cs240a-partitioning

# cs240a-partitioning - CS 240A Graph and hypergraph...

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CS 240A: Graph and hypergraph partitioning Thanks to Aydin Buluc, Umit Catalyurek, Alan Edelman, and Kathy Yelick for some of these slides.

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CS 240A: Graph and hypergraph partitioning Motivation and definitions Motivation from parallel computing Theory of graph separators Heuristics for graph partitioning Iterative swapping Spectral Geometric Multilevel Beyond graphs Shortcomings of the graph partitioning model Hypergraph models of communication in MatVec Parallel methods for partitioning hypergraphs
CS 240A: Graph and hypergraph partitioning Motivation and definitions Motivation from parallel computing Theory of graph separators Heuristics for graph partitioning Iterative swapping Spectral Geometric Multilevel Beyond graphs Shortcomings of the graph partitioning model Hypergraph models of communication in MatVec Parallel methods for partitioning hypergraphs

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Sparse Matrix Vector Multiplication
Definition of Graph Partitioning Given a graph G = (N, E, WN, WE) N = nodes (or vertices), E = edges WN = node weights WE = edge weights Often nodes are tasks, edges are communication, weights are costs Choose a partition N = N1 U N2 U … U NP such that Total weight of nodes in each part is “about the same” Total weight of edges connecting nodes in different parts is small Balance the work load, while minimizing communication Special case of N = N1 U N2: Graph Bisection 1 (2) 2 (2) 3 (1) 4 (3) 5 (1) 6 (2) 7 (3) 8 (1) 5 4 6 1 2 1 2 1 2 3

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Applications Telephone network design Original application, algorithm due to Kernighan Load Balancing while Minimizing Communication Sparse Matrix times Vector Multiplication Solving PDEs N = {1,…,n}, (j,k) in E if A(j,k) nonzero, W N (j) = #nonzeros in row j, W E (j,k) = 1 VLSI Layout N = {units on chip}, E = {wires}, W E (j,k) = wire length Sparse Gaussian Elimination Used to reorder rows and columns to increase parallelism, and to decrease “fill-in” Data mining and clustering Physical Mapping of DNA
Partitioning by Repeated Bisection To partition into 2k parts, bisect graph recursively k times

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Separators in theory If G is a planar graph with n vertices, there exists a set of at most sqrt(6n) vertices whose removal leaves no connected component with more than 2n/3 vertices. (“Planar graphs have sqrt(n)-separators.”) “Well-shaped” finite element meshes in 3 dimensions have n 2/3 - separators. Also some others – trees, graphs of bounded genus, chordal graphs, bounded-excluded-minor graphs, … Mostly these theorems come with efficient algorithms, but they aren’t used much. “Random graphs” don’t have good separators.
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