Let S be a point set in the plane such that each of its ele-1 ments is colored either red or blue. A matching of S with rectangles is 2 any set of pairwise-disjoint axis-aligned rectangles such that each rectan-3 gle contains exactly two points of S. Such a matching is monochromatic 4 if every rectangle contains points of the same color, and is bichromatic if 5 every rectangle contains points of different colors. In this paper we study 6 the following two problems: 7 1. Find a maximum monochromatic matching of S with rectangles. 8 2. Find a maximum bichromatic matching of S with rectangles. 9 For each problem we provide a polynomial-time approximation algorithm 10 that constructs a matching with at least 1/4 of the number of rectan-11 gles of an optimal matching. We show that the first problem is NP-hard 12 even if either the matching rectangles are restricted to axis-aligned seg-13 ments or S is in general position, that is, no two points of S share the 14 same x or y coordinate. We further show that the second problem is 15 also NP-hard, even if S is in general position. These NP-hardness results 16 follow by showing that deciding the existence of a perfect matching is 17 NP-complete in each case. The approximation results are based on a rela-18 tion of our problem with the problem of finding a maximum independent 19 set in a family of axis-aligned rectangles. With this paper we extend pre-20 vious ones on matching one-colored points with rectangles and squares, 21 and matching two-colored points with segments. Furthermore, using our 22 techniques, we prove that it is NP-complete to decide a perfect matching 23 with rectangles in the case where all points have the same color, solving 24 an open problem of Bereg, Mutsanas, and Wolff [CGTA (2009)]. 25 1 Introduction 26 Matching points in the plane with geometric objects consists in, given an input 27 point set S and a class C of geometric objects, finding a collection M ⊆ C such 28 that each element of M contains exactly two points of S and every point of 29 S lies in at most one element of M . This kind of geometric matching problem 30 was introduced byÁbrego et al. [1], calling a geometric matching strong if the 31 geometric objects are disjoint, and perfect if every point of S belongs to some 32 element of M . They studied the existence and properties of matchings for point 33 sets in the plane when C is the set of axis-aligned squares, or the family of disks. 34 Bereg et al. [6] continued the study of this class of problems. They proved by 35 a constructive proof that if C is the class of axis-aligned rectangles, then every 36 point set of n points in the plane admits a strong matching that matches at least 37 2 n/3 of the points; and leaved open the computational complexity of finding 38 such a maximum strong matching. They assume that there can be points with 39the same x or y coordinate, condition that makes the optimization problem hard. 40In the case in which C is the class of axis-aligned squares, they proved that it is 41 NP-hard to decide whether a...

The predecessor problem is a key component of the fundamental sorting-and-searching core of algorithmic problems. While binary search is the optimal solution in the comparison model, more realistic machine models on integer sets open the door to a rich universe of data structures, algorithms, and lower bounds. In this article, we review the evolution of the solutions to the predecessor problem, focusing on the important algorithmic ideas, from the famous data structure of van Emde Boas to the optimal results of Patrascu and Thorup. We also consider lower bounds, variants, and special cases, as well as the remaining open questions.

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Motivated by the analysis of range queries in databases, we introduce the computation of the Depth Distribution of a set B of axis aligned boxes, whose computation generalizes that of the Klee's Measure and of the Maximum Depth. In the worst case over instances of fixed input size n, we describe an algorithm of complexity within O(n d+1 2 log n), using space within O(n log n), mixing two techniques previously used to compute the Klee's Measure. We refine this result and previous results on the Klee's Measure and the Maximum Depth for various measures of difficulty of the input, such as the profile of the input and the degeneracy of the intersection graph formed by the boxes.A preliminary version of these results were presented at the 23 rd Annual International Computing and Combinatorics Conference (COCOON'17) [5]