Partitioning Spatially Located Load with Rectangles: Algorithms and Simulations
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1 Partitioning Spatially Located Load with Rectangles: Algorithms and Simulations, Erdeniz Ozgun Bas, Umit V. Catalyurek Department of Biomedical Informatics, The Ohio State University Frejus 2010 :: 1 / 32
2 A load distribution problem Load matrix In parallel computing, the load can be spatially located. The computation should be distributed accordingly. Applications Particles in Cell (stencil). Sparse Matrices. Direct Volume Rendering. Metrics Load balance. Communication. Stability. Introduction:: 2 / 32
3 Outline 1 Introduction 2 Preliminaries Notation In One Dimension Simulation Setting 3 Rectilinear Partitioning Nicol s Algorithm 4 Jagged Partitioning PxQ jagged partitioning m-way Jagged Partitioning 5 Hierarchical Bisection Recursive Bisection Dynamic Programming 6 Final thoughts Summing up Conclusion and Perspective Introduction:: 3 / 32
4 The Rectangular Partitioning Problem Definition Let A be a n 1 n 2 matrix of non-negative values. The problem is to partition the [1, 1] [n 1, n 2 ] rectangle into a set S of m rectangles. The load of rectangle r = [x, y] [x, y ] is L(r) = x i x,y j y A[i][j]. The problem is to minimize L max = max r S L(r). Prefix Sum Algorithms are rarely interested in the value of a particular element but rather interested in the load of a rectangle. The matrix is given as a 2D prefix sum array Pr such as Pr[i][j] = i i,j j A[i ][j ]. By convention Pr[0][j] = Pr[i][0] = 0. We can now compute the load of rectangle r = [x, y] [x, y ] as L(r) = Pr[x ][y ] + Pr[x 1][y 1] Pr[x ][y 1] Pr[x 1][y ]. Preliminaries::Notation 4 / 32
5 In One Dimension Heuristic : Direct Cut [MP97] Greedily set the first interval at the first i such as i i A[i ] Complexity: O(m log n m ). Guarantees : L max(dc) P i A[i ] m Optimal : Nicol s algorithm [Nic94] (improved by [PA04]) P i A[i ] m. + max i A[i]. Use Probe(B) which tries to build a solution of value less than B. It loads greedily the processors up with the largest interval of load less than B. It exploits the property that there exists a solution so that the first interval [1, i] is either the smallest such that Probe(L([1, i])) is true or the largest such that Probe(L([1, i])) is false. Complexity: O((m log n m )2 ). Note: it works on more than load matrices, as long as the load of intervals are non-decreasing (by inclusion). Preliminaries::In One Dimension 5 / 32
6 Simulation Setting Classes (Some inspired by [MS96]) Processors Simulation are perform with different number of processors: most squared numbers up to 10,000. Metric Load imbalance is the presented metric : P L max i,j A[i][j] m 1. Preliminaries::Simulation Setting 6 / 32
7 Outline of the Talk 1 Introduction 2 Preliminaries Notation In One Dimension Simulation Setting 3 Rectilinear Partitioning Nicol s Algorithm 4 Jagged Partitioning PxQ jagged partitioning m-way Jagged Partitioning 5 Hierarchical Bisection Recursive Bisection Dynamic Programming 6 Final thoughts Summing up Conclusion and Perspective Rectilinear Partitioning:: 7 / 32
8 Rectilinear Partitioning Rectilinear Partitioning:: 8 / 32
9 Known results on rectilinear partitioning NP Complete [GM96] and there is no (2 ɛ)-approximation algorithm (unless P = NP). [Nic94]: a θ(m)-approximation algorithm based on iterative refinement. O(n 1 n 2 ) iterations in O(Q(P log n 1 P )2 + P(Q log n 2 Q )2 ). [AHM01](refinement of [Nic94]): a θ(m 1/4 )-approximation algorithm for squared matrices. [KMS97]: a 120-approximation algorithm of complexity O(n 1 n 2 ). [GIK02]: 4-approximation algorithm (from rectangle stabbing) of complexity O(log( i,j A[i][j])n10 ) (heavy linear programming). 1 n10 2 [MS05]: (4 + ɛ)-approximation algorithm that runs in O((n 1 + n 2 + PQ)P log(n 1 n 2 )). Rectilinear Partitioning:: 9 / 32
