CS133 Computational Geometry Computational Geometry on Big Data 1 - PowerPoint PPT Presentation

CS133 Computational Geometry Computational Geometry on Big Data 1

Big Geometric Data Geotagged Satellite Imagery Check ins Tweets Billions of check ins More than 17 PB Billions of tweets Millions more every day 2

MapReduce Map Reduce Map Reduce Map Reduce Map Reduce Input Big Data Split Input Shuffle Output 3

CG Algorithms on big data Utilize divide and conquer algorithms 1. Partition the input across machines 2. (Optional) prune partitions that do not contribute to answer 3. Apply the algorithm locally in each partition 4. Combine the partial answers to compute the final result

Examples Convex hull algorithm Closest pair Farthest pair Voronoi diagram/Delaunay triangulation 5

Data Partitioning 6

Spatial Partitioning 7

Skyline (Maximal Vectors) Select all non-dominated points Input Output 𝑦 1 , 𝑧 1 ≻ 𝑦 2 , 𝑧 2 ⟺ 𝑦 1 ≥ 𝑦 2 ∧ 𝑧 1 ≥ 𝑧 2 8

Skyline in MapReduce Non-spatial partitioning Spatial partitioning  Partition  Pruning  Local skyline  Global skyline

Skyline Pruning 𝑞 1 𝑞 1 𝑞 3 𝑞 2 𝑞 2 Partition domination rules 𝑞 1 . 𝑦𝑛𝑗𝑜, 𝑞 1 . 𝑧𝑛𝑗𝑜 ≻ 𝑞 2 . 𝑦𝑛𝑏𝑦, 𝑞 2 . 𝑧𝑛𝑏𝑦 𝑞 1 . 𝑦𝑛𝑗𝑜, 𝑞 1 . 𝑧𝑛𝑏𝑦 ≻ 𝑞 3 . 𝑦𝑛𝑏𝑦, 𝑞 3 . 𝑧𝑛𝑏𝑦 𝑞 1 . 𝑦𝑛𝑏𝑦, 𝑞 1 . 𝑧𝑛𝑗𝑜 ≻ 𝑞 2 . 𝑦𝑛𝑏𝑦, 𝑞 2 . 𝑧𝑛𝑏𝑦 10

Convex Hull 11

Convex Hull in MapReduce Non-spatial partitioning Spatial partitioning  Partition  Pruning  Local hull  Global hull

Pruning The intersection of the four skyline pruning rules with all directions 13

Closest Pair Find the pair of points that have the shortest Euclidean distance Input Output

Closest Pair in MapReduce Non-spatial partitioning Spatial partitioning  Partition  Local closest pair  Global closest pair

Farthest Pair Find the pair of points that have the largest Euclidean distance Input Output

Farthest Pair in MapReduce Non-spatial partitioning Spatial partitioning  Partition  Pruning  Local farthest pair  Global farthest pair

Voronoi Diagram Partitioning Local VD Pruning Vertical Merge Pruning Horizontal Merge Final output

Voronoi Diagram Pruning 19

Conclusion Computational geometry algorithms can be parallelized Both non-spatial and spatial partitioning can be used Spatial partitioning enables some pruning techniques This method applies to several computational geometry algorithms