Efficient operations on discrete paths

We present linear time and space operations on discrete paths. First, we compute the outer hull of any discrete path. As a consequence, a linear time and space algorithm is obtained for computing the convex hull. Next, we provide a linear algorithm computing the overlay graph of two simple closed paths. From this overlay graph, one can easily compute the intersection, union and difference of two Jordan polyominoes, i.e. polyominoes whose boundary is a Jordan curve. The linear complexity is obtained by using an enriched version of a data structure introduced by Brlek, Koskas and Provençal: a quadtree for representing points in the discrete plane augmented with neighborhood links, which was introduced in particular to decide in linear time if a discrete path is self-intersecting

Dimensional verification and correction of five-axis numerically controlled milling tool paths

A system of algorithms is presented for material removal simulation, automatic dimensional verification and integrated error correction of numerically controlled (NC) milling tool paths. Several different approaches to these problems have been proposed including direct solid modeling, discrete vector intersection, and spatial partitioning. However, each of these methods suffer inherent restrictions that limit their practical application. This dissertation presents a discrete dexel NC verification algorithm based on a spatial partitioning technique (dexel representation) which incorporates the advantages of the discrete vector intersection approach. Hence, real-time animated five-axis milling simulation is supported by efficient regularized Boolean set operations, and dimensional milling errors are verified simultaneously with the simulation process. Based on intermediate dimensional verification results, a reduction of intersection volume algorithm is developed to eliminate detected gouges on the part surface. In addition, a technique for detection and elimination of unexpected collisions between the tool assembly and the workpiece is developed. These combined algorithms automatically correct tool paths to avoid gouges and collisions resulting in tool paths that are ready for immediate industrial application. A major disadvantage of dexel-based spatial partitioning, as originally proposed, is view dependency, i.e., dexels are constructed along a specific viewing vector so reconstruction of dexels is required for each new viewing direction. To overcome this problem, a contour display method is developed to transform dexel-based objects into a set of parallel planar contours thus enabling dynamic viewing transformations. In summary, this dissertation describes a unique hybrid approach to NC milling verification which provides for efficient, accurate and automatic assessment and correction of five-axis milling tool paths

The boolean map distance: theory and efficient computation

We propose a novel distance function, the boolean map distance (BMD), that defines the distance between two elements in an image based on the probability that they belong to different components after thresholding the image by a randomly selected threshold value. This concept has been explored in a number of recent publications, and has been proposed as an approximation of another distance function, the minimum barrier distance (MBD). The purpose of this paper is to introduce the BMD as a useful distance function in its own right. As such it shares many of the favorable properties of the MBD, while offering some additional advantages such as more efficient distance transform computation and straightforward extension to multi-channel images

Techniques for the Fast Simulation of Models of Highly dependable Systems

With the ever-increasing complexity and requirements of highly dependable systems, their evaluation during design and operation is becoming more crucial. Realistic models of such systems are often not amenable to analysis using conventional analytic or numerical methods. Therefore, analysts and designers turn to simulation to evaluate these models. However, accurate estimation of dependability measures of these models requires that the simulation frequently observes system failures, which are rare events in highly dependable systems. This renders ordinary Simulation impractical for evaluating such systems. To overcome this problem, simulation techniques based on importance sampling have been developed, and are very effective in certain settings. When importance sampling works well, simulation run lengths can be reduced by several orders of magnitude when estimating transient as well as steady-state dependability measures. This paper reviews some of the importance-sampling techniques that have been developed in recent years to estimate dependability measures efficiently in Markov and nonMarkov models of highly dependable system

Rank, select and access in grammar-compressed strings

Given a string $S$ of length $N$ on a fixed alphabet of $\sigma$ symbols, a grammar compressor produces a context-free grammar $G$ of size $n$ that generates $S$ and only $S$. In this paper we describe data structures to support the following operations on a grammar-compressed string: \mbox{rank}_c(S,i) (return the number of occurrences of symbol $c$ before position $i$ in $S$); \mbox{select}_c(S,i) (return the position of the $i$th occurrence of $c$ in $S$); and \mbox{access}(S,i,j) (return substring $S[i,j]$). For rank and select we describe data structures of size $O(n\sigma\log N)$ bits that support the two operations in $O(\log N)$ time. We propose another structure that uses $O(n\sigma\log (N/n)(\log N)^{1+\epsilon})$ bits and that supports the two queries in $O(\log N/\log\log N)$, where $\epsilon>0$ is an arbitrary constant. To our knowledge, we are the first to study the asymptotic complexity of rank and select in the grammar-compressed setting, and we provide a hardness result showing that significantly improving the bounds we achieve would imply a major breakthrough on a hard graph-theoretical problem. Our main result for access is a method that requires $O(n\log N)$ bits of space and $O(\log N+m/\log_\sigma N)$ time to extract $m=j-i+1$ consecutive symbols from $S$. Alternatively, we can achieve $O(\log N/\log\log N+m/\log_\sigma N)$ query time using $O(n\log (N/n)(\log N)^{1+\epsilon})$ bits of space. This matches a lower bound stated by Verbin and Yu for strings where $N$ is polynomially related to $n$.Comment: 16 page