Interval Selection in the Streaming Model

A set of intervals is independent when the intervals are pairwise disjoint. In the interval selection problem we are given a set $\mathbb{I}$ of intervals and we want to find an independent subset of intervals of largest cardinality. Let $\alpha(\mathbb{I})$ denote the cardinality of an optimal solution. We discuss the estimation of $\alpha(\mathbb{I})$ in the streaming model, where we only have one-time, sequential access to the input intervals, the endpoints of the intervals lie in $\{1,...,n \}$, and the amount of the memory is constrained. For intervals of different sizes, we provide an algorithm in the data stream model that computes an estimate $\hat\alpha$ of $\alpha(\mathbb{I})$ that, with probability at least $2/3$, satisfies $\tfrac 12(1-\varepsilon) \alpha(\mathbb{I}) \le \hat\alpha \le \alpha(\mathbb{I})$. For same-length intervals, we provide another algorithm in the data stream model that computes an estimate $\hat\alpha$ of $\alpha(\mathbb{I})$ that, with probability at least $2/3$, satisfies $\tfrac 23(1-\varepsilon) \alpha(\mathbb{I}) \le \hat\alpha \le \alpha(\mathbb{I})$. The space used by our algorithms is bounded by a polynomial in $\varepsilon^{-1}$ and $\log n$. We also show that no better estimations can be achieved using $o(n)$ bits of storage. We also develop new, approximate solutions to the interval selection problem, where we want to report a feasible solution, that use $O(\alpha(\mathbb{I}))$ space. Our algorithms for the interval selection problem match the optimal results by Emek, Halld{\'o}rsson and Ros{\'e}n [Space-Constrained Interval Selection, ICALP 2012], but are much simpler.Comment: Minor correction

Bayesian inference through encompassing priors and importance sampling for a class of marginal models for categorical data

We develop a Bayesian approach for selecting the model which is the most supported by the data within a class of marginal models for categorical variables formulated through equality and/or inequality constraints on generalised logits (local, global, continuation or reverse continuation), generalised log-odds ratios and similar higher-order interactions. For each constrained model, the prior distribution of the model parameters is formulated following the encompassing prior approach. Then, model selection is performed by using Bayes factors which are estimated by an importance sampling method. The approach is illustrated through three applications involving some datasets, which also include explanatory variables. In connection with one of these examples, a sensitivity analysis to the prior specification is also considered

On the Equivalence Between a Minimal Codomain Cardinality Riesz Basis Construction, a System of Hadamard–Sylvester Operators, and a Class of Sparse, Binary Optimization Problems

Piecewise, low-order polynomial, Riesz basis families are constructed such that they share the same coefficient functionals of smoother, orthonormal bases in a localized indexing subset. It is shown that a minimal cardinality basis codomain can be realized by inducing sparsity, via l1 regularization, in the distributional derivatives of the basis functions and that the optimal construction can be found numerically by constrained binary optimization over a suitably large dictionary. Furthermore, it is shown that a subset of these solutions are equivalent to a specific, constrained analytical solution, derived via Sylvester-type Hadamard operators