalogs which are registered with the community and it is not possible to exactly determine which peer communities can answer it).. Our approach for e-catalog selection develops novel and more advanced query rewriring techniques for flexible and effective e-catalogs selection. Existing query rewriting approaches, developped in the context of information integration systems (see [10] for a survey), assume the existence of global mediated schema, as well as availability of the description of the underlying data source capabilities. Given a user query expressed over the global schema, the data sources that are relevant to answer the query are selected by means of a rewriting algorithm that allows to reformulate the user query into an equivalent or maximally subsumed (contained) query whose definition refers only to source descriptions. It should be noted that, an underlying assumption behind using such kind of operators (i.e., subsumption or equivalence) in a rewriting algorithm is that the source descriptions are known a priori (i.e., they are published in the mediator system).. The query rewriting techniques proposed in this paper go beyond subsumption and equivalence between queries and e-catalog descriptions. First, our approach enables partial matching between e-catalog descriptions and queries, thereby allowing to cater for flexible selection of e-catalogs in environments where : (i) the existence of one centralized schema or ontology is not realistic (ii) descriptions of external e-catalogs (members of communities which are accessible via peer relationships) are not known when processing a query at a given community. In WS-CatalogNet, peer communities do not share the descriptions of their respective members. In addition, a community may form a peer relationship with another community without necessarily providing precise description of the mapping between its ontology and the ontology of the peer community. As a consequence, processing a user query in such a context requires the ability to select among the local members of a community those e-catalogs that are able to answer a part of the query, as well as the ability to compute the remaining part of the query that cannot be answered by the actual community (i.e, cannot be answered by e-catalogs which are registered with the community). The remaining part of a query can be forwarded to peer communities for further processing. Second, we argue that in the presence of multiple possible alternative e-catalogs with overlapping or identical functionality, users will discriminate these alternatives based on their Quality of Service (QoS). QoS is a broad concept that encompasses a number of non-functional properties such as price and reputation [12] . A QoS-aware and user-centric approach to e-catalogs selection is therefore needed.. Our main contribution in this context is the characterization of several types of flexible query rewritings and analysis of their associated computational complexity. Given a user query expressed over a community ontology:. • We consider first the problem of computing those rewritings, called best cover rewritings, that minimize the part of the query that cannot be answered by the community. We show that such a rewriting, as well as a best cover rewriting that minimize the number of the e-catalogs selected for answering the query (called non redundant rewriting), can be computed efficiently (i.e., in polynomial time).. • We show that computing all the non redundant rewritings can be done in incremental subexponential time in the combined size of the query and the number of the e-catalogs.. • We investigate the problem of computing the best cover rewritings that maximize the user satisfaction with respect to a given Quality of Service (QoS) criteria (e.g., communication cost, price, etc). We call such rewritings best quality rewritings. We show that, the general problem of computing a best quality rewriting is NP-hard. However in some particular cases which depend on the characteristics of the QoS function, a best quality rewriting can be computed efficiently.. • We propose a hypergraph-based algorithm to effectively compute all the best quality rewritings of a given request and we describe first experimental results that show the performance of our algorithm using up to 660 e-catalogs.. The paper proceeds as follows. Section 2 presents the design overview of WSCatalogNet. Section 3 and 4 discuss the flexible query rewriting algorithms. The implementation and evaluation of WS-CatalogNet are explained in section 5, followed by application scenarios in section 6. Discussions on the related work are presented in section 7 and finally, section 8 concludes the paper with final remarks. Our work provides complementary contributions to related work on Web information sharing and querying. We focus on providing support for achieving effective and efficient access to e-catalogs resident data. Since scalability is of great importance in e-catalog environments, the information space is organised in communities that are inter-related using peer relationships. The model that we use to describe community ontologies relies on simple concepts (categories and attributes) that we found to be useful and commonly used to describe ecatalogs content and capabilities. We consider different types of peer relationships between communities (i.e, companionship and similarity relationships). Such flexibility allows to establish different interaction types among communities. The specification of mappings in our approach combines: (i) IR style where no explicit mapping description is provided (synonym-based matching approach) and (ii) full mapping description when it is possible or desired (e.g., in the case of companionship relationships). We formalised e-catalogs selection as a rewriting process for e-catalog communities. Query routing among peer communities is based on forwarding policies. We proposed a novel hypergraphbased algorithm to effectively select relevant e-catalogs for a given query. The proposed algorithm finds rewritings that 'best match' a given query with respect to a quality function, where the relationships between a query and its rewritings goes beyond containment or equivalence. We believe that there is a need for such a flexible query rewriting approach to cope with the high dynamicity and heterogeneity of the Web-based environments.. Our ongoing work concerns developing and evaluating self-monitoring and adaptation techniques. We believe that in large and dynamic environments, e-catalog communities should be self-adaptive in the sense that, mappings among their ontologies should be restructured over time to improve the effectiveness of the overall communities network. This will enable e-catalogs networks to be adaptive and responsive to the ways users interact with the e-catalogs and communities thereby increasing their usability. Another interesting future research direction of our is to study load balancing among services communities in order to improve the performance of query answering in a communities network. In this section, we examine work done information sharing and querying across multiple sources. We also overview related efforts in Web services and semantic Web areas.

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


Characterizations

of

polynomial

complexity

classes

with

a

better

intensionality

In

this

paper,

we

study

characterizations

of

polynomial

complexity

classes

using

first

order

functional

programs

and

we

try

to

improve

their

intensionality,

that

is

the

number

of

natural

algorithms

captured.

We

use

polynomial

assignments

over

the

reals.

The

polynomial

assignments

used

are

inspired

by

the

notions

of

quasiinterpretation

and

sup-interpretation,

and

are

decidable

when

considering

polynomials

of

bounded

degree

ranging

over

real

numbers.

