Computational Geometry-Theory and Applications最新文献

We consider a special variant of a pursuit-evasion game called lions and contamination. In a graph whose vertices are originally contaminated, a set of lions walks around the graph and each lion clears the contamination from every vertex it visits. The contamination, however, simultaneously spreads to any adjacent vertex not occupied by a lion. We study the relationship between different types of clearings of graphs, such as clearings which do not allow recontamination, clearings where at most one lion moves at each time step and clearings where lions are forbidden to be stacked on the same vertex. We answer several questions raised by Adams et al. [1].

We investigate the problem of partitioning a rectilinear polygon P with n vertices and no holes into rectangles using disjoint line segments drawn inside P under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside P is minimized. We present an $O\left(n^3\right)$-time algorithm using $O\left(n^2\right)$ space that returns a minimum ink partition of P. In the thick partition, the minimum side length over all resulting rectangles is maximized. We present an $O(n^3{\log }^{2}n)$-time algorithm using $O\left(n^3\right)$ space that returns a thick partition using line segments incident to vertices of P, and an $O(n^6{\log }^{2}n)$-time algorithm using $O\left(n^6\right)$ space that returns a thick partition using line segments incident to the boundary of P. We also show that if the input rectilinear polygon has holes, the corresponding decision problem for the thick partition problem using line segments incident to vertices of the polygon is NP-complete. We also present an $O\left(m^3\right)$-time 3-approximation algorithm for the minimum ink partition for a rectangle containing m point holes.

Spiral Galaxies is a pencil-and-paper puzzle played on a grid of unit squares: given a set of points called centers, the goal is to partition the grid into polyominoes such that each polyomino contains exactly one center and is ${180}^{\circ }$ rotationally symmetric about its center. We show that this puzzle is NP-complete, ASP-complete, and #P-complete even if (a) all solutions to the puzzle have rectangles for polyominoes; or (b) the polyominoes are required to be rectangles and all solutions to the puzzle have just $1\times 1$, $1\times 3$, and $3\times 1$ rectangles. The proof for the latter variant also implies NP/ASP/#P-completeness of finding a noncrossing perfect matching in distance-2 grid graphs where edges connect vertices of Euclidean distance 2. Moreover, we prove NP-completeness of the design problem of minimizing the number of centers such that there exists a set of galaxies that exactly cover a given shape.

We consider the following Euclidean 2-center problem. Given n disks in the plane, find two smallest congruent disks such that every input disk intersects at least one of the two congruent disks. We present a deterministic algorithm for the problem that returns an optimal pair of congruent disks in $O(n^2{\log }^{3}n/\log \log n)$ time. We also present a randomized algorithm with $O(n^2{\log }^{2}n/\log \log n)$ expected time. These results improve upon the previously best deterministic and randomized algorithms, making a step closer to the optimal algorithms for the problem. We show that the same algorithms also work for two variants of the problem, the 2-piercing problem and the restricted 2-cover problem on disks. We also consider the 2-center problem and its two variants on n convex polygons, each with $O\left(1\right)$ vertices in the plane and present efficient algorithms for them.

We study online matching in the Euclidean 2-dimensional plane with the non-crossing constraint. The offline version was introduced by Atallah in 1985 and the online version was introduced and studied more recently by Bose et al. The input to the monochromatic non-crossing matching (MNM) problem consists of a sequence of points. Upon arrival of a point, an algorithm can decide to match it with a previously unmatched point or leave it unmatched. The line segments corresponding to the edges in the matching should not cross each other, and the goal is to maximize the size of the matching. The decisions are irrevocable, and while an optimal offline solution always matches all the points, an online algorithm cannot match all the points in the worst case, unless it is given some additional information, i.e., advice. In the bichromatic version (BNM), blue points are given in advance and the same number of red points arrive online. The goal is to maximize the number of red points matched to blue points without creating any crossings.

We show that the advice complexity of solving BNM optimally on a circle (or, more generally, on inputs in a convex position) is tightly bounded by the logarithm of the $n^{\text{th}}$ Catalan number from above and below. This result corrects the previous claim of Bose et al. that the advice complexity is $\log (n!)$. At the heart of the result is a connection between the non-crossing constraint in online inputs and the 231-avoiding property of permutations of n elements. We also show a lower bound of $n/3-1$ and an upper bound of 3n on the advice complexity for MNM on a plane. This gives an exponential improvement over the previously best-known lower bound and an improvement in the constant of the leading term in the upper bound. In addition, we establish a lower bound of $\frac{\alpha }{2}D\left(\frac{2(1-\alpha )}{\alpha }\right||1/4)n$ on the advice complexity for achieving competitive ratio $\alpha \in (16/17,1)$ for MNM on a circle where $D\left(p\right|\left|q\right)$ is the relative entropy between two Bernoulli random variables with parameters p and q. Standard tools from advice complexity, such as partition trees and reductions from string guessing problems, do not seem to apply to MNM/BNM, so we have to design our lower bounds fr