Computational geometry

The Ball-Pivoting Algorithm (BPA) computes a triangle mesh interpolating a given point cloud. Typically, the points are surface samples acquired with multiple range scans of an object. The principle of the BPA is very simple: Three points form a triangle if a ball of a user-specified radius p touches them without containing any other point. Starting with a seed triangle, the ball pivots around an edge (i.e., it revolves around the edge while keeping in contact with the edge's endpoints) until it touches another point, forming another triangle. The process continues until all reachable edges have been tried, and then starts from another seed triangle, until all points have been considered. The process can then be repeated with a ball of larger radius to handle uneven sampling densities. We applied the BPA to datasets of millions of points representing actual scans of complex 3D objects. The relatively small amount of memory required by the BPA, its time efficiency, and the quality of the results obtained compare favorably with existing techniques.

/pdf/the-ball-pivoting-algorithm-for-surface-reconstruction-12jv6xu280.pdf

The ball-pivoting algorithm for surface reconstruction

Complexity and Approximation

: Current systems for graph computation require a distributed computing cluster to handle very large real-world problems, such as analysis on social networks or the web graph. While distributed computational resources have become more accessible developing distributed graph algorithms still remains challenging, especially to non-experts. In this work, we present GraphChi, a disk-based system for computing efficiently on graphs with billions of edges. By using a well-known method to break large graphs into small parts, and a novel Parallel Sliding Windows algorithm, GraphChi is able to execute several advanced data mining, graph mining and machine learning algorithms on very large graphs, using just a single consumer-level computer. We show, through experiments and theoretical analysis, that GraphChi performs well on both SSDs and rotational hard drives. We build on the basis of Parallel Sliding Windows to propose a new data structure Partitioned Adjacency Lists, which we use to design an online graph database GraphChi-DB.We demonstrate that, on a single PC, GraphChi-DB can process over one hundred thousand graph updates per second, while simultaneously performing computation. GraphChi-DB compares favorably to existing graph databases, particularly on data that is much larger than the available memory. We evaluate our work both experimentally and theoretically. Based on the Parallel Sliding Windows algorithm, we propose new I/O efficient algorithms for solving fundamental graph problems. We also propose a novel algorithm for simulating billions of random walks in parallel on a single computer. By repeating experiments reported for existing distributed systems we show that with only fraction of the resources, GraphChi can solve the same problems in a very reasonable time. Our work makes large-scale graph computation available to anyone with a modern PC.

Large-scale Graph Computation on Just a PC

Current systems for graph computation require a distributed computing cluster to handle very large real-world problems, such as analysis on social networks or the web graph. While distributed computational resources have become more accessible, developing distributed graph algorithms still remains challenging, especially to non-experts.In this work, we present GraphChi, a disk-based system for computing efficiently on graphs with billions of edges. By using a well-known method to break large graphs into small parts, and a novel parallel sliding windows method, GraphChi is able to execute several advanced data mining, graph mining, and machine learning algorithms on very large graphs, using just a single consumer-level computer. We further extend GraphChi to support graphs that evolve over time, and demonstrate that, on a single computer, GraphChi can process over one hundred thousand graph updates per second, while simultaneously performing computation. We show, through experiments and theoretical analysis, that GraphChi performs well on both SSDs and rotational hard drives.By repeating experiments reported for existing distributed systems, we show that, with only fraction of the resources, GraphChi can solve the same problems in very reasonable time. Our work makes large-scale graph computation available to anyone with a modern PC.

/pdf/graphchi-large-scale-graph-computation-on-just-a-pc-c5qonwv7c3.pdf

GraphChi: large-scale graph computation on just a PC

We present a collection of new techniques for designing and analyzing e cient external-memory algorithms for graph problems and illustrate how these techniques can be applied to a wide variety of speci c problems. Our results include: Proximate-neighboring. We present a simple method for deriving external-memory lower bounds via reductions from a problem we call the \proximate neighbors" problem. We use this technique to derive non-trivial lower bounds for such problems as list ranking, expression tree evaluation, and connected components. PRAM simulation. We give methods for e ciently simulating PRAM computations in external memory, even for some cases in which the PRAM algorithm is not work-optimal. We apply this to derive a number of optimal (and simple) external-memory graph algorithms. Time-forward processing. We present a general technique for evaluating circuits (or \circuit-like" computations) in external memory. We also use this in a deterministic list ranking algorithm. Department of Computer Science, Box 1910, Brown University, Providence, RI 02912{1910. y Supported in part by the National Science Foundation, by the U.S. Army Research O ce, and by the Advanced Research

