Monadic predicates play a prominent role in many decidable cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting is quantifier-free formulas whose satisfiability is decidable, such as linear arithmetic. Here we develop a semidecision procedure for extracting a monadic decomposition of a formula when it exists.
This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M of the same set of n points,such that each M i is compatible with M i+1 . This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.
Abstract. Barrier coverage in a sensor network has the goal of ensuring that all paths through the surveillance domain joining points in some start region to some target region will intersect the coverage region associated with at least one sensor. In this paper, we revisit a notion of redundant barrier coverage known as -barrier coverage. We describe two different notions of width, or impermeability, of the barrier provided by the sensors in to paths joining two arbitrary regions to .The first, what we refer to as the thickness of the barrier, counts the minimum number of sensor region intersections, over all paths from to . The second, what we refer to as the resilience of the barrier, counts the minimum number of sensors whose removal permits a path from to with no sensor region intersections. Of course, a configuration of sensors with resilience has thickness at least and constitutes a -barrier for and . One of our two main results demonstrates that any (Euclidean) shortest path from to that intersects a fixed number of distinct sensors, never intersects any one sensor more than three times. It follows that the resilience of (with respect to and ) is at most three times the thickness of (with respect to and ). (Furthermore, if points in and are moderately separated (relative to the radius of individual sensor coverage) then no shortest path intersects any one sensor more than two times, and hence the resilience of is at most two times the thickness of .) Our second main result, which we are only able to sketch here, shows that the approximation bounds can be tightened (to 1.666 in the case of moderately separated and ) by exploiting topological properties of simple paths that make double visits to a collection of disks.
This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M of the same set of n points, for some k ∈ O(log n), there is a sequence of perfect matchings M = M 0 , M 1 ,. .. , M k = M , such that each M i is compatible with M i+1. This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.
Let M (n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M (n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and P GL(2, q) with Frobenius maps to obtain new, improved lower bounds for M (n, d). We give new randomized algorithms. We give better lower bounds for M (n, d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower bound for M (n, n − 2) for all n ≡ 2 (mod 3) such that n + 1 is a prime power.
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