Contrarily

to

quasi-interpretations,

the

considered

assignments

are

not

required

to

have

the

subterm

property.

Consequently,

they

capture

a

strictly

larger

number

of

natural

algorithms

(including

quotient,

gcd,

duplicate

elimination

from

a

list)

than

previous

characterizations

using

quasi-interpretations.

The

resource

control

is

a

fundamental

issue

for

critical

systems

which

is

studied

by

the

implicit

computational

complexity

community

in

four

distinct

approaches

that

we

briefly

review.

The

first

one

deals

with

linear

type

disciplines

in

order

to

restrict

computational

time

(Girard

Lafont

Baillot

and

Mogbil

Gaboardi

and

Ronchi

Della

Rocca

Gaboardi

et

al.

The

second

approach

(Hofmann

introduced

a

resource

atomic

type,

into

linear

type

systems,

for

higher-order

functional

programming.

The

third

one

considers

imperative

programming

languages

in

(Niggl

and

Wunderlich

Kristiansen

and

Jones

and

complexity

analysis

of

Java

(Albert

et

al.

a,b)

The

fourth

approach

is

the

one

on

which

we

focus

in

this

paper

which

considers

polynomial

assignments,

quasi-interpretations

(Bonfante

et

al.

and

sup-interpretations

(Marion

and

Péchoux

It

is

related

to

polynomial

interpretations

(Lankford

One

of

the

key

challenges

is

to

create

complexity

tools

in

order

to

analyze

the

computational

complexity

of

programs

automatPermission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page.

To

copy

otherwise,

to

republish,

to

post

on

servers

or

to

redistribute

to

lists,

requires

prior

specific

permission

and/or

a

fee.

July

Valencia,

Spain.

Copyright

c

ically.

In

this

paper,

we

focus

on

polynomial

upper

bounds

on

program

outputs

(quasi-interpretations

and

sup-interpretation).

In

order

to

predict

the

complexity

of

the

function

computed

by

a

program,

we

first

have

to

prove

that

it

terminates

and,

then,

to

put

drastic

restrictions

on

either

its

syntax

or

its

semantics.

Nowadays,

the

motivation

behind

such

an

analysis

is

to

increase

the

intensionality

of

program

complexity

static

analysis,

that

is,

to

enlarge

the

number

of

captured

algorithms.

In

particular,

suitable

characterizations

of

complexity

classes

have

to

include

a

broad

set

of

programs

which

are

relevant

from

a

programmer

perspective..

Polynomial

assignments

are

strongly

related

to

polynomial

interpretations

of

(Lankford

Manna

and

Ness

However,

we

no

longer

consider

polynomials

over

natural

numbers

but

polynomials

over

real

numbers

using

the

work

of

(Dershowitz

We

compensate

for

the

loss

of

well-foundedness

over

real

numbers

by

putting

restriction

on

the

considered

quasi-orderings.

Polynomial

assignments

are

also

related

to

quasi-interpretations

(Bonfante

et

al.

and

sup-interpretations

(Marion

and

Péchoux

These

two

notions

provide

upper

bounds

on

the

size

of

the

values

computed

by

a

function

symbol.

It

is

demonstrated

in

(Bonfante

et

al.

that

these

notions

combined

with

the

product

extension

and

the

lexicographic

extension

of

Recursive

Path

Orderings

(RPO)

characterize

the

set

of

functions

computable

in

polynomial

time

and,

respectively,

the

set

of

functions

computable

in

polynomial

space..

In

this

paper,

we

make

a

step

forward

in

the

study

of

intensional

computational

complexity

by

giving

new

characterizations

of

the

sets

of

functions

computable

in

polynomial

time

and

in

polynomial

space.

These

characterizations

allow

us

to

capture

distinct

natural

algorithms

(quotient,

gcd,

duplicate

elimination

from

a

list...)

which

where

not

captured

by

previous

methods

of

(Bonfante

et

al.

Moreover,

we

present

characterizations

having

more

flexiblity

than

the

ones

of

(Bonfante

et

al.

since

they

rely

on

polynomials

over

reals

and

without

the

subterm

property

(i.e.

without

∀i,

P

X

i,

Xi).

As

a

consequence,

the

considered

assignment

have

a

stronger

intensionality

than

quasi-interpretations

which

are

subterm

assignments..

Using

a

result

of

Tarski

(Tarski

we

have

demonstrated

in

(Bonfante

et

al.

that

the

polynomial

assignment

synthesis,

which

consists

in

finding

an

assignment

for

a

given

program,

is

exponential

in

the

number

of

variables.

From

a

practical

point

of

view,

this

is

not

a

serious

drawback

since

our

study

is

related

to

program

static

analysis.

Consequently,

the

search

for

suitable

assignments

can

be

performed

offline

and

without

time

restrictions.

Some

heuristics

for

quasi-interpretation

synthesis

were

already

developed

in

a

software

called

CROCUS,

available

at

which

captures

a

lot

of

programs

efficiently

(about

of

the

TPDB

pro-grams

are

analyzed

in

less

than

second

The

paper

is

organized

as

follows.

Section

describes

the

syntax

and

the

semantics

of

the

language

and

some

preliminary

notions

on

first

order

functional

programs.

Section

defines

the

polynomial

assignments

and

the

notion

of

quasi-interpretation.

Section

improves

the

results

of

(Bonfante

et

al.

by

providing

better

intensional

characterizations

of

the

sets

of

functions

computable

in

polynomial

space

and

polynomial

time

using

dependency

pairs,

quasi-interpretations

and

Recursive

Path

Orderings.

Finally,

section

provides

two

characterizations

of

the

sets

of

functions

computable

in

polynomial

space

and

in

polynomial

time

only

using

polynomial

assignments

without

the

subterm

property.

