We demonstrate how graph decomposition techniques can be employed for the visualization of hierarchical co-occurrence patterns between medical data items. Our research is based on Gaifman graphs (a mathematical concept introduced in Logic), on specific variants of this concept, and on existing graph decomposition notions, specifically, graph modules and the clan decomposition of so-called 2-structures. The construction of the Gaifman graphs from a dataset is based on co-occurrence, or lack of it, of items in the dataset. We may select a discretization on the edge labels to aim at one among several Gaifman graph variants. Then, the decomposition of the graph may provide us with visual information about the data co-occurrences, after which one can proceed to more traditional statistical analysis.

The authors have recently proposed the usage of modular decompositions of Gaifman graphs as an exploratory data analysis tool. We describe how these techniques allow for a compact, hierarchical visualization of the patterns of cooccurrence between data items, in the context of medical data corresponding to simultaneous diagnostics of patients.

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective, the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises. By exploiting a previously noted alternative view of this entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises. As a result, we obtain decision algorithms of better complexity, additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion.

Some existing notions of redundancy among association rules allow for a logical-style characterization and lead to irredundant bases of absolutely minimum size. We push the intuition of redundancy further to find an intuitive notion of novelty of an association rule, with respect to other rules. Namely, an irredundant rule is so because its confidence is higher than what the rest of the rules would suggest; then, one can ask: how much higher? We propose to measure such a sort of novelty through the confidence boost of a rule. Acting as a complement to confidence and support, the confidence boost helps to obtain small and crisp sets of mined association rules and solves the well-known problem that, in certain cases, rules of negative correlation may pass the confidence bound. We analyze the properties of two versions of the notion of confidence boost, one of them a natural generalization of the other. We develop algorithms to filter rules according to their confidence boost, compare the concept to some similar notions in the literature, and describe the results of some experimentation employing the new notions on standard benchmark datasets. We describe an open source association mining tool that embodies one of our variants of confidence boost in such a way that the data mining process does not require the user to select any value for any parameter.

Support Vector Machines are a family of algorithms for the analysis of data based on convex Quadratic Programming. We derive randomized algorithms for training SVMs, based on a variation of Random Sampling Techniques; these have been successfully used for similar problems. We formally prove an upper bound on the expected running time which is quasilinear with respect to the number of data points and polynomial with respect to the other parameters, i.e., the number of attributes and the inverse of a chosen soft margin parameter. [This is the combined journal version of the conference papers (Balcázar, J.L. et al. in Proceedings of 12th International Conference on Algorithmic Learning Theory (ALT’01), pp. 119–134, [ 2001 ]; Balcázar, J.L. et al. in Proceedings of First IEEE International Conference on Data Mining (ICDM’01), pp. 43–50, [ 2001 ]; and Balcázar, J.L. et al. in Proceedings of SIAM Workshop in Discrete Mathematics and Data Mining, pp. 19–29, [ 2002 ]).]

We show that for most complexity classes of interest, all sets complete under first-order projections (fops) are isomorphic under first-order isomorphisms. That is, a very restricted version of the Berman--Hartmanis conjecture holds. Since "natural" complete problems seem to stay complete via fops, this indicates that up to first-order isomorphism there is only one "natural" complete problem for each "nice" complexity class.

The computational power of recurrent neural networks is shown to depend ultimately on the complexity of the real constants (weights) of the network. The complexity, or information contents, of the weights is measured by a variant of resource-bounded Kolmogorov (1965) complexity, taking into account the time required for constructing the numbers. In particular, we reveal a full and proper hierarchy of nonuniform complexity classes associated with networks having weights of increasing Kolmogorov complexity.

Sequential algorithms given by Angluin 1987 and Schapire 1992 learn deterministic nite automata DFA exactly from Membership and Equivalence queries.These algorithms are feasible, in the sense that they take time polynomial in n and m, where n is the number of states of the automaton and m is the length of the longest counterexample to an Equivalence query.This paper studies whether parallelism can lead to substantially more e cient algorithms for the problem.We show that no CRCW PRAM machine using a number of processors polynomial in n and m can identify DFA in on= log n time.Furthermore, this lower bound is tight up to constant factors: we develop a CRCW PRAM learning algorithm that uses polynomially many processors and exactly learns DFA in time On= log n.

