CS 267 Applications of Parallel Computers Lecture 14: Graph Partitioning - IOutlineDefinition of Graph PartitioningApplicationsSparse Matrix Vector MultiplicationCost of Graph PartitioningFirst Heuristic: Repeated Graph BisectionOverview of Partitioning Heuristics for BisectionEdge Separators vs. Vertex Separators of G(N,E)Graphs with Nodal Coordinates - Planar graphsGraphs with Nodal Coordinates: Inertial PartitioningInertial Partitioning: Choosing LInertial Partitioning: choosing L (continued)Graphs with Nodal Coordinates: Random lines and circlesRandom lines and circles: Well Shaped GraphsGeneralizing Lipton/Tarjan to higher dimensionsStereographic ProjectionChoosing a Random SpherePartitioning with Nodal Coordinates - SummaryCS267 L14 Graph Partitioning I.1Demmel Sp 1999CS 267 Applications of Parallel ComputersLecture 14: Graph Partitioning - IJames Demmelhttp://www.cs.berkeley.edu/~demmel/cs267_Spr99CS267 L14 Graph Partitioning I.2Demmel Sp 1999Outline°Review definition of Graph Partitioning problem°Outline of Heuristics°Partitioning with Nodal Coordinates°Partitioning without Nodal Coordinates°Multilevel Acceleration•BIG IDEA, will appear often in course°Available Software•good sequential and parallel software availble°Comparison of Methods°ApplicationsCS267 L14 Graph Partitioning I.3Demmel Sp 1999Definition of Graph Partitioning°Given a graph G = (N, E, WN, WE)•N = nodes (or vertices), E = edges•WN = node weights, WE = edge weights°Ex: N = {tasks}, WN = {task costs}, edge (j,k) in E means task j sends WE(j,k) words to task k°Choose a partition N = N1 U N2 U … U NP such that•The sum of the node weights in each Nj is “about the same”•The sum of all edge weights of edges connecting all different pairs Nj and Nk is minimized°Ex: balance the work load, while minimizing communication°Special case of N = N1 U N2: Graph BisectionCS267 L14 Graph Partitioning I.4Demmel Sp 1999Applications°Load Balancing while Minimizing Communication°Sparse Matrix time Vector Multiplication•Solving PDEs•N = {1,…,n}, (j,k) in E if A(j,k) nonzero, •WN(j) = #nonzeros in row j, WE(j,k) = 1°VLSI Layout•N = {units on chip}, E = {wires}, WE(j,k) = wire length°Telephone network design•Original application, algorithm due to Kernighan°Sparse Gaussian Elimination•Used to reorder rows and columns to increase parallelism, decrease “fill-in”°Physical Mapping of DNA°Evaluating Bayesian Networks?CS267 L14 Graph Partitioning I.5Demmel Sp 1999Sparse Matrix Vector MultiplicationCS267 L14 Graph Partitioning I.6Demmel Sp 1999Cost of Graph Partitioning°Many possible partitionings to search:°n choose n/2 ~ sqrt(2n/pi)*2n bisection possibilities°Choosing optimal partitioning is NP-complete•Only known exact algorithms have cost = exponential(n)°We need good heuristicsCS267 L14 Graph Partitioning I.7Demmel Sp 1999First Heuristic: Repeated Graph Bisection°To partition N into 2k parts, bisect graph recursively k times•Henceforth discuss mostly graph bisectionCS267 L14 Graph Partitioning I.8Demmel Sp 1999Overview of Partitioning Heuristics for Bisection°Partitioning with Nodal Coordinates•Each node has x,y,z coordinates•Partition nodes by partitioning space°Partitioning without Nodal Coordinates•Sparse matrix of Web: A(j,k) = # times keyword j appears in URL k°Multilevel acceleration (BIG IDEA)•Approximate problem by “coarse graph”, do so recursivelyCS267 L14 Graph Partitioning I.9Demmel Sp 1999Edge Separators vs. Vertex Separators of G(N,E)°Edge Separator: Es (subset of E) separates G if removing Es from E leaves two ~equal-sized, disconnected components of N: N1 and N2 °Vertex Separator: Ns (subset of N) separates G if removing Ns and all incident edges leaves two ~equal-sized, disconnected components of N: N1 and N2°Making an Ns from an Es: pick one endpoint of each edge in Es•How big can |Ns| be, compared to |Es| ?°Making an Es from an Ns: pick all edges incident on Ns•How big can |Es| be, compared to |Ns| ?°We will find Edge or Vertex Separators, as convenientEs = green edges or blue edgesNs = red verticesCS267 L14 Graph Partitioning I.10Demmel Sp 1999Graphs with Nodal Coordinates - Planar graphs°Planar graph can be drawn in plane without edge crossings°Ex: m x m grid of m2 nodes: vertex separatorNs with |Ns| = m = sqrt(|N|) (see last slide for m=5 )°Theorem (Tarjan, Lipton, 1979): If G is planar, Ns such that •N = N1 U Ns U N2 is a partition,•|N1| <= 2/3 |N| and |N2| <= 2/3 |N|•|Ns| <= sqrt(8 * |N|)°Theorem motivates intuition of following algorithmsCS267 L14 Graph Partitioning I.11Demmel Sp 1999Graphs with Nodal Coordinates: Inertial Partitioning°For a graph in 2D, choose line with half the nodes on one side and half on the other•In 3D, choose a plane, but consider 2D for simplicity°Choose a line L, and then choose an L perpendicular to it, with half the nodes on either side°Remains to choose L1) L given by a*(x-xbar)+b*(y-ybar)=0, with a2+b2=1; (a,b) is unit vector to L 2) For each nj = (xj,yj), compute coordinate Sj = -b*(xj-xbar) + a*(yj-ybar) along L3) Let Sbar = median(S1,…,Sn)4) Let nodes with Sj < Sbar be in N1, rest in N2CS267 L14 Graph Partitioning I.12Demmel Sp 1999Inertial Partitioning: Choosing L°Clearly prefer L on left below°Mathematically, choose L to be a total least squares fit of the nodes•Minimize sum of squares of distances to L (green lines on last slide)•Equivalent to choosing L as axis of rotation that minimizes the moment of inertia of nodes (unit weights) - source of nameCS267 L14 Graph Partitioning I.13Demmel Sp 1999Inertial Partitioning: choosing L (continued)j (length of j-th green line)2 = j [ (xj - xbar)2 + (yj - ybar)2 - (-b*(xj - xhar) + a*(yj - ybar))2 ] … Pythagorean Theorem = a2 * j (xj - xbar)2 + 2*a*b* j (xj - xbar)*(xj - ybar) + b2 j (yj - ybar)2 = a2 * X1 + 2*a*b* X2 + b2 * X3 = [a b] * X1 X2 * a X2 X3 bMinimized by choosing (xbar , ybar) = (j xj , j yj) / N = center of mass (a,b) = eigenvector of smallest eigenvalue of X1 X2 X2 X3(a,b) is unit vectorperpendicular to LCS267 L14 Graph Partitioning I.14Demmel Sp 1999Graphs with Nodal Coordinates: Random lines and circles°Emulate Lipton/Tarjan in higher dimensions than planar (2D)°Take an n by n by n mesh
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