We

illustrate

by

many

examples

that

algorithms

captured

are

distinct

from

the

ones

of

section

The

characterizations

using

quasi-interpretations

and

RPO

presented

in

section

strictly

generalize

the

ones

of

(Bonfante

et

al.

as

illustrated

by

example

still

allowing

to

capture

programs

having

an

exponential

derivation

length

but

being

computable

in

polynomial

time

using

memozation

techniques.

Consequently,

these

results

confer

a

new

approach

with

the

notion

of

reduction

pair

and

ameliorate

the

intensionality

of

the

QI

method..

On

the

other

hand,

the

second

approach

of

section

uses

only

polynomial

assignments.

It

allows

to

obtain

non

subterm

assignment

and

increases

consequently

the

intensionality

of

our

study

(even

if

the

programs

having

an

exponential

derivation

length

are

not

captured

in

DP

(TIME)).

This

is

a

real

improvement

in

the

study

of

semantics

interpretations:

Indeed,

we

are

now

able

to

take

the

interpretation

of

the

symbol

minus

to

be

minus

(X,

Y

X

(instead

of

max(X,

Y

The

closer

interpretations

are

from

the

size

of

the

computed

functions,

the

better

intensionality

our

study

has.

We

claim

that

the

following

example

which

computes

has

no

quasiinterpretation:.

half(S(S(x)))

S(half(x))

log(S(x))

S(log(half(S(x)))).

because

of

the

last

rule.

However,

it

is

in

DP

(TIME)

by

taking

S

(X)

X

half

(X)

and

log

(X)

X.

Moreover,

this

last

section

also

allows

to

capture

programs

having

a

quasi-interpretation

but

which

are

not

terminating

under

RPO.

Indeed,

all

the

program

including

a

rule

of

the

shape

f(x)

f(g(x)),

with

g

a

non

size

increasing

function,

will

admit

a

quasiinterpretation

(just

take

f

(X)

g

(X)

X).

However,

since

x

rpo

g(x)

does

not

hold,

they

are

not

terminating

by

RPO..

To

conclude,

the

non-subterm

assignments

studied

in

the

criterion

for

DP

(SPACE)

are

inspired

by

the

assignments

of

(Lucas

and

correspond

to

the

notion

of

sup-interpretation

introduced

in

(Marion

and

Péchoux

Since

the

criteria

using

DP

is

decidable

and

so

are

the

inequalities

of

DP

(SPACE)

using

polynomial

assignments

of

bounded

degree

(as

demonstrated

in

(Bonfante

et

al.

we

obtain

a

decidable

criterion

to

synthesize

non-subterm

assignments,

which

is

a

new

result.

