Pattern Matching for sets of segments

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200

Gravity Gradiometry and Map Matching: An Aid to Aircraft Inertial Navigation Systems

Inertial navigation systems (INS) offer passive, all-weather, and undeniable navigation information, which military customers often view as especially appealing strengths. Unfortunately, Airmen and engineers still struggle with INS’s drifting position errors, and navigation aids generally detract from INS’s strengths. At this year’s Air, Space, and Cyberspace in the 21st Century Conference, the Chief of Staff of the Air Force identified the Global Positioning System (GPS) as a widely-known and exploitable vulnerability, saying that it’s critical the Joint force reduce GPS dependence. Recent advances provide an opportunity for gravity gradient instruments (GGI), which measure spatial derivatives of the gravity vector, to aid an INS and preserve its strengths. This thesis shows that a GGI and map matching enhanced (GAME) INS improves navigation accuracy, presents the conditions that make GAME feasible for aircraft, and identifies opportunities for improvement. The methodology includes computer models and algorithms, where a GGI and map matching aid an INS through a Kalman filter. Simulations cover different terrains, altitudes, velocities, flight durations, INS drifts, update rates, components of the gravity gradient tensor, GGI and map noise levels, map resolutions, and levels of interpolation. Although GAME with today’s technology only appears worthwhile for long range and long endurance flights, the technologies expected in 10 years promise a broad spectrum of scenarios where GAME potentially provides great returns on investments and dominates the market for secure and covert navigation

Approximate nearest neighbor problem in high dimensions

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 47-49).We investigate the problem of finding the approximate nearest neighbor when the data set points are the substrings of a given text T. The exact version of this problem is defined as follows. Given a text T of length n, we want to build a data structure that supports the following operation: given a pattern P, find the substring of T that is the closest to P. Since the exact version of this problem is surprisingly difficult, we address the approximate version, in which we are allowed to return a substring of T that is at most c times further than the actual closest substring of T. This problem occurs, for example, in computational biology [4, 5]. In particular, we study the case where the length of the pattern P, denoted by m, is not known in advance, which is the most natural scenario. We present a data structure that uses O(n1+1/c) space and has 0 (nl/c + mn⁰(l)) query time' when the distance between two strings is the Hamming distance. These bounds essentially match the earlier bounds of [12], which assumed that the pattern length m is fixed in advance. Furthermore, our data structure can be constructed in O (n1+1/c + n1+⁰(1)M1/3) time, where M is an upper bound for m. This time essentially matches the preprocessing time of [12] as long as the term O(n1+1/c) dominates the running time, which is the case when, for example, c < 3. We also extend our results to the case where the distances are measured according to the lI distance. The query time and the space bound are essentially the same, while the preprocessing time becomes 6 (n'+/c + nl+o(l)M2/3) (We use notation f(n) = O(g(n)) to denote f(n) = g(n) logO(1) g(n)).by Alexandr Andoni.M.Eng

Geometric Optimization Problem Solving: Matching Sets of Line Segments and Multi-robot Path Planning

Department of Computer Science and EngineeringWe study two geometric optimization problems: Line segments pattern matching and multi-robot path planning. We give approximation algorithms for matching two sets of line segments in constant dimension. We consider several versions of the problem: Hausdorff distance, bottleneck distance and largest common subset. We study these similarity measures under several sets of transformations: translations in arbitrary dimension, rotations about a fixed point and rigid motions in two dimensions. As opposed to previous theoretical work on this problem, we match segments individually, in other words we regard our two input sets as sets of segments rather than unions of segments. Then we consider a multi-robot path planning problem. A collection of square robots need to move on the integer grid, from their given starting points to their target points, and without collision between robots, or between robots and a set of input obstacles. We designed and implemented three algorithms for this problem. First, we computed a feasible solution by placing middle-points outside of the minimum bounding box of the starting positions, the target positions and the obstacles, and moving each robot from its starting point to its target point through a middle-point. Second, we applied a simple local search approach where we repeatedly delete and insert again a random robot through an optimal path. It improves the quality of the solution, as the robots no longer need to go through the middle-points. Finally, we used simulated annealing to further improve this feasible solution.ope