On the general position subset selection problem

Let $f(n,\ell)$ be the maximum integer such that every set of $n$ points in the plane with at most $\ell$ collinear contains a subset of $f(n,\ell)$ points with no three collinear. First we prove that if $\ell \leq O(\sqrt{n})$ then $f(n,\ell)\geq \Omega(\sqrt{\frac{n}{\ln \ell}})$. Second we prove that if $\ell \leq O(n^{(1-\epsilon)/2})$ then $f(n,\ell) \geq \Omega(\sqrt{n\log_\ell n})$, which implies all previously known lower bounds on $f(n,\ell)$ and improves them when $\ell$ is not fixed. A more general problem is to consider subsets with at most $k$ collinear points in a point set with at most $\ell$ collinear. We also prove analogous results in this setting

Multi-Constellation GNSS: New Bounds on DOP and a Related Satellite Selection Process

GPS receivers convert the measured pseudoranges from the visible GPS satellites into an estimate of the position and clock offset of the receiver. For various reasons receivers might only track and process a subset of the visible satellites. It would be desired, of course, to use the best subset. In general selecting the best subset is a combinatorics problem; selecting m objects from a choice of n allows for n m potential subsets. And since the GDOP performance criterion is nonlinear and non-separable, finding the best subset is a brute force procedure; hence, a number of authors have described sub-optimal algorithms for choosing satellites. This paper revisits this problem, especially in the context of multiple GNSS constellations, for the GDOP and PDOP criteria. Included are a discussion of optimum constellations (based upon parallel work of these authors on achievable lower bounds to GDOP and PDOP), musings on how the non-separableness of DOP makes it impossible to rank order the satellites, and a review/discussion of subset selection algorithms. Our long term goal is the development of better selection algorithms for multi-constellation GNSS

A Temporal Algorithm for Satellite Subset Selection in Multi-Constellation GNSS

GNSS receivers convert the measured pseudoranges from the visible GNSS satellites into an estimate of the position and clock offset of the receiver. For various reasons receivers might want to process only a subset of the visible satellites; it would be desired, of course, to use the best subset. In general, selecting the best subset is a combinatorics problem; selecting m objects from a choice of n allows for (n m) potential subsets. And since the typical performance criterion (e.g. Geometric Dilution of Precision) is nonlinear and non-separable in the satellites’ locations in the sky, finding the best subset is a brute force procedure; hence, a number of authors have described sub-optimal algorithms for choosing satellites. This paper revisits this problem, especially in the context of multiple GNSS constellations. The paper begins with a review of the existing subset selection algorithms. We note that all of these algorithms are what might be called “snapshot” in nature, selecting a subset for a single, fixed skyview of satellites. Through an example with the GPS constellation, we examine the time-sequential, or temporal, characteristics of the best subset selection noting: That the best subset at a particular point (snapshot) in time is also the best subset for a significant time interval around that point (typically measured in minutes). That the changes in the best subset over time are primarily, but not always, due to the loss or gain of a satellite crossing the horizon (or, more precisely, the receiver’s mask angle). Based upon these observations this paper develops several time-sequential, or temporal, algorithms that attempt to track the optimum subset of satellites over time at low computational cost. The accuracy and complexity of the algorithms are assessed with GPS constellation data. On a larger scale, these algorithms are then tested on combined GPS, GLONASS, and Galileo constellations with the resulting performance compared to optimal solutions found via exhaustive search

Tractable Orders for Direct Access to Ranked Answers of Conjunctive Queries

We study the question of when we can provide logarithmic-time direct access to the k-th answer to a Conjunctive Query (CQ) with a specified ordering over the answers, following a preprocessing step that constructs a data structure in time quasilinear in the size of the database. Specifically, we embark on the challenge of identifying the tractable answer orderings that allow for ranked direct access with such complexity guarantees. We begin with lexicographic orderings and give a decidable characterization (under conventional complexity assumptions) of the class of tractable lexicographic orderings for every CQ without self-joins. We then continue to the more general orderings by the sum of attribute weights and show for it that ranked direct access is tractable only in trivial cases. Hence, to better understand the computational challenge at hand, we consider the more modest task of providing access to only a single answer (i.e., finding the answer at a given position) - a task that we refer to as the selection problem. We indeed achieve a quasilinear-time algorithm for a subset of the class of full CQs without self-joins, by adopting a solution of Frederickson and Johnson to the classic problem of selection over sorted matrices. We further prove that none of the other queries in this class admit such an algorithm.Comment: 17 page

Sensor selection for fine-grained behavior verification that respects privacy

A useful capability is that of classifying some agent's behavior using data from a sequence, or trace, of sensor measurements. The sensor selection problem involves choosing a subset of available sensors to ensure that, when generated, observation traces will contain enough information to determine whether the agent's activities match some pattern. In generalizing prior work, this paper studies a formulation in which multiple behavioral itineraries may be supplied, with sensors selected to distinguish between behaviors. This allows one to pose fine grained questions, e.g., to position the agent's activity on a spectrum. In addition, with multiple itineraries, one can also ask about choices of sensors where some behavior is always plausibly concealed by (or mistaken for, or conflated with) another. Using sensor ambiguity to limit the acquisition of knowledge is a strong privacy guarantee, and one which some earlier work has examined. By concretely formulating privacy requirements for sensor selection, this paper connects both lines of work: privacy -- where there is a bound from above, and behavior verification -- where sensors are bounded from below. We examine the worst case computational complexity that results from both types of bounds, proving that upper bounds are more challenging under standard computational complexity assumptions. The problem is intractable in general, but we give a novel approach to solving this problem that can exploit interrelationships between constraints, and we see opportunities for a few optimizations. Case studies are presented to demonstrate the usefulness and scalability of our proposed solution, and to assess the impact of the optimizations

Phase Transitions and Symmetry Breaking in Genetic Algorithms with Crossover

In this paper, we consider the role of the crossover operator in genetic algorithms. Specifically, we study optimisation problems that exhibit many local optima and consider how crossover affects the rate at which the population breaks the symmetry of the problem. As an example of such a problem, we consider the subset sum problem. In so doing, we demonstrate a previously unobserved phenomenon, whereby the genetic algorithm with crossover exhibits a critical mutation rate, at which its performance sharply diverges from that of the genetic algorithm without crossover. At this critical mutation rate, the genetic algorithm with crossover exhibits a rapid increase in population diversity. We calculate the details of this phenomenon on a simple instance of the subset sum problem and show that it is a classic phase transition between ordered and disordered populations. Finally, we show that this critical mutation rate corresponds to the transition between the genetic algorithm accelerating or preventing symmetry breaking and that the critical mutation rate represents an optimum in terms of the balance of exploration and exploitation within the algorithm