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


Mantaining Dynamic Matrices for Fully Dynamic Transitive Closure In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive closure. In particular, we devise a deterministic algorithm for general directed graphs that achieves O(n 2 ) amortized time for updates, while preserving unit worst-case cost for queries. In case of deletions only, our algorithm performs updates faster in O(n) amortized time. We observe that fully dynamic transitive closure algorithms with O(1) query time maintain explicitly the transitive closure of the input graph, in order to answer each query with exactly one lookup (on its adjacency matrix). Since an update may change as many as (n 2 ) entries of this matrix, no better bounds are possible for this class of algorithms. In this paper we present fully dynamic algorithms for maintaining the transitive closure of a directed graph. A dynamic graph algorithm maintains a given property on a graph subject to dynamic updates, such as edge insertions and edge deletions. We say that an algorithm is fully dynamic if it can handle both edge insertions and edge deletions. A partially dynamic algorithm can handle either edge insertions or edge deletions, but not both: we say that it is incremental if it supports insertions only, and decremental if it supports deletions only. In the fully dynamic transitive closure problem we wish to maintain a directed graph G = (V , E) under an intermixed sequence of the following operations:. Insert(x, y): insert an edge from x to y in G; Delete(x, y): delete the edge from x to y in G; Query(x, y): report yes if there is a path from x to y in G, and no otherwise.. Throughout the paper, we denote by m and by n the number of edges and vertices in G, respectively.. Previous Work Research on dynamic transitive closure spans over two decades. Before describing the results known, we list the bounds obtainable with simple-minded methods. If we do nothing during each update, then we have to explore the whole graph in order to answer reachability queries: this gives O(n 2 ) time per query and O(1) time per update in the worst case. On the other extreme, we could recompute the transitive closure from scratch after each update; as this task can be accomplished via matrix multiplication [1, 18] , this approach yields O(1) time per query and O(n ω ) time per update in the worst case, where ω is the best known exponent for matrix multiplication (currently ω < 2.38 [2] ).. For the incremental version of the problem, the first algorithm was proposed by Ibaraki and Katoh [10] in 1983: its running time was O(n 3 ) over any sequence of insertions. This bound was later improved to O(n) amortized time per insertion by Italiano [11] and also by La Poutré and van Leeuwen [16] . Yellin [22] gave an O(m * δ max ) algorithm for m edge insertions, where m * is the number of edges in the final transitive closure and δ max is the maximum out-degree of the final graph. All these algorithms maintain explicitly the transitive closure, and so their query time is O (1) .. The first decremental algorithm was again given by Ibaraki and Katoh [10] , with a running time of O(n 2 ) per deletion. This was improved to O(m) per deletion by La Poutré and van Leeuwen [16] . Italiano [12] presented an algorithm that achieves O(n) amortized time per deletion on directed acyclic graphs. Yellin [22] gave an O(m * δ max ) algorithm for m edge deletions, where m * is the initial number of edges in the transitive closure and δ max is the maximum out-degree of the initial graph. Again, the query time of all these algorithms is O (1) . More recently, Henzinger and King [9] gave a randomized decremental transitive closure algorithm for general directed graphs with a query time of O(n/ log n) and an amortized update time of O(n log 2 n).. The first fully dynamic transitive closure algorithm was devised by Henzinger and King [9] in 1995: they gave a randomized Monte Carlo algorithm with onesided error supporting a query time of O(n/ log n) and an amortized update time of O(nm 0.58 log 2 n), wherem is the average number of edges in the graph throughout the whole update sequence. Sincem can be as large as O(n 2 ), their update time is O(n 2.16 log 2 n). Khanna , Motwani and Wilson [13] proved that, when a lookahead of (n 0.18 ) in the updates is permitted, a deterministic update bound of O(n 2.18 ) can be achieved. Recently, King and Sagert [15] showed how to support queries in O (1) time and updates in O(n 2.26 ) time for general directed graphs and O(n 2 ) time for directed acyclic graphs; their algorithm is randomized with one-sided error. In a major breakthrough [14] , King improved the bounds of King and Sagert, by exhibiting a deterministic algorithm on general digraphs with O(1) query time and O(n 2 log n) amortized time per update operations, where updates are insertions of a set of edges incident to the same vertex and deletions of an arbitrary subset of edges. We remark that all these algorithms (except [14] ) use fast matrix multiplication as a subroutine.. We observe that fully dynamic transitive closure algorithms with O(1) query time maintain explicitly the transitive closure of the input graph, in order to answer each query with exactly one lookup (on its adjacency matrix). Since an update may change as many as (n 2 ) entries of this matrix, O(n 2 ) seems to be the best update bound that one could hope for this class of algorithms. It is thus quite natural to ask whether the O(n 2 ) update bound can be actually realized for fully dynamic transitive closure on general directed graphs while maintaining one lookup per query.. Our Results In this paper, we affirmatively answer this question. We exhibit a deterministic algorithm for fully dynamic transitive closure on general digraphs that does exactly one matrix look-up per query and supports updates in O(n 2 ) amortized time, thus improving over [14] . Our algorithm can also support within the same time bounds the generalized updates of [14] , i.e., insertion of a set of edges incident to the same vertex and deletion of an arbitrary subset of edges. In the special case of deletions only, our algorithm achieves O(n) amortized time for deletions and O(1) time for queries: this generalizes to directed graphs the bounds of [12] , and improves over [9] . In [5] , we show how to break through the O(n 2 ) barrier in the case of directed acyclic graphs.. Our results are based on a novel technique: we introduce a general framework for maintaining polynomials defined over matrices, and we cast fully dynamic transitive closure into this framework. In particular, our algorithm hinges upon the equivalence between transitive closure and matrix multiplication on a closed semiring; this relation has been known for over 30 years (see, e.g., the results of Munro [18] , Furman [8] and Fischer and Meyer [7] ) and yields the fastest known static algorithm for transitive closure. Surprisingly, no one before seems to have exploited this equivalence in the dynamic setting: some recent algorithms [9, 13, 15 ] make use of fast matrix multiplication, but only as a subroutine for fast updates. Differently from other approaches, the crux of our method is to use dynamic reevaluation of products of Boolean matrices as the kernel for solving dynamic transitive closure.. The remainder of this paper is organized as follows. In Sect. 2 we first define the fully dynamic transitive closure problem and show that it is equivalent to a fully dynamic Boolean matrix closure problem. In Sect. 3 we address the problem of maintaining polynomials defined over Boolean matrices during updates of their variables. In Sect. 4 we present a first method for casting fully dynamic transitive closure into the dynamic problem on matrices considered in Sect. 3: as a warm-up, we revisit within this framework the algorithm of King [14] from a completely different perspective. In Sect. 5 we show how the algebraic approach of Sect. 3 can be further refined: this yields our improved algorithm for fully dynamic transitive closure. Finally, in Sect. 6 we list some concluding remarks. In this paper we have presented new and improved algorithms for maintaining the transitive closure of a directed graph under edge insertions and edge deletions. As a main contribution, we have introduced a general framework for casting fully dynamic transitive closure into the problem of dynamically reevaluating polynomials over matrices when updates of variables are performed. Such technique has turned out to be very flexible and powerful, leading both to revisit the best known algorithm for fully dynamic transitive closure [14] from a completely different perspective, and to design new and faster algorithms for this problem. We remark that the algebraic approach to dynamic data structures for graph problems introduced in this paper has sparkled a wealthy of new results [17, 20, 21] . Designing efficient dynamic data structures for maintaining polynomials over Boolean matrices allowed us to devise the fairly complex deterministic algorithm described in Sect. 5, which supports updates in quadratic amortized time and queries with just one matrix lookup. Our algorithm improves the best bounds for fully dynamic transitive closure achieved in [14] . After our work, Roditty [19] reduced the Init bound from O(n 3 ) to O(mn), and Sankowski [20] made our bounds worst case. In the following, we discuss the correctness and the complexity of our implementation of operations Init * , Set * , Reset * , and Lookup * presented in Sect. 4.2. We recall that X is an n × n Boolean matrix and P k , 0 ≤ k ≤ log 2 n, are the polynomials introduced in Definition 14.. Theorem 5 If at any time during a sequence of operations there is a path of length up to 2 k between x and y in X, then P k [x, y] = 1.. Proof We proceed by induction. The base is trivial. We assume that the claim holds inductively for P k−1 , and we show that, after any operation, the claim holds also for P k .. • Init * : since any Init * operation rebuilds from scratch P k , the claim holds from Lemma 2. • Set * : let us assume that a Set * operation is performed on the i-th row and column of X and a new path π of length up to 2 k , say π = x, . . . , i, . . . , y , appears in X due to this operation. We prove that P k [x, y] = 1 after the operation. Observe that P k .LazySet( P k−1 , P k−1 ) puts in place any new 1's in any occurrence of the variable P k−1 in data structure P k . We remark that, although the maintained value of P k in data structure P k is not updated by LazySet and therefore the correctness of the current operation is not affected, this step is very important: indeed, new 1's corresponding to new paths of length up to 2 k−1 that appear in X will be useful in future Set * operations for detecting the appearance of new paths of length up to 2 k .. If both the portions x i and i y of π have length up to 2 k−1 , then π gets recorded in P 2 k−1 , and therefore in P k , thanks to one of P k .SetRow(i, P k−1 , P k−1 ) or P k .SetCol(i, P k−1 , P k−1 ). On the other hand, if i is close to (but does not coincide with) one endpoint of π , the appearance of π may be recorded in P 3 k−1 , but not in P 2 k−1 . This is the reason why degree 2 does not suffice for P k in this dynamic setting. • Reset * : by inductive hypothesis, we assume that P k−1 [x, y] flips to zero after a Reset * operation only if no path of length up to 2 k−1 remains in X between x and y. Since any P k .Reset operation on P k leaves it as if cleared 1's in P k−1 were never set to 1, P k [x, y] flips to zero only if no path of length up to 2 k remains in X.. We remark that the condition stated in Theorem 5 is only sufficient because P k may keep track of paths having length strictly more than 2 k , though no longer than 3 k . However, for k = log 2 n the condition is also necessary as no shortest path can be longer than n = 2 k . Thus, it is straightforward to see that a path of any length between x and y exists at any time in X if and only if P log 2 n [x, y] = 1.. The following theorem addresses the running time and space requirements of operations Init * , Set * and Reset * . Proof Since we maintain O(log n) polynomials with constant degree k = 3 and constant number of terms h = 3, it follows from Theorem 4 that the total space usage is O(n 2 · log n).. We now discuss the time bounds. Lookup * requires one Lookup operation, while Init * requires O(log n) Init operations. Set * can be implemented via O(log n) LazySet, SetRow, and SetCol operations. Thus their amortized bound follows from Theorem 4. As in Theorem 4, the cost of Reset * can be amortized against previous Init * and Set * operations.. Now we discuss the correctness and the complexity of our implementation. Before providing the main claims, we give some preliminary definitions and lemmas that are useful for capturing algebraic properties of the changes that polynomials in our data structure undergo during a Set * operation. The next definition recalls a property of Boolean update matrices that is related to the operational concept of i-centered update.

