Testing For Membership Research Articles

Knowledge Refinement via Rule Selection

In several different applications, including data transformation and entity resolution, rules are used to capture aspects of knowledge about the application at hand. Often, a large set of such rules is generated automatically or semi-automatically, and the challenge is to refine the encapsulated knowledge by selecting a subset of rules based on the expected operational behavior of the rules on available data. In this paper, we carry out a systematic complexity-theoretic investigation of the following rule selection problem: given a set of rules specified by Horn formulas, and a pair of an input database and an output database, find a subset of the rules that minimizes the total error, that is, the number of false positive and false negative errors arising from the selected rules. We first establish computational hardness results for the decision problems underlying this minimization problem, as well as upper and lower bounds for its approximability. We then investigate a bi-objective optimization version of the rule selection problem in which both the total error and the size of the selected rules are taken into account. We show that testing for membership in the Pareto front of this bi-objective optimization problem is DP-complete. Finally, we show that a similar DP-completeness result holds for a bi-level optimization version of the rule selection problem, where one minimizes first the total error and then the size.

On the analysis of Bloom filters

The Bloom filter is a simple random binary data structure which can be efficiently used for approximate set membership testing. When testing for membership of an object, the Bloom filter may give a false positive, whose probability is the main performance figure of the structure. We complete and extend the analysis of the Bloom filter available in the literature by means of the γ-transform approach. Known results are confirmed and new results are provided, including the variance of the number of bits set to 1 in the filter. We consider the choice of bits to be set to 1 when an object is inserted both with and without replacement, in what we call standard and classic Bloom filter, respectively. Simple iterative schemes for the computation of the false positive probability and a new non-iterative approximation, taking into account the variance of bits set to 1, are also provided.

Edge intersection graphs of [formula omitted]-shaped paths in grids

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞,⌜,⌟,⌝, and we consider zero bend paths (i.e., ∣ and–) to be degenerate ⌞’s. These graphs, called B1-EPG graphs, were first introduced by Golumbic et al. (2009). We consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {⌞},{⌞,⌜},{⌞,⌝}, and {⌞,⌜,⌝}) and we denote the classes by [⌞],[⌞,⌜],[⌞,⌝], and [⌞,⌜,⌝] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [⌞]⊊[⌞,⌜],[⌞,⌝]⊊[⌞,⌜,⌝]⊊B1-EPG and [⌞,⌜] is incomparable with [⌞,⌝]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩[⌞].

Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞,⌜,⌟,⌝). These graphs, called B1- EPG graphs, were first introduced by Golumbic et al (2009) [Golumbic, M. C., M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid, Networks 54:3 (2009), 130–138]. We focus on the class [⌞] (the edge intersection graphs of ⌞-shapes) and show that testing for membership in [⌞] is NP-complete. We then give a characterization and polytime recognition algorithm for special subclasses of Split∩[⌞]. We also consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {⌞},{⌞,⌜},{⌞,⌝},{⌞,⌜,⌝} – note: all other subsets are isomorphic to these up to 90 degree rotation). We observe the expected strict inclusions and incomparability (i.e., [⌞]⊊[⌞,⌜],[⌞,⌝]⊊[⌞,⌜,⌝]⊊B1-EPG and [⌞,⌜] is incomparable with [⌞,⌝]).

Testing composite hypotheses about discrete ergodic processes

Given a discrete-valued sample X1,…,Xn, we wish to decide whether it was generated by a distribution belonging to a family H0, or it was generated by a distribution belonging to a family H1. In this work we assume that all distributions are stationary ergodic and do not make any further assumptions (in particular, no independence or mixing rate assumptions). We find some necessary and some sufficient conditions, formulated in terms of the topological properties of H0 and H1, for the existence of a consistent test. For the case where H1 is the complement of H0 (to the set of all stationary ergodic processes), these necessary and sufficient conditions coincide, thereby providing a complete characterization of families of processes membership to which can be consistently tested, against their complement, based on sampling. This criterion includes as special cases several known and some new results on testing for membership to various parametric families, as well as testing identity, independence, and other hypotheses.

Two-tier Bloom filter to achieve faster membership testing

Testing for element membership in a Bloom filter requires hashing of a test element (e.g. a string) and multiple lookups in memory. A design of a new two-tier Bloom filter with on-chip hash functions and cache is described. For elements with a heavy-tailed distribution for popularity, membership testing time can be significantly reduced.

Sampling to provide or to bound: With applications to fully dynamic graph algorithms

In dynamic graph algorithms the following provide-or-bound problem has to be solved quickly: Given a set S containing a subset R and a way of generating random elements from S testing for membership in R, either (i) provide an element of R, or (ii) give a (small) upper bound on the size of R that holds with high probability. We give an optimal algorithm for this problem. This algorithm improves the time per operation for various dynamic graph algorithms by a factor of O(log n). For example, it improves the time per update for fully dynamic connectivity from O(log3n) to O(log2n). © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11, 369–379 (1997)

Lower bounds for the lengths of refutations

This paper was motivated by a need to consider the time efficiency of Prolog programs. In the context of logic programming, we consider the minimal lengths of refutations of a goal with respect to a program. We present proofs of a number of results of this type. We are especially interested in the special case of testing for membership in a given set of terms. Concomitantly, we are led to classify sets of terms in a way appropriate for these considerations. Since our results for logic programming provide lower bounds, they will immediately imply corresponding lower bounds for pure Prolog.

Testing for class membership in multi-parent hierarchies

Several modern systems have hierarchies of classes of objects, where each class may be a member of more than one higher-level class. If the hierarchy forms a tree (or forest), we can test class membership efficiently by labeling each leaf with an integer and each higher-level class with a pair of integers such that a leaf is in a class if and only if its label is in the range defined by the class label. Such labelings are not possible for arbitrary hierarchies. This article shows how to apply an algorithm from graph theory to determine such a labeling, if it exists, for a given hierarchy. If the algorithm fails to find a labeling, then no labeling exists.

Combinatorial Configurations

Previous article Next article Combinatorial ConfigurationsH. J. RyserH. J. Ryserhttps://doi.org/10.1137/0117056PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (2), 51 (1950), 161–166 MR0032578 0038.02003 CrossrefISIGoogle Scholar[2] D. R. Fulkerson and , O. A. Gross, Incidence matrices and interval graphs, Pacific J. Math., 15 (1965), 835–855 MR0186421 0132.21001 CrossrefISIGoogle Scholar[3] Hugo Hadwiger and , Hans Debrunner, Combinatorial geometry in the plane, Translated by Victor Klee. With a new chapter and other additional material supplied by the translator, Holt, Rinehart and Winston, New York, 1964vii+113 MR0164279 Google Scholar[4] Marshall Hall, Jr., Combinatorial theory, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1967x+310 MR0224481 0196.02401 Google Scholar[5] Paul R. Halmos and , Herbert E. 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