Searching edges in the overlap of two plane graphs

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains one of which is convex in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n) time and O(n+m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n axis-aligned rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure

Towards Visual Ego-motion Learning in Robots

Many model-based Visual Odometry (VO) algorithms have been proposed in the past decade, often restricted to the type of camera optics, or the underlying motion manifold observed. We envision robots to be able to learn and perform these tasks, in a minimally supervised setting, as they gain more experience. To this end, we propose a fully trainable solution to visual ego-motion estimation for varied camera optics. We propose a visual ego-motion learning architecture that maps observed optical flow vectors to an ego-motion density estimate via a Mixture Density Network (MDN). By modeling the architecture as a Conditional Variational Autoencoder (C-VAE), our model is able to provide introspective reasoning and prediction for ego-motion induced scene-flow. Additionally, our proposed model is especially amenable to bootstrapped ego-motion learning in robots where the supervision in ego-motion estimation for a particular camera sensor can be obtained from standard navigation-based sensor fusion strategies (GPS/INS and wheel-odometry fusion). Through experiments, we show the utility of our proposed approach in enabling the concept of self-supervised learning for visual ego-motion estimation in autonomous robots.Comment: Conference paper; Submitted to IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2017, Vancouver CA; 8 pages, 8 figures, 2 table

Real-time Batched Distance Computation for Time-Optimal Safe Path Tracking

In human-robot collaboration, there has been a trade-off relationship between the speed of collaborative robots and the safety of human workers. In our previous paper, we introduced a time-optimal path tracking algorithm designed to maximize speed while ensuring safety for human workers. This algorithm runs in real-time and provides the safe and fastest control input for every cycle with respect to ISO standards. However, true optimality has not been achieved due to inaccurate distance computation resulting from conservative model simplification. To attain true optimality, we require a method that can compute distances 1. at many robot configurations to examine along a trajectory 2. in real-time for online robot control 3. as precisely as possible for optimal control. In this paper, we propose a batched, fast and precise distance checking method based on precomputed link-local SDFs. Our method can check distances for 500 waypoints along a trajectory within less than 1 millisecond using a GPU at runtime, making it suited for time-critical robotic control. Additionally, a neural approximation has been proposed to accelerate preprocessing by a factor of 2. Finally, we experimentally demonstrate that our method can navigate a 6-DoF robot earlier than a geometric-primitives-based distance checker in a dynamic and collaborative environment

Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing

Motivated by the desire to cope with data imprecision, we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(n alpha(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(n alpha(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (<= k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Omega(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "standard" convex hull and sorting problems, in which the two problems require Theta(n log n) steps in the worst case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at SoCG 201

The {RDF}-3X Engine for Scalable Management of {RDF} Data

RDF is a data model for schema-free structured information that is gaining momentum in the context of Semantic-Web data, life sciences, and also Web 2.0 platforms. The ``pay-as-you-go'' nature of RDF and the flexible pattern-matching capabilities of its query language SPARQL entail efficiency and scalability challenges for complex queries including long join paths. This paper presents the RDF-3X engine, an implementation of SPARQL that achieves excellent performance by pursuing a RISC-style architecture with streamlined indexing and query processing. The physical design is identical for all RDF-3X databases regardless of their workloads, and completely eliminates the need for index tuning by exhaustive indexes for all permutations of subject-property-object triples and their binary and unary projections. These indexes are highly compressed, and the query processor can aggressively leverage fast merge joins with excellent performance of processor caches. The query optimizer is able to choose optimal join orders even for complex queries, with a cost model that includes statistical synopses for entire join paths. Although RDF-3X is optimized for queries, it also provides good support for efficient online updates by means of a staging architecture: direct updates to the main database indexes are deferred, and instead applied to compact differential indexes which are later merged into the main indexes in a batched manner. Experimental studies with several large-scale datasets with more than 50 million RDF triples and benchmark queries that include pattern matching, manyway star-joins, and long path-joins demonstrate that RDF-3X can outperform the previously best alternatives by one or two orders of magnitude

Real-Time Data Processing With Lambda Architecture

Data has evolved immensely in recent years, in type, volume and velocity. There are several frameworks to handle the big data applications. The project focuses on the Lambda Architecture proposed by Marz and its application to obtain real-time data processing. The architecture is a solution that unites the benefits of the batch and stream processing techniques. Data can be historically processed with high precision and involved algorithms without loss of short-term information, alerts and insights. Lambda Architecture has an ability to serve a wide range of use cases and workloads that withstands hardware and human mistakes. The layered architecture enhances loose coupling and flexibility in the system. This a huge benefit that allows understanding the trade-offs and application of various tools and technologies across the layers. There has been an advancement in the approach of building the LA due to improvements in the underlying tools. The project demonstrates a simplified architecture for the LA that is maintainable