10 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Rectilinear Partitioning::Nicol s Algorithm 10 / 32
11 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
12 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
13 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
14 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
15 Nicol s Rectilinear Algorithm [Nic94] max PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
16 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Partition it in Q. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
17 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Partition it in Q. Get a PxQ rectilinear partitioning. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
18 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Partition it in Q. Get a PxQ rectilinear partitioning. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
19 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Partition it in Q. Get a PxQ rectilinear partitioning. Ignore the row partition. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
20 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Partition it in Q. Get a PxQ rectilinear partitioning. Ignore the row partition. Iterate if improve. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
21 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Sum the columns to make a 1d instance. Partition it in P parts. Get a Px1 rectilinear partitioning. Sum the rows in each part. Build a 1d instance by taking the maximum for each interval. Partition it in Q. Get a PxQ rectilinear partitioning. Ignore the row partition. Iterate if improve. Rectilinear Partitioning::Nicol s Algorithm 10 / 32
22 Nicol s Rectilinear Algorithm [Nic94] PxQ rectilinear partitioning Complexity: O(n 1 n 2 ) iterations (around 10 in practice) 1 iteration : O(Q(P log n 1 P )2 + P(Q log n 2 Q )2 ). Rectilinear Partitioning::Nicol s Algorithm 10 / 32
23 Outline of the Talk 1 Introduction 2 Preliminaries Notation In One Dimension Simulation Setting 3 Rectilinear Partitioning Nicol s Algorithm 4 Jagged Partitioning PxQ jagged partitioning m-way Jagged Partitioning 5 Hierarchical Bisection Recursive Bisection Dynamic Programming 6 Final thoughts Summing up Conclusion and Perspective Jagged Partitioning:: 11 / 32
24 PxQ Jagged Partitioning Jagged Partitioning::PxQ jagged partitioning 12 / 32
25 PxQ heuristic PxQ Jagged Partitioning Jagged Partitioning::PxQ jagged partitioning 13 / 32
26 PxQ heuristic PxQ Jagged Partitioning Sum on columns to generate a 1D problem. Jagged Partitioning::PxQ jagged partitioning 13 / 32
27 PxQ heuristic PxQ Jagged Partitioning Sum on columns to generate a 1D problem. Partition it in P parts. Jagged Partitioning::PxQ jagged partitioning 13 / 32
28 PxQ heuristic PxQ Jagged Partitioning Sum on columns to generate a 1D problem. Partition it in P parts. For the first stripe, sum on rows. Jagged Partitioning::PxQ jagged partitioning 13 / 32
29 PxQ heuristic PxQ Jagged Partitioning Sum on columns to generate a 1D problem. Partition it in P parts. For the first stripe, sum on rows. Partition it in Q parts. Jagged Partitioning::PxQ jagged partitioning 13 / 32
30 PxQ heuristic PxQ Jagged Partitioning Sum on columns to generate a 1D problem. Partition it in P parts. For the first stripe, sum on rows. Partition it in Q parts. Treat all stripes. Complexity : O((P log n 1 P )2 + P (Q log n 2 Q )2 ). Jagged Partitioning::PxQ jagged partitioning 13 / 32
31 How good is that? Theorem If there are no zero in the array, the heuristic P Q-way partitioning is a (1 + P n 1 )(1 + Q n 2 )-approximation algorithm where = max A min A, P < n 1, Q < n 2. Jagged Partitioning::PxQ jagged partitioning 14 / 32