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


Broad expertise retrieval in sparse data environments Expertise retrieval has been largely unexplored on data other than the W3C collection. At the same time, many intranets of universities and other knowledge-intensive organisations offer examples of relatively small but clean multilingual expertise data, covering broad ranges of expertise areas. We first present two main expertise retrieval tasks, along with a set of baseline approaches based on generative language modeling, aimed at finding expertise relations between topics and people. For our experimental evaluation, we introduce (and release) a new test set based on a crawl of a university site. Using this test set, we conduct two series of experiments. The first is aimed at determining the effectiveness of baseline expertise retrieval methods applied to the new test set. The second is aimed at assessing refined models that exploit characteristic features of the new test set, such as the organizational structure of the university, and the hierarchical structure of the topics in the test set. Expertise retrieval models are shown to be robust with respect to environments smaller than the W3C collection, and current techniques appear to be generalizable to other settings. An organization's intranet provides a means for exchanging information between employees and for facilitating employee collaborations. To efficiently and effectively achieve this, it is necessary Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. to provide search facilities that enable employees not only to access documents, but also to identify expert colleagues.. At the TREC Enterprise Track [22] the need to study and understand expertise retrieval has been recognized through the introduction of Expert Finding tasks. The goal of expert finding is to identify a list of people who are knowledgeable about a given topic. This task is usually addressed by uncovering associations between people and topics [10] ; commonly, a co-occurrence of the name of a person with topics in the same context is assumed to be evidence of expertise. An alternative task, which using the same idea of people-topic associations, is expert profiling, where the task is to return a list of topics that a person is knowledgeable about [3] .. The launch of the Expert Finding task at TREC has generated a lot of interest in expertise retrieval, with rapid progress being made in terms of modeling, algorithms, and evaluation aspects. However, nearly all of the expert finding or profiling work performed has been validated experimentally using the W3C collection [24] from the Enterprise Track. While this collection is currently the only publicly available test collection for expertise retrieval tasks, it only represents one type of intranet. With only one test collection it is not possible to generalize conclusions to other realistic settings.. In this paper we focus on expertise retrieval in a realistic setting that differs from the W3C setting-one in which relatively small amounts of clean, multilingual data are available, that cover a broad range of expertise areas, as can be found on the intranets of universities and other knowledge-intensive organizations. Typically, this setting features several additional types of structure: topical structure (e.g., topic hierarchies as employed by the organization), organizational structure (faculty, department, . . . ), as well as multiple types of documents (research and course descriptions, publications, and academic homepages). This setting is quite different from the W3C setting in ways that might impact upon the performance of expertise retrieval tasks.. We focus on a number of research questions in this paper: Does the relatively small amount of data available on an intranet affect the quality of the topic-person associations that lie at the heart of expertise retrieval algorithms? How do state-of-the-art algorithms developed on the W3C data set perform in the alternative scenario of the type described above? More generally, do the lessons from the Expert Finding task at TREC carry over to this setting? How does the inclusion or exclusion of different documents affect expertise retrieval tasks? In addition to, how can the topical and organizational structure be used for retrieval purposes?. To answer our research questions, we first present a set of baseline approaches, based on generative language modeling, aimed at finding associations between topics and people. This allows us to formulate the expert finding and expert profiling tasks in a uniform way, and has the added benefit of allowing us to understand the relations between the two tasks. For our experimental evaluation, we introduce a new data set (the UvT Expert Collection) which is representative of the type of intranet that we described above. Our collection is based on publicly available data, crawled from the website of Tilburg University (UvT). This type of data is particularly interesting, since (1) it is clean, heterogeneous, structured, and focused, but comprises a limited number of documents; (2) contains information on the organizational hierarchy; (3) it is bilingual (English and Dutch); and (4) the list of expertise areas of an individual are provided by the employees themselves. Using the UvT Expert collection, we conduct two sets of experiments. The first is aimed at determining the effectiveness of baseline expertise finding and profiling methods in this new setting. A second group of experiments is aimed at extensions of the baseline methods that exploit characteristic features of the UvT Expert Collection; specifically, we propose and evaluate refined expert finding and profiling methods that incorporate topicality and organizational structure.. Apart from the research questions and data set that we contribute, our main contributions are as follows. The baseline models developed for expertise finding perform well on the new data set. While on the W3C setting the expert finding task appears to be more difficult than profiling, for the UvT data the opposite is the case. We find that profiling on the UvT data set is considerably more difficult than on the W3C set, which we believe is due to the large (but realistic) number of topical areas that we used for profiling: about 1,500 for the UvT set, versus 50 in the W3C case. Taking the similarity between topics into account can significantly improve retrieval performance. The best performing similarity measures are content-based, therefore they can be applied on the W3C (and other) settings as well. Finally, we demonstrate that the organizational structure can be exploited in the form of a context model, improving MAP scores for certain models by up to 70%.. The remainder of this paper is organized as follows. In the next section we review related work. Then, in Section 3 we provide detailed descriptions of the expertise retrieval tasks that we address in this paper: expert finding and expert profiling. In Section 4 we present our baseline models, of which the performance is then assessed in Section 6 using the UvT data set that we introduce in Section 5. Advanced models exploiting specific features of our data are presented in Section 7 and evaluated in Section 8. We formulate our conclusions in Section 9. In this paper we focused on expertise retrieval (expert finding and profiling) in a new setting of a typical knowledge-intensive organization in which the available data is of high quality, multilingual, and covering a broad range of expertise area. Typically, the amount of available data in such an organization (e.g., a university, a research institute, or a research lab) is limited when compared to the W3C collection that has mostly been used for the experimental evaluation of expertise retrieval so far.. To examine expertise retrieval in this setting, we introduced (and released) the UvT Expert collection as a representative case of such knowledge intensive organizations. The new collection reflects the typical properties of knowledge-intensive