# Courseware - Geometric Algorithms

• Line segments: properties, intersections
• Convex hull finding algorithms

## Parallel algorithm for finding intersections of line segments in 3-D (Bradley Kuszmaul)

 Material Type: Lecture / Presentation, Coding example, Problem set Technical Format: text file, zip archive, PDF document Location: Go to materials Date Added: 03/17/2010 Date Modified: 03/17/2010 Author Bradley Kuszmaul, MIT Computer Science and Artificial Intelligence Laboratory Description: The included source code implements a parallel search for intersections of input line segments within a 3-D space, as described in the included problem description text file. Three different methods of solution are initially considered. Complexity analysis and potential parallelization of the first two (brute force search, sweep-line algorithm) are considered and used to eliminate each from further consideration. The third method, Tree Decomposition, is chosen and explained in detail. The idea behind Tree Decomposition is to spatially divide the line segments into three classifications: left, right, and straddle, where the lines in the left and right portions are completely contained within a classification (and cannot intersect) and those in the straddle branch have endpoints in both left and right spaces. Each branch can be similarly divided using a plane that has not been used until the set of lines in a leaf node is small enough to compute possible intersections. There are 5 points within the algorithm that can be executed in parallel. The write-up describes the parallelization details of each. Cilk++ is used for implementing the parallelism. The code was intended for Linux OS and includes a makefile to build the application. DISCLAIMER: This code is provided by the author as a submitted contest entry, and is intended for educational use only. The code is not guaranteed to solve all instances of the input data sets and may require modifications to work in your own specific environment. Recommended Audience: Advanced programmers, Undergraduate students Language: English Keywords: 3D Line segment, line intersection, Cilk++, tree decomposition, parallel merge sort, computational geometry, Intel Threading Challenge

## Parallel algorithm for finding intersections of line segments in 3-D (Dmitry Vyukov)

 Material Type: Lecture / Presentation, Coding example, Problem set Technical Format: text file, Word document, zip archive Location: Go to materials Date Added: 03/17/2010 Date Modified: 03/17/2010 Author Dmitry Vyukov, Intel Threading Challenge Description: The included source code implements a parallel search for intersections of input line segments within a 3-D space, as described in the included problem description text file. Three different methods of solution are initially considered. Complexity analysis and potential parallelization of the first two (brute force search, sweep-line algorithm) are considered and used to eliminate each from further consideration. The third method, Tree Decomposition, is chosen and explained in detail. The idea behind Tree Decomposition is to spatially divide the line segments into three classifications: left, right, and straddle, where the lines in the left and right portions are completely contained within a classification (and cannot intersect) and those in the straddle branch have endpoints in both left and right spaces. Each branch can be similarly divided using a plane that has not been used until the set of lines in a leaf node is small enough to compute possible intersections. There are 5 points within the algorithm that can be executed in parallel. The write-up describes the parallelization details of each. Cilk++ is used for implementing the parallelism. The code was intended for Linux OS and includes a makefile to build the application. DISCLAIMER: This code is provided by the author as a submitted contest entry, and is intended for educational use only. The code is not guaranteed to solve all instances of the input data sets and may require modifications to work in your own specific environment. Recommended Audience: Advanced programmers, Undergraduate students Language: English Keywords: 3D Line segment, line intersection, Intel Threading Building Blocks, TBB, SSE, Streaming SIMD Execution, parallelsort algorithm, computational geometry

## Parallel algorithm to 3-D Convex Hull Problem (Bradley Kuszmaul)

 Material Type: Coding example, Problem set, Lecture / Presentation Technical Format: text file, zip archive, PDF document Location: Go to materials Date Added: 03/17/2010 Date Modified: 03/17/2010 Author Bradley Kuszmaul, Intel Description: The included code and white paper provides a parallel solution for the 3-D Convex Hull problem, as described in the included problem description text file. Parallelism is achieved using Cilk++. Possible points for the convex hull are found by repeatedly selecting four points from the input set and finding the largest volumes of the formed tetrahedrons (using matrix determinants). An O(n^4) algorithm confirms those points that are actually on the convex hull from the possible points previously found. DISCLAIMER: This code is provided by the author as a submitted contest entry, and is intended for educational use only. The code is not guaranteed to solve all instances of the input data sets and may require modifications to work in your own specific environment. Recommended Audience: Advanced programmers, Undergraduate students Language: English Keywords: C, Cilk++, computational geometry, C++, Convex Hull, 3D Convex Hull, Threading Challenge Contest, matrix determinant, tetrahedron volume
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