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Dense Matrix Algorithms. Carl Tropper Department of Computer Science McGill University. Dense. Few non-zero entries Topics Matrix-Vector Multiplication Matrix-Matrix Multiplication Solving a System of Linear Equations . Introductory ramblings. I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. Dense Matrix Algorithms Carl Tropper Department of Computer Science McGill University

2. Dense • Few non-zero entries • Topics • Matrix-Vector Multiplication • Matrix-Matrix Multiplication • Solving a System of Linear Equations

3. Introductory