Join-Reachability Problems in Directed Graphs

For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs

Constructions of Snake-in-the-Box Codes for Rank Modulation

Snake-in-the-box code is a Gray code which is capable of detecting a single error. Gray codes are important in the context of the rank modulation scheme which was suggested recently for representing information in flash memories. For a Gray code in this scheme the codewords are permutations, two consecutive codewords are obtained by using the "push-to-the-top" operation, and the distance measure is defined on permutations. In this paper the Kendall's $\tau$-metric is used as the distance measure. We present a general method for constructing such Gray codes. We apply the method recursively to obtain a snake of length $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$ for permutations of $S_{2n+1}$, from a snake of length $M_{2n-1}$ for permutations of~$S_{2n-1}$. Thus, we have $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4338$, improving on the previous known ratio of $\lim\limits_{n\to \infty} \frac{1}{\sqrt{\pi n}}$. By using the general method we also present a direct construction. This direct construction is based on necklaces and it might yield snakes of length $\frac{(2n+1)!}{2} -2n+1$ for permutations of $S_{2n+1}$. The direct construction was applied successfully for $S_7$ and $S_9$, and hence $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4743$.Comment: IEEE Transactions on Information Theor

One-class classifiers based on entropic spanning graphs

One-class classifiers offer valuable tools to assess the presence of outliers in data. In this paper, we propose a design methodology for one-class classifiers based on entropic spanning graphs. Our approach takes into account the possibility to process also non-numeric data by means of an embedding procedure. The spanning graph is learned on the embedded input data and the outcoming partition of vertices defines the classifier. The final partition is derived by exploiting a criterion based on mutual information minimization. Here, we compute the mutual information by using a convenient formulation provided in terms of the $\alpha$-Jensen difference. Once training is completed, in order to associate a confidence level with the classifier decision, a graph-based fuzzy model is constructed. The fuzzification process is based only on topological information of the vertices of the entropic spanning graph. As such, the proposed one-class classifier is suitable also for data characterized by complex geometric structures. We provide experiments on well-known benchmarks containing both feature vectors and labeled graphs. In addition, we apply the method to the protein solubility recognition problem by considering several representations for the input samples. Experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state-of-the-art approaches.Comment: Extended and revised version of the paper "One-Class Classification Through Mutual Information Minimization" presented at the 2016 IEEE IJCNN, Vancouver, Canad

Equivalence Classes and Conditional Hardness in Massively Parallel Computations

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model