Graph partitioning Prof. Richard Vuduc Georgia Institute of Technology CSE/CS 8803 PNA: Parallel Numerical Algorithms [L.27] Tuesday, April 22, 2008 1
Today’s sources CS 194/267 at UCB (Yelick/Demmel) “Intro to parallel computing” by Grama, Gupta, Karypis, & Kumar 2
Review: Dynamic load balancing 3
Parallel efficiency: 4 scenarios Consider load balance , concurrency , and overhead 4
Summary Unpredictable loads → online algorithms Fixed set of tasks with unknown costs → self-scheduling Dynamically unfolding set of tasks → work stealing Theory ⇒ randomized should work well Other scenarios: What if… … locality is of paramount importance? ⇒ Diffusion-based models? … processors are heterogeneous? ⇒ Weighted factoring? … task graph is known in advance? ⇒ Static case; graph partitioning (today) 5
Graph partitioning 6
Problem definition Weighted graph c:1 G = ( V, E, W V , W E ) 2 1 1 4 d:3 Find partitioning of nodes s.t.: a:2 b:2 2 3 2 Sum of node-weights ~ even 5 f:2 e:1 Sum of inter-partition edge- 1 weights minimized 6 g:3 h:1 V = V 1 ∪ V 2 ∪ · · · ∪ V p 7
c:1 2 1 1 4 d:3 a:2 b:2 2 3 2 5 f:2 e:1 1 6 g:3 h:1 8
c:1 2 1 1 4 d:3 a:2 b:2 2 3 2 5 f:2 e:1 1 6 g:3 h:1 9
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Cost of graph partitioning Many possible partitions Consider V = V 1 U V 2 � � n � 2 π n · 2 n ≈ n 2 Problem is NP-Complete, so need heuristics 11
First heuristic: Repeated graph bisection To get 2 k partitions, bisect k times 12
Edge vs. vertex separators Edge separator : E s ⊂ E, s.t. removal creates two disconnected components Vertex separator : V s ⊂ V, s.t. removing Vs and its incident edges creates two disconnected components E s → V s : | V s | | E s | ≤ V s V s → E s : | E s | d · | V s | , ≤ d = max degree E s E s 13
Overview of bisection heuristics With nodal coordinates: Spatial partitioning Without nodal coordinates Multilevel acceleration: Use coarse graphs 14
Partitioning with nodal coordinates 15
Intuition: Planar graph theory Planar graph : Can draw G in the plane w/o edge crossings Theorem (Lipton & Tarjan ’79): Planar G ⇒ ∃ V s s.t. (1) V = V 1 ∪ V s ∪ V 2 | V 1 | , | V 2 | ≤ 2 V s (2) 3 | V | � (3) | V s | ≤ 8 | V | 16
Inertial partitioning 17
Inertial partitioning Choose line L L 18
Inertial partitioning Choose line L L 19
Inertial partitioning Choose line L L 20
Inertial partitioning Choose line L L (¯ x, ¯ y ) ( a, b ) 21
Inertial partitioning Choose line L L : a · ( x − ¯ x ) + b · ( y − ¯ y ) = 0 L a 2 + b 2 = 1 (¯ x, ¯ y ) ( a, b ) 22
Inertial partitioning Choose line L L : a · ( x − ¯ x ) + b · ( y − ¯ y ) = 0 L a 2 + b 2 = 1 Project points onto L (¯ x, ¯ y ) ( a, b ) 23
Inertial partitioning Choose line L ( x k , y k ) L : a · ( x − ¯ x ) + b · ( y − ¯ y ) = 0 L a 2 + b 2 = 1 s k Project points onto L (¯ x, ¯ y ) ( a, b ) 24
Inertial partitioning Choose line L ( x k , y k ) L : a · ( x − ¯ x ) + b · ( y − ¯ y ) = 0 L a 2 + b 2 = 1 s k Project points onto L (¯ x, ¯ y ) s k = − b · ( x k − ¯ x ) + a · ( y k − ¯ y ) ( a, b ) 25
Inertial partitioning Choose line L ( x k , y k ) L : a · ( x − ¯ x ) + b · ( y − ¯ y ) = 0 L a 2 + b 2 = 1 s k Project points onto L (¯ x, ¯ y ) s k = − b · ( x k − ¯ x ) + a · ( y k − ¯ y ) ( a, b ) Compute median and separate s = median( s 1 , . . . , s n ) ¯ 26
How to choose L? N 1 N 2 L L N 1 N 2 27
How to choose L? ( x k , y k ) L s k Least-squares fit: Minimize sum-of-square distances (¯ x, ¯ y ) � ( d k ) 2 ( a, b ) k x ) 2 + ( y k − ¯ � y ) 2 − ( s k ) 2 � � = ( x k − ¯ k x ) 2 + ( y k − ¯ � y ) 2 − ( − b ( x k − ¯ y )) 2 � � = ( x k − ¯ x ) + a ( y k − ¯ k y ) 2 + 2 ab a 2 � � y ) + b 2 � x ) 2 = ( y k − ¯ ( x k − ¯ x )( y k − ¯ ( x k − ¯ k k k a 2 · α 1 + 2 ab · α 2 + b 2 · α 3 = � � � � a α 1 α 2 b ) · = ( a · b α 2 α 3 28
How to choose L? ( x k , y k ) L s k Least-squares fit: Minimize sum-of-square distances (¯ x, ¯ y ) Interpretation: ( a, b ) Equivalent to choosing L as axis of rotation that minimizes moment of inertia. � � a � ( d k ) 2 Minimize: = ( a b ) · A (¯ x, ¯ y ) · b k 1 � = ¯ = x x k ⇒ n k 1 � ¯ = y y k n k � � a = Eigenvector of smallest eigenvalue of A b 29