institutes noted above and also includes several features which may are potentially useful for expertise retrieval, such as topical and organizational structure.. We evaluated how current state-of-the-art models for expert finding and profiling performed in this new setting and then refined these models in order to try and exploit the different characteristics within the data environment (language, topicality, and organizational structure). We found that current models of expertise retrieval generalize well to this new environment; in addition we found that refining the models to account for the differences results in significant improvements, thus making up for problems caused by data sparseness issues.. Future work includes setting up manual assessments of automatically generated profiles by the employees themselves, especially in cases where the employees have not provided a profile themselves. Looking at Table 1 we see that Model 2 performs the best across the board. However, when the data is clean and very focused (RD), Model 3 outperforms it in a number of cases. Model 1 has the best coverage of candidates (%ca) and topics (%q). The various document types differ in their characteristics and how they improve the finding and profiling tasks. Expert profiling benefits much from the clean data present in the RD and CD document types, while the publications contribute the most to the expert finding task. Adding the homepages does not prove to be particularly useful.. When we compare the results across languages, we find that the coverage of English topics (%q) is higher than of the Dutch ones for expert finding. Apart from that, the scores fall in the same range for both languages. For the profiling task the coverage of the candidates (%ca) is very similar for both languages. However, the performance is substantially better for the English topics.. While it is hard to compare scores across collections, we conclude with a brief comparison of the absolute scores in Table 1 to those reported in [3, 4] on the W3C test set (2005 edition). For expert finding the MAP scores for Model 2 reported here are about 50% higher than the corresponding figures in [4] , while our MRR scores are slightly below those in [4] . For expert profiling, the differences are far more dramatic: the MAP scores for Model 2 reported here are around 50% below the scores in [3] , while the (best) MRR scores are about the same as those in [3] . The cause for the latter differences seems to reside in the number of knowledge areas considered here-approx. 30 times more than in the W3C setting. Initial approaches to expertise finding often employed databases containing information on the skills and knowledge of each individual in the organization [11] . Most of these tools (usually called yellow pages or people-finding systems) rely on people to self-assess their skills against a predefined set of keywords. For updating profiles in these systems in an automatic fashion there is a need for intelligent technologies [5] . More recent approaches use specific document sets (such as email [6] or software [18] ) to find expertise. In contrast with focusing on particular document types, there is also an increased interest in the development of systems that index and mine published intranet documents as sources of evidence for expertise. One such published approach is the P@noptic system [9] , which builds a representation of each person by concatenating all documents associated with that person-this is similar to Model 1 of Balog et al. [4] , who formalize and compare two methods. Balog et al.'s Model 1 directly models the knowledge of an expert from associated documents, while their Model 2 first locates documents on the topic and then finds the associated experts. In the reported experiments the second method performs significantly better when there are sufficiently many associated documents per candidate. Most systems that took part in the 2005 and 2006 editions of the Expert Finding task at TREC implemented (variations on) one of these two models; see [10, 20] . Macdonald and Ounis [16] propose a different approach for ranking candidate expertise with respect to a topic based on data fusion techniques, without using collectionspecific heuristics; they find that applying field-based weighting models improves the ranking of candidates. Petkova and Croft [19] propose yet another approach, based on a combination of the above Model 1 and 2, explicitly modeling topics.. Turning to other expert retrieval tasks that can also be addressed using topic-people associations, Balog and de Rijke [3] addressed the task of determining topical expert profiles. While their methods proved to be efficient on the W3C corpus, they require an amount of data that may not be available in the typical knowledge-intensive organization. Balog and de Rijke [2] study the related task of finding experts that are similar to a small set of experts given as input.. As an aside, creating a textual "summary" of a person shows some similarities to biography finding, which has received a considerable amount of attention recently; see e.g., [13] .. We use generative language modeling to find associations between topics and people. In our modeling of expert finding and profiling we collect evidence for expertise from multiple sources, in a heterogeneous collection, and integrate it with the co-occurrence of candidates' names and query terms-the language modeling setting allows us to do this in a transparent manner. Our modeling proceeds in two steps. In the first step, we consider three baseline models, two taken from [4] (the Models 1 and 2 mentioned above), and one a refined version of a model introduced in [3] (which we refer to as Model 3 below); this third model is also similar to the model described by Petkova and Croft [19] . The models we consider in our second round of experiments are mixture models similar to contextual language models [1] and to the expanded documents of Tao et al. [21] ; however, the features that we use for definining our expansions-including topical structure and organizational structure-have not been used in this way before. The output of our models was evaluated against the self-assigned topic labels, which were treated as relevance judgements. Results were evaluated separately for English and Dutch. For English we only used topics for which the Dutch translation was available; for Dutch all topics were considered. The results were averaged for the queries in the intersection of relevance judgements and results; missing queries do not contribute a value of 0 to the scores.. We use standard information retrieval measures, such as Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR). We also report the percentage of topics (%q) and candidates (%ca) covered, for the expert finding and profiling tasks, respectively. Table 1 shows the performance of Model 1, 2, and 3 on the expert finding and profiling tasks. The rows of the table correspond to the various document types (RD, CD, PUB, and HP) and to their combinations. RD+CD+PUB+HP is equivalent to the full collection and will be referred as the BASELINE of our experiments. Given that the self-assessments are also sparse in our collection, in order to be able to measure differences between the various models, we selected a subset of topics, and evaluated (some of the) runs only on this subset. This set is referred as main topics, and consists of topics that are located at the top level of the topical hierarchy. (A main topic has subtopics, but is not a subtopic of any other topic.) This main set consists of 132 Dutch and 119 English topics. The relevance judgements were restricted to the main topic set, but were not expanded with subtopics. wise mutual information (PMI), log-likelihood (LL), and distance within topic hierarchy (HDIST). We managed to improve upon the baseline in all cases, but the improvement is more noticeable for the profiling task. For both tasks, the LL method performed best. The content-based approaches performed consistently better than HDIST.

