Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$

Colored Ray Configurations

We study the cyclic sequences induced at infinity by pairwise-disjoint colored rays with apices on a given balanced bichromatic point set, where the color of a ray is inherited from the color of its apex. We derive a lower bound on the number of color sequences that can be realized from any fixed point set. We also examine sequences that can be realized regardless of the point set and exhibit negative examples as well. In addition, we provide algorithms to decide whether a sequence is realizable from a given point set on a line or in convex position

Adaptive computation of the swap-insert correction distance

The Swap-Insert Correction distance from a string S of length n to another string L of length m≥n on the alphabet [1.δ] is the minimum number of insertions, and swaps of pairs of adjacent symbols, converting S into L. Contrarily to other correction distances, computing it is NP-Hard in the size δ of the alphabet. We describe an algorithm computing this distance in time within O(δ2nmtδ.1), where for each [1.δ] there are occurrences of in S,mϵoccurrences of ¿ in L, and where "t = max [1.δ] min is a new parameter of the analysis, measuring one aspect of the difficulty of the instance. The difficulty "t is bounded by above by various terms, such as the length n of the shortest string S, and by the maximum number of occurrences of a single character in S (max[1.δ]). This result illustrates how, in many cases, the correction distance between two strings can be easier to compute than in the worst case scenario

Cross-sections of line configurations in R-3 and (d-2)-flat configurations in R-d

We consider sets L = {l(1),..., l(n)} of n labeled lines in general position in R-3, and study the order types of point sets {p(1),..., p(n)} that stem from the intersections of the lines in L with (directed) planes Pi, not parallel to any line of L, that is, the proper cross-sections of L. As two main results, we show that the number of different order types that can be obtained as cross-sections of G is O(n(9)) when considering all possible planes Pi, and O(n3) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R-2 moving with constant (but possibly different) speeds along straight lines forms at most O(n(3)) different order types over time. We further generalize the setting from R-3 to R-d with d > 3, showing that the number of order types that can be obtained as cross-sections of a set of n labeled (d - 2)-flats in Rd with planes is O ((((n3) + n)(d(d-2)))). (C) 2018 Elsevier B.V. All rights reserved