The authors introduce and study a family of interconnection schemes, the Midimew networks, based on circulant graphs of degree 4. A family of such circulants is determined and shown to be optimal with respect to two distance parameters simultaneously, namely maximum distance and average distance, among all circulants of degree 4.. These graphs are regular, point-symmetric, and maximally connected, and one such optimal graph exists for any given number of nodes. The proposed interconnection schemes consist of mesh-connected networks with wrap-around links, and are isomorphic to the optimal distance circulants previously considered. Ways to construct one such network for any number of nodes are shown, their good properties to build interconnection schemes for multicomputers are examined, and some interesting particular cases are discussed. The problem of routing is also addressed, and a basic algorithm is provided which is adequate for implementing the routing policy required to convey messages, traversing shortest paths between nodes.>

A class of mesh networks with wrap-around links is obtained from a class of circulant graphs by means of a graph isomorphism. We demonstrate how to obtain, from the adjacency pattern of the graph, simple parameters that serve to construct a planar design of the network. Several performance parameters are evaluated: in particular, we show that diameter and average distance are simultaneously minimized. This implies a minimization of the network communication delays. Due to its easy implementation and good behavior characteristics, the proposed interconnection scheme is appropriate in several architectural environments. Specifically, this topology is suitable as an interconnection subsystem for message passing MIMD architectures, as well as for SIMD machines with a static interconnection scheme. In the particular case of SIMD machines, a comparison is made with the ILLIAC IV-type networks. As a consequence, we propose still another topology, when the number of processing elements is an even power of 2; for this solution, we show that a reduction in the network distances is achieved, without losing speed in performing arbitrary permutations.

Questions about the polynomial-time hierarchy are studied. In particular, the questions, “Does the polynomial-time hierarchy collapse?” and “Is the union of the hierarchy equal to PSPACE?” are considered, along with others comparing the union of the hierarchy with certain probabilistic classes. In each case it is shown that the answer is “yes” if and only if for every sparse set S , the answer is “yes” when the classes are relativized to S if and only if there exists a sparse set S such that the answer is “yes” when the classes are relativized to S . Thus, in each case the question is answered if it is answered for any arbitrary sparse oracle set. Long and Selman first proved that the polynomial-time hierarchy collapses if and only if for every sparse set S , the hierarchy relative to S collapses. This result is re-proved here by a different technique.

We develop the notions of “generalized lowness” for sets in PH (the union of the polynomial-time hierarchy) and of “generalized highness” for arbitrary sets. Also, we develop the notions of “extended lowness” and “extended highness” for arbitrary sets. These notions extend the decomposition of NP into low sets and high sets developed by Schöning [15] and studied by Ko and Schöning [9]. We show that either every sparse set in PH is generalized high or no sparse set in PH is generalized high. Further, either every sparse set is extended high or no sparse set is extended high. In both situations, the former case corresponds to the polynomial-time hierarchy having only finitely many levels while the latter case corresponds to the polynomial-time hierarchy extending infinitely many levels.

Relativizations of complexity classes in which simple sets exist are considered. A recursive oracle is constructed relative to which a simple set exists for NP. Some other general theorems are proven, showing that the time bounds are not a crucial hypothesis; bounds on the way in which the oracle is accessible, namely, the number of queries and/or the number of nondeterministic steps, are shown to be the fundamental hypothesis. As a result, simple sets are shown to exist in many different relativized complexity classes.

We show that several questions about the polynomial-time hierarchy can be answered by answering their counterparts for the polynomial-time hierarchy relativized to an arbitrary sparse oracle set. For each of these questions, the answer will be the same for the hierarchy relativized to S/sub 1/ as it will be for the hierarchy relativized to S/sub 2/ for any choice of S/sub 1/ and S/sub 2/ that are sparse sets, including the choice of S/sub 1/ being empty and S/sub 2/ being nonempty but sparse.