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Anti-jamming timing channels for wireless networks Wireless communication is susceptible to radio interference, which prevents the reception of communications. Although evasion strategies have been proposed, such strategies are costly or ineffective against broadband jammers. In this paper, we explore an alternative to evasion strategies that involves the establishment of a timing channel that exists in spite of the presence of jamming. The timing channel is built using failed packet reception times. We first show that it is possible to detect failed packet events in spite of jamming. We then explore single sender and multi-sender timing channel constructions that may be used to build a low-rate overlay link-layer. We discuss implementation issues that we have overcome in constructing such jammingresistant timing channel, and present the results of validation efforts using the MICA2 platform. Finally, we examine additional error correction and authentication mechanisms that may be used to cope with adversaries that both jam and seek to corrupt our timing channel. Wireless technologies depend on the opportunities provided by a broadcast medium. However, the factors that facilitate tetherless, ubiquitous communication also can be used to disrupt such communication, thereby undermining the availability of a wireless networks. There are many Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. strategies that may be employed to disrupt wireless connectivity. Some strategies, such as deassociation attacks or association flooding [1] , represent network-oriented attacks that have been employed against commodity wireless systems (e.g. 802.11). Although such availability threats are powerful, they can (and should) be addressed through authentication. An alternative strategy to disrupt wireless communications is to directly interfere with communications by jamming the communication channel [2] . In fact, intentionally interfering with wireless communications is simple: an RF source (e.g. a waveform generator) may broadcast energy on a broad spectral band to disrupt signal reception [2] , or the medium access control may be cleverly exploited to disrupt link-layer decoding [3, 4] , or medium access control mechanisms may be bypassed to prevent other devices from accessing the channel [1, 3] .. RF interference has traditionally been addressed at the physical layer through modulation approaches (such as spread spectrum) with sufficiently powerful anti-jam margins to make disruption difficult. Unfortunately, such physical layer approaches are typically representative of military wireless systems and, for reasons of cost, are generally not employed in commodity wireless platforms 1 . For commodity networks, several defense strategies have been proposed to repair network connectivity in the presence of interference. In [5] , protocols are presented that allow an ad hoc/sensor network to adjust its operating channel to evade interference. Unfortunately, the interference is limited to jammers operating on one or a few channels, and it should be noted that it is harder for an entire network to adjust its channel allocations than for an aggressive adversary to scan the spectrum and interfere on subsequent channels. Similarly, constantly adapting channels, as proposed in [6] , might emulate a link-layer frequency hopping strategy, but also cannot survive in the presence of broadband adversaries. Spatial evasion strategies [7] are costly and require complex reparation protocols to prevent network partitioning.. Such evasion strategies are limited and, although it is impossible to overcome a broadband adversary with unlimited transmission power, it is nonetheless desirable to have mechanisms to communicate in the presence of broadband interference when the adversary has transmission power limitations. In this paper, we propose an alternative to evasion strategies. Rather, we seek to communicate in the presence of a powerful RF adversary 2 . Our objective is to create a low bit-rate overlay that exists on top of the conventional physical/link-layers and that would survive in the presence of a persistent broadband interferer. Our low bit-rate channel is constructed as a timing-channel that exists in spite of the jammer being present. We emphasize that this timing channel is not a panacea-it restores the availability of a communication link in the presence of interference, but other security issues, such as authentication, must be addressed through complimentary methods.. We begin the paper in Section 2 by providing an overview of our timing channel overlay. In Section 3, we show that detecting failed packet receptions is feasible, and thus our timing channel can be constructed. We next examine a simple single sender scenario in Section 4. Then we extend our timing channel to a multiple sender scenario, in Section 5. Finally, using our multi-sender timing channel, we explore the construction of an overlay link-layer in Section 6, and discuss mechanisms that enhance the reliability of our overlay. We wrap up the paper by discussing related work in Section 7, and provide concluding remarks in Section 8. We have proposed the creation of a jamming-resistant timing channel to restore the availability of communication links in the presence of interference. The timing channel is built as a low-rate physical layer overlay on top of the traditional physical/link-layers. Our timing channel uses the detection and timing of failed packet receptions at the receiver, which we have shown is possible by time stamping CRC failures or by monitoring the signal strength. We then proposed interarrival codes for building a single-sender, single-receiver timing channel, and validated three different packet sequencing methods on the MICA2 platform. To cope with multiple sender, single-receiver cases, we presented an asynchronous code construction and a decoding procedure that employs a bank of parallel correlators. We evaluated our multi-sender construction in several cases and found that we could achieve an overlay link reliability better than 99%. Following the construction of the timing channel bit pipe, we presented our construction of an overlay data link layer, which provides framing, error detection/correction, and authentication. Our timing channel can resume communications and, although the data rate may be low, this accomplishment is significant when compared to the alternative of the jammer causing there to be no communication capabilities. Finally, we note that the timing channel construction is intended primarily to restore the availability of communication functions and that additional mechanisms, such as authentication, may be used in the overlay or at higher layers to assure the new channel is secure. Coping with jamming and interference is usually a topic that is addressed through conventional PHY-layer communication techniques. In these systems, spreading techniques (e.g. frequency hopping) are commonly used to provide resilience to interference [13, 14] . Although such PHY-layer techniques can address the challenges of an RF interferer, they require advanced transceivers.. The issue of detecting and mapping jamming for sensor networks was studied by Wood and Stankovic in [8] . The problem of jamming detection was further studied by Xu et al. in [3] , where the authors presented several jamming models and explored the need for more advanced form of detection algorithms to identify jamming. Additional jamming strategies were studied by Law et al. [4] , and the efficiency of these methods was quantified in terms of the amount of resources needed to conduct an attack. Further work on jamming has studied MAC-layer jamming attacks on reservation-based medium access control schems [15] .. Countermeasures for coping with jammed regions in wireless networks have been studied in [2, 7, 16, 17] . In [16] , the use of error correcting codes is proposed to cope with jamming. In [2] , channel surfing and spatial retreats are proposed. Spatial retreats was studied in more detail in [7] , while a channel surfing system was built and studied in [5] . A variation of channel surfing was studied in [6] , where the channel assignments are constantly adapted to emulate link-layer frequency hopping. Channel surfing strategies are not a suitable defense against broadband jamming. In [18] , a protocol suite is proposed to cope with jamming that reduces the impact of jamming. Wormhole-based antijamming techniques have been proposed as a means to allow the delivery of important alarm messages [17] .. Our work takes a different strategy to communicating in the presence of interference. We create a timing channel that serves as the basis for a new link-layer overlay, that can allow communication of important messages in spite of the presence of broadband interference. Timing channels have traditionally been studied in the context of covert communication, e.g. [19, 20] . Techniques to mitigate the effectiveness of covert channels have been studied in the context of network pumps [21] . More generally, the fundamental information-bearing capacity of timing and queuing channels has been studied in [22, 23] . In the jamming context, we assume the adversary knows the presence of communication, and our work differs from covert timing channels in that we seek availability instead of covertness.

