No abstract

A collection of simple closed Jordan curves in the plane is called a family of pseudo-circles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudo-circles is said to be an empty lens if it does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of Ò pseudo-circles with the property that any two curves intersect precisely twice. This bound implies that any collection of Ò Ü -monotone pseudo-circles can be cut into Ç´Ò µ arcs so that any two intersect at most once; this improves a previous bound of Ç´Ò ¿ µ due to Tamaki and Tokuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced to Ç´Ò ¿ ¾´Ð Ó Òµ Ç´« ×´Ò µµ µ, where «´Òµ is the inverse Ackermann function, and × is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudocircles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to Ç´Ò ¿ µ. As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, of pairwise intersecting pseudo-circles, of arbitrary Ü-monotone pseudo-circles, of parabolas, and of homothetic copies of any fixed simply-shaped convex curve. We also obtain a variant of the Gallai-Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound on the number of distinct distances under any well-behaved norm.

We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√ n) colors in the worst case. We then derive several efficient algorithms. The first algorithm is randomized and simple to analyze; it requires an expected number of at most O(log 2 n) colors, and produces a coloring which is valid with high probability. The second algorithm is deterministic, and is a variant of the initial simple algorithm; it uses a maximum of Θ(log 2 n) colors. The third algorithm is a randomized variant of the second algorithm; it requires an expected number of at most O(log n log log n) colors and always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case.

We construct weak -nets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1 r -nets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in R d we construct weak 1 r -nets of size r · 2 poly(α(r)) , where the degree of the polynomial in the exponent depends (quadratically) on d.Our constructions result from a reduction to a new problem, which we call stabbing interval chains with j-tuples. Given the range of integers N = [1, n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N . A j-tuple p = (p 1 , . . . , p j ) is said to stab an interval chain C = I 1 · · · I k if each p i falls on a different interval of C. The problem is to construct a small-size family Z of j-tuples that stabs all k-interval chains in N .Let z (j) k (n) denote the minimum size of such a family Z. We derive almosttight upper and lower bounds for z (j) k (n) for every fixed j; our bounds involve functions α m (n) of the inverse Ackermann hierarchy. Specifically, we show that for j = 3 we have z (3) k (n) = Θ nα k/2 (n) for all k ≥ 6. For each j ≥ 4 we derive a pair of functions P j (m), Q j (m), almost equal asymptotically, such that z (j) P j (m) (n) = O(nα m (n)) and z (j) Q j (m) (n) = Ω(nα m (n)).

Let H = (V, E) be a hypergraph. A conflict-free coloring of H is an assignment of colors to V such that, in each hyperedge e ∈ E, there is at least one uniquely-colored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols, and several other fields. Conflict-free coloring has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.