External-memory graph algorithms

Dynamic algorithms and data structures in the area of computational geometry are surveyed. The work has a twofold purpose: it introduces the area to the nonspecialist and reviews the state of the art for the specialist. Fundamental data structures, such as balanced search trees and general techniques for dynamization, are reviewed. Range searching, intersections, point location, convex hull, and proximity are discussed. Problems that do not fall into these categories are also discussed. Open problems are given. >

Dynamic algorithms in computational geometry

The authors give I/O-optimal techniques for the extraction of isosurfaces from volumetric data, by a novel application of the I/O-optimal interval tree of Arge and Vitter (1996). The main idea is to preprocess the data set once and for all to build an efficient search structure in disk, and then each time one wants to extract an isosurface, they perform an output-sensitive query on the search structure to retrieve only those active cells that are intersected by the isosurface. During the query operation, only two blocks of main memory space are needed, and only those active cells are brought into the main memory, plus some negligible overhead of disk accesses. This implies that one can efficiently visualize very large data sets on workstations with just enough main memory to hold the isosurfaces themselves. The implementation is delicate but not complicated. They give the first implementation of the I/O-optimal interval tree, and also implement their methods as an I/O filter for Vtk's isosurface extraction for the case of unstructured grids. They show that, in practice, the algorithms improve the performance of isosurface extraction by speeding up the active-cell searching process so that it is no longer a bottleneck. Moreover, this search time is independent of the main memory available. The practical efficiency of the techniques reflects their theoretical optimality.

/pdf/i-o-optimal-isosurface-extraction-mqew9ma5ln.pdf

I/O optimal isosurface extraction

In this paper, we propose a novel external-memory algorithm to support view-dependent simplification for datasets much larger than main memory. In the preprocessing phase, we use a new spanned sub-meshes simplification technique to build view-dependence trees I/O-efficiently, which preserves the correct edge collapsing order and thus assures the run-time image quality. We further process the resulting view-dependence trees to build the meta-node trees, which can facilitate the run-time level-of-detail rendering and is kept in disk. During run-time navigation, we keep in main memory only the portions of the meta-node trees that are necessary to render the current level of details, plus some prefetched portions that are likely to be needed in the near future. The prefetching prediction takes advantage of the nature of the run-time traversal of the meta-node trees, and is both simple and accurate. We also employ the implicit dependencies for preventing incorrect foldovers, as well as main-memory buffer management and parallel processes scheme to separate the disk accesses from the navigation operations, all in an integrated manner. The experiments show that our approach scales well with respect to the main memory size available, with encouraging preprocessing and run-time rendering speeds and without sacrificing the image quality.

/pdf/external-memory-view-dependent-simplification-1q240e9euo.pdf

External Memory View-Dependent Simplification

We consider the problems of conflict detection and resolution in air traffic management (ATM) from the perspective of computational geometry and give algorithms for solving these problems efficiently. For conflict resolution, we propose a simple method that can route multiple aircraft, conflict-free, through a cluttered airspace, using a prioritized routing scheme in space-time. Our algorithms have been implemented into a simulation system that tracks a large set of flights, having multiple conflicts, and proposes modified routes to resolve them. We report on the preliminary results from an extensive set of experiments that are under may to determine the effectiveness of our methods.

/pdf/geometric-algorithms-for-conflict-detection-resolution-in-gpcdmrvl9i.pdf

Yi-Jen Chiang

Papers

External-memory graph algorithms

Dynamic algorithms in computational geometry

I/O optimal isosurface extraction

External Memory View-Dependent Simplification

Geometric algorithms for conflict detection/resolution in air traffic management