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On the complexity of finding common approximate substrings Problems associated with ÿnding strings that are within a speciÿed Hamming distance of a given set of strings occur in several disciplines. In this paper, we use techniques from parameterized complexity to assess non-polynomial time algorithmic options and complexity for the COMMON APPROXIMATE SUBSTRING (CAS) problem. Our analyses indicate under which parameter restrictions useful algorithms are possible, and include both class membership and parameterized reductions to prove class hardness. In order to achieve ÿxed-parameter tractability, either a ÿxed string length or both a ÿxed size alphabet and ÿxed substring length are su cient. Fixing either the string length or the alphabet size and Hamming distance is shown to be necessary, unless W [1] = FPT . An assortment of parameterized class membership and hardness results cover all other parameterized variants, showing in particular the e ect of ÿxing the number of strings. Finding strings that are approximately present in each of a given set of strings is a problem common to many areas of application, including computational biology.. What "approximately present" means can vary, including the existence of di erent and arbitrarily-sized gaps in instances of the found string. In this paper, we restrict the approximate presence of a string C in a given string S to mean the existence of a substring of S that is within a speciÿed Hamming distance of C. That substring of S is considered to be an (approximate) instance of C, and the string C is considered to be an approximate substring of S. If C is an approximate substring of each string in a set F of strings, then it is a common approximate substring for F.. Formally, given two strings x and y of the same length over an alphabet , the Hamming distance between x and y is the number of positions at which the symbols in x and y di er. Over the last several years, a number of results have been derived for restricted versions of the following problem involving Hamming distance:

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