What about 3D (or higher dimensions)? Intuition : Regular n x n x n mesh Edges to 6 nearest neighbors Partition using planes General graphs: Need notion of “ well-shaped ” like a mesh n 3 | V | = n 2 | V s | = 2 2 3 ) = O ( | E | 3 ) = O ( | V | 30
Random spheres “Separators for sphere packings and nearest neighbor graphs.” Miller, Teng, Thurston, Vavasis (1997), J. ACM Definition: A k-ply neighborhood system in d dimensions = set {D 1 , …,D n } of closed disks in R d such that no point in R d is strictly interior to more than k disks Example : 3-ply system 31
Random spheres “Separators for sphere packings and nearest neighbor graphs.” Miller, Teng, Thurston, Vavasis (1997), J. ACM Definition: A k-ply neighborhood system in d dimensions = set {D 1 , …,D n } of closed disks in R d such that no point in R d is strictly interior to more than k disks Definition: An ( α ,k) overlap graph , for α >= 1 and a k-ply neighborhood: Node = D j Edge j → i if expanding radius of smaller Example : (1,1) overlap graph disk by > α causes two disks to overlap for a 2D mesh. 32
Random spheres (cont’d) Theorem (Miller, et al.): Let G = (V, E) be an ( α , k) overlap graph in d dimensions, with n = |V|. Then there is a separator V s s.t.: V V 1 ∪ V s ∪ V 2 = d + 1 | V 1 | , | V 2 | d + 2 · n < � � d − 1 1 d · n | V s | α · k O = d In 2D, same as Lipton & Tarjan 33
Random spheres: An algorithm Choose a sphere S in R d Edges that S “cuts” form edge separator E s Build V s from E s Choose S “randomly,” s.t. satisfies theorem with high probability 34
Random spheres algorithm Partition 1: All disks inside S Partition 2: All disks outside S S Separator 35
Choosing a random sphere: Stereographic projections Given p in plane, project to p’ on sphere. p’ 1. Draw line from p to north pole. 2. p’ = intersection. p = ( x, y ) p (2 x, 2 y, x 2 + y 2 − 1) p ′ = x 2 + y 2 + 1 36
Random spheres separator algorithm (Miller, et al .) Do stereographic projection from R d to sphere S in R d+1 Find center-point of projected points Center-point c: Any hyperplane through c divides points ~ evenly There is a linear programming algorithm & cheaper heuristics Conformally map points on sphere Rotate points around origin so center-point at (0, 0, …, 0, r) for some r Dilate points: Unproject; multiply by sqrt((1-r)/(1+r)); project Net effect: Maps center-point to origin & spreads points around S Pick a random plane through the origin; intersection of plane and sphere S = “circle” Unproject circle, yielding desired circle C in R d Create V s : Node j in V s if if α *D j intersections C 37
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Summary: Nodal coordinate-based algorithms Other variations exist Algorithms are efficient: O(points) Implicitly assume nearest neighbor connectivity: Ignores edges! Common for graphs from physical models Good “initial guess” for other algorithms Poor performance on non-spatial graphs 44
Partitioning without nodal coordinates 45
A coordinate-free algorithm: Breadth-first search Choose root r and run BFS, which produces: root L0 Subgraph T of G (same nodes, N1 1 subset of edges) 2 T rooted at r 3 Level of each node = distance N2 4 from r Tree edges Horizontal edges Inter-level edges 46
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Kernighan/Lin (1970): Iteratively refine Given edge-weighted graph and partitioning: = ( V, E, W E ) G = | A | = | B | V A ∪ B, = { ( u, v ) ∈ E : u ∈ A, v ∈ B } E s � T ≡ cost( A, B ) w ( e ) ≡ e ∈ E s Find equal-sized subsets X, Y of A, B s.t. swapping reduces cost Need ability to quickly compute cost for many possible X, Y 48
K-L refinement: Definitions Definition : “External” and “internal” costs of a ∈ A, and their difference; I ( a ) similarly for B: � E ( a ) w ( a, b ) ≡ E ( b ) E ( a ) ( a,b ) ∈ E s � w ( a, a ′ ) I ( a ) ≡ I ( b ) ( a,a ′ ) ∈ A D ( a ) E ( a ) − I ( a ) ≡ 49
Consider swapping two nodes Swap X = { a } and Y = { b }: I ( a ) A ′ = ( A − a ) ∪ b B ′ = ( B − b ) ∪ a E ( b ) E ( a ) Cost changes: I ( b ) T ′ = T − ( D ( a ) + D ( b ) − 2 w ( a, b )) T − gain( a, b ) ≡ 50
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