32 How good is that? Theorem If there are no zero in the array, the heuristic P Q-way partitioning is a (1 + P n 1 )(1 + Q n 2 )-approximation algorithm where = max A min A, P < n 1, Q < n 2. Proof. P i One dimension guarantee (upper bound) L max (DC) A[i ] m + max i A[i] P can be rewritten as L max (DC) A[i] m (1 + m n ). It allows to bound P the imbalance of a stripe : Load stripe A[i][j] P (1 + P n 1 ). And finally of a processor : L max (1 + P n 1 )(1 + Q n 2 ). Jagged Partitioning::PxQ jagged partitioning 14 / 32
33 How good is that? Theorem If there are no zero in the array, the heuristic P Q-way partitioning is a (1 + P n 1 )(1 + Q n 2 )-approximation algorithm where = max A min A, P < n 1, Q < n 2. Proof. P i One dimension guarantee (upper bound) L max (DC) A[i ] m + max i A[i] P can be rewritten as L max (DC) A[i] m (1 + m n ). It allows to bound P the imbalance of a stripe : Load stripe A[i][j] P (1 + P n 1 ). And finally of a processor : L max (1 + P n 1 )(1 + Q n 2 ). Theorem The approximation ratio is minimized by P = m n 1 n 2. Jagged Partitioning::PxQ jagged partitioning 14 / 32
34 An optimal PxQ jagged partitioning A Dynamic Programming Formulation L max (n 1, P) = min 1 k<n1 max L max (k 1, P 1), 1D(k, n 1, Q) L max (0, P) = 0 L max (n 1, 0) = +, n 1 1 O(n 1 m) L max functions. O(n1 2 ) 1D functions. For a 512x512 matrix and 1000 processors, that s 512, ,144 values. On 64-bit values, that s 6MB. Jagged Partitioning::PxQ jagged partitioning 15 / 32
35 An optimal PxQ jagged partitioning A Dynamic Programming Formulation L max (n 1, P) = min 1 k<n1 max L max (k 1, P 1), 1D(k, n 1, Q) L max (0, P) = 0 L max (n 1, 0) = +, n 1 1 O(n 1 m) L max functions. O(n1 2 ) 1D functions. For a 512x512 matrix and 1000 processors, that s 512, ,144 values. On 64-bit values, that s 6MB. Not all values need to be stored Binary search on k. Lower bound/upper bound on L max and 1D. Tree pruning. Jagged Partitioning::PxQ jagged partitioning 15 / 32
36 Performance of PxQ jagged Partitioning 1 iteration Nicol Heuristic PxQ Jagged Optimal PxQ Jagged 0.1 load imbalance nb proc Jagged Partitioning::PxQ jagged partitioning 16 / 32
37 m-way Jagged Partitioning Jagged Partitioning::m-way Jagged Partitioning 17 / 32
38 m-way jagged partitioning heuristic Algorithm Cut in P stripes. Distribute P processors in each stripe proportionally to the i,j stripe s load : alloc j = A[i][j] load j (m P). Jagged Partitioning::m-way Jagged Partitioning 18 / 32
39 m-way jagged partitioning heuristic Algorithm Cut in P stripes. Distribute P processors in each stripe proportionally to the i,j stripe s load : alloc j = A[i][j] load j (m P). Theorem If there are no zero in A, the approximation ratio of the described m algorithm is m P (1 + n 2 ) + m Pn 2 (1 + P n 1 ). Proof. Same kind of proof than for heuristic PxQ jagged partitioning. Recall that the guarantee of heuristic PxQ jagged partitioning was: (1 + P n 1 )(1 + Q n 2 ). m-way is better for large m values. Jagged Partitioning::m-way Jagged Partitioning 18 / 32
40 An optimal m-way partitioning A Dynamic Programming Formulation L max (n 1, m) = min 1 k<n1,1 x m max L max (k 1, m x), 1D(k, n 1, x) L max (0, m) = 0 L max (n 1, 0) = +, n 1 1 O(n 1 m) L max functions. O(n1 2 m) 1D functions. The same kind of optimizations apply. For a 512x512 matrix on 1,000 processors. That s 512, ,144,000 values, if they are 64-bits, about 2GB (and takes 30 minutes). Jagged Partitioning::m-way Jagged Partitioning 19 / 32
41 Performance of m-way 1 iteration Nicol Heuristic PxQ Jagged Heuristic m-way Jagged Optimal m-way Jagged 0.1 load imbalance nb proc Jagged Partitioning::m-way Jagged Partitioning 20 / 32
42 Outline of the Talk 1 Introduction 2 Preliminaries Notation In One Dimension Simulation Setting 3 Rectilinear Partitioning Nicol s Algorithm 4 Jagged Partitioning PxQ jagged partitioning m-way Jagged Partitioning 5 Hierarchical Bisection Recursive Bisection Dynamic Programming 6 Final thoughts Summing up Conclusion and Perspective Hierarchical Bisection:: 21 / 32
43 Hierarchical Bisection Partitioning Hierarchical Bisection:: 22 / 32
44 Recursive Bisection [BB87] m = 8 Algorithm m processors to partition a rectangle. Complexity: O(m log max n 1, n 2 ). Hierarchical Bisection::Recursive Bisection 23 / 32
45 Recursive Bisection [BB87] m = 8 Algorithm m processors to partition a rectangle. Cut to balance the load evenly. Complexity: O(m log max n 1, n 2 ). Hierarchical Bisection::Recursive Bisection 23 / 32
46 Recursive Bisection [BB87] m = 4 m = 4 Algorithm m processors to partition a rectangle. Cut to balance the load evenly. Allocate half the processors to each side. Complexity: O(m log max n 1, n 2 ). Hierarchical Bisection::Recursive Bisection 23 / 32
47 Recursive Bisection [BB87] m = 4 m = 2 m = 2 Algorithm m processors to partition a rectangle. Cut to balance the load evenly. Allocate half the processors to each side. Complexity: O(m log max n 1, n 2 ). Hierarchical Bisection::Recursive Bisection 23 / 32
48 Recursive Bisection [BB87] m = 4 m = 1 m = 1 m = 2 Algorithm m processors to partition a rectangle. Cut to balance the load evenly. Allocate half the processors to each side. Cut the dimension that balances the load best. Complexity: O(m log max n 1, n 2 ). Hierarchical Bisection::Recursive Bisection 23 / 32
49 Recursive Bisection [BB87] Algorithm m processors to partition a rectangle. Cut to balance the load evenly. Allocate half the processors to each side. Cut the dimension that balances the load best. Complexity: O(m log max n 1, n 2 ). Hierarchical Bisection::Recursive Bisection 23 / 32
50 Performance of Recursive Bisection 1 iteration Nicol Heuristic PxQ Jagged Heuristic m-way Jagged Recursive Bisection 0.1 load imbalance nb proc Hierarchical Bisection::Recursive Bisection 24 / 32
51 An Optimal Hierarchical Bisection Algorithm A Dynamic Programming Formulation L max (x 1, x 2, y 1, y 2, m) = min j min (min x max L max (x 1, x, y 1, y 2, j), L max (x + 1, x 2, y 1, y 2, m j)), (min y max L max (x 1, x 2, y 1, y, j), L max (x 1, x 2, y + 1, y 2, m j)) O(n 2 1 n2 2 m) L max functions. For a 512x512 matrix and 1000 processors, that s 68,719,476,736,000 values. On 64-bit values, that s 544TB. Hierarchical Bisection::Dynamic Programming 25 / 32
52 An Optimal Hierarchical Bisection Algorithm A Dynamic Programming Formulation L max (x 1, x 2, y 1, y 2, m) = min j min (min x max L max (x 1, x, y 1, y 2, j), L max (x + 1, x 2, y 1, y 2, m j)), (min y max L max (x 1, x 2, y 1, y, j), L max (x 1, x 2, y + 1, y 2, m j)) O(n 2 1 n2 2 m) L max functions. For a 512x512 matrix and 1000 processors, that s 68,719,476,736,000 values. On 64-bit values, that s 544TB. The Relaxed Hierarchical Heuristic Build the solution according to L max (x 1, x 2, y 1, y 2, m) = min j min (min x max L(x 1,x,y 1,y 2 ) j, L(x+1,x 2,y 1,y 2 ) m j ), (min y max L(x 1,x 2,y 1,y) j, L(x 1,x 2,y+1,y 2 ) m j ) Hierarchical Bisection::Dynamic Programming 25 / 32
53 Performance of Relaxed Hierarchical iteration Nicol Heuristic PxQ Jagged Heuristic m-way Jagged Recursive Bisection Relaxed Hierarchical load imbalance nb proc Hierarchical Bisection::Dynamic Programming 26 / 32
54 Outline of the Talk 1 Introduction 2 Preliminaries Notation In One Dimension Simulation Setting 3 Rectilinear Partitioning Nicol s Algorithm 4 Jagged Partitioning PxQ jagged partitioning m-way Jagged Partitioning 5 Hierarchical Bisection Recursive Bisection Dynamic Programming 6 Final thoughts Summing up Conclusion and Perspective Final thoughts:: 27 / 32
55 More General? Recursively Defined Partitioning Most of them are polynomial by Dynamic Programming Final thoughts:: 28 / 32
56 More General? Recursively Defined Partitioning Most of them are polynomial by Dynamic Programming Arbitrary Rectangles NP-Complete with a 5 4 non-approximability result [KMP98]. There is a known 2-approximation of complexity O(n 1 n 2 + m log n 1 n 2 ) which heavily relies on linear programming [Pal06]. Final thoughts:: 28 / 32
57 Performance Over the Execution processors Nicol Heuristic PxQ Jagged Heuristic m-way Jagged Recursive Bisection Relaxed Hierarchical load imbalance iteration Final thoughts::summing up 29 / 32
58 Relaxed Hierarchical Might Be Unstable processors Nicol Heuristic PxQ Jagged Heuristic m-way Jagged Recursive Bisection Relaxed Hierarchical load imbalance iteration Final thoughts::summing up 30 / 32
59 Conclusion and Perspective Conclusion Proposed new classes of partitioning. Proved that most recursively defined classes are polynomial: Proposed two new well-founded heuristics which outperform state-of-the-art algorithm. Theoretically analyzed two heuristics. Perspective Better m-way jagged partitioning algorithm. Integration into real physic simulation codes. Include communication models.. Final thoughts::conclusion and Perspective 31 / 32
60 Thank you Collaborators Thanks to H. Karimabadi, A. Majumdar, Y.A. Omelchenko and K.B. Quest, collaborators of the Petaapps NSF OCI grant, for providing the particle-in-cell dataset. More information contact : esaule@bmi.osu.edu visit: Research at HPC lab is funded by Final thoughts::conclusion and Perspective 32 / 32
61 Bengt Aspvall, Magnús M. Halldórsson, and Fredrick Manne. Approximations for the general block distribution of a matrix. Theor. Comput. Sci., 262(1-2): , Marsha Berger and Shahid Bokhari. A partitioning strategy for nonuniform problems on multiprocessors. IEEE Transaction on Computers, C36(5): , Daya Ram Gaur, Toshihide Ibaraki, and Ramesh Krishnamurti. Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. J. Algorithms, 43(1): , Michelangelo Grigni and Fredrik Manne. On the complexity of the generalized block distribution. In IRREGULAR 96: Proceedings of the Third International Workshop on Parallel Algorithms for Irregularly Structured Problems, pages , London, UK, Springer-Verlag. S. Khanna, S. Muthukrishnan, and M. Paterson. Final thoughts::conclusion and Perspective 32 / 32
62 On approximating rectangle tiling and packaging. In proceedings of the 19th SODA, pages , Sanjeev Khanna, S. Muthukrishnan, and Steven Skiena. Efficient array partitioning. In ICALP 97: Proceedings of the 24th International Colloquium on Automata, Languages and Programming, pages , London, UK, Springer-Verlag. Serge Miguet and Jean-Marc Pierson. Heuristics for 1d rectilinear partitioning as a low cost and high quality answer to dynamic load balancing. In HPCN Europe 97: Proceedings of the International Conference and Exhibition on High-Performance Computing and Networking, pages , London, UK, Springer-Verlag. Fredrik Manne and Tor Sørevik. Partitioning an array onto a mesh of processors. Final thoughts::conclusion and Perspective 32 / 32
63 In PARA 96: Proceedings of the Third International Workshop on Applied Parallel Computing, Industrial Computation and Optimization, pages , London, UK, Springer-Verlag. S. Muthukrishnan and Torsten Suel. Approximation algorithms for array partitioning problems. Journal of Algorithms, 54:85 104, David Nicol. Rectilinear partitioning of irregular data parallel computations. Journal of Parallel and Distributed Computing, 23: , Ali Pinar and Cevdet Aykanat. Fast optimal load balancing algorithms for 1d partitioning. Journal of Parallel and Distributed Computing, 64: , K. Paluch. A new approximation algorithm for multidimensional rectangle tiling. In Proceedings of ISAAC, Final thoughts::conclusion and Perspective 32 / 32
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