strip

We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,\delta]$ depends on the strip width $\delta$. We obtain two main results. First, for the case where the points have distinct integer $x$-coordinates, we prove that a shortest bitonic tour (which can be computed in $O(n\log^2 n)$ time using an existing algorithm) is guaranteed to be a shortest tour overall when $\delta\leq 2\sqrt{2}$, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to $\delta$. More precisely, our algorithm has running time $2^{O(\sqrt{\delta})} n^2$ for sparse point sets, where each $1\times\delta$ rectangle inside the strip contains $O(1)$ points. For random point sets, where the points are chosen uniformly at random from the rectangle~$[0,n]\times [0,\delta]$, it has an expected running time of $2^{O(\sqrt{\delta})} n^2 + O(n^3)$

A quasi-polynomial algorithm for well-spaced hyperbolic TSP

We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature $-1$. Let $\alpha$ denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an $n^{O(\log^2 n)\max(1,1/\alpha)}$ algorithm for Hyperbolic TSP. This is quasi-polynomial time if $\alpha$ is at least some absolute constant, and it grows to $n^{O(\sqrt{n})}$ as $\alpha$ decreases to $\log^2 n/\sqrt{n}$. (For even smaller values of $\alpha$, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of $n^{O(\sqrt{n})}$.)Comment: SoCG 202

Rectilinear Steiner Trees in Narrow Strips

A rectilinear Steiner tree for a set $P$ of points in $\mathbb{R}^2$ is a tree that connects the points in $P$ using horizontal and vertical line segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of Minimal Rectilinear Steiner Tree for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,\delta]$ depends on the strip width $\delta$. We obtain two main results. 1) We present an algorithm with running time $n^{O(\sqrt{\delta})}$ for sparse point sets, that is, point sets where each $1\times\delta$ rectangle inside the strip contains $O(1)$ points. 2) For random point sets, where the points are chosen randomly inside a rectangle of height $\delta$ and expected width $n$, we present an algorithm that is fixed-parameter tractable with respect to $\delta$ and linear in $n$. It has an expected running time of $2^{O(\delta \sqrt{\delta})} n$.Comment: 21 pages, 13 figure

Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on Computational Geometry 200

Image Processing Applications in Real Life: 2D Fragmented Image and Document Reassembly and Frequency Division Multiplexed Imaging

In this era of modern technology, image processing is one the most studied disciplines of signal processing and its applications can be found in every aspect of our daily life. In this work three main applications for image processing has been studied. In chapter 1, frequency division multiplexed imaging (FDMI), a novel idea in the field of computational photography, has been introduced. Using FDMI, multiple images are captured simultaneously in a single shot and can later be extracted from the multiplexed image. This is achieved by spatially modulating the images so that they are placed at different locations in the Fourier domain. Finally, a Texas Instruments digital micromirror device (DMD) based implementation of FDMI is presented and results are shown. Chapter 2 discusses the problem of image reassembly which is to restore an image back to its original form from its pieces after it has been fragmented due to different destructive reasons. We propose an efficient algorithm for 2D image fragment reassembly problem based on solving a variation of Longest Common Subsequence (LCS) problem. Our processing pipeline has three steps. First, the boundary of each fragment is extracted automatically; second, a novel boundary matching is performed by solving LCS to identify the best possible adjacency relationship among image fragment pairs; finally, a multi-piece global alignment is used to filter out incorrect pairwise matches and compose the final image. We perform experiments on complicated image fragment datasets and compare our results with existing methods to show the improved efficiency and robustness of our method. The problem of reassembling a hand-torn or machine-shredded document back to its original form is another useful version of the image reassembly problem. Reassembling a shredded document is different from reassembling an ordinary image because the geometric shape of fragments do not carry a lot of valuable information if the document has been machine-shredded rather than hand-torn. On the other hand, matching words and context can be used as an additional tool to help improve the task of reassembly. In the final chapter, document reassembly problem has been addressed through solving a graph optimization problem

Contributions to the solution of the symmetric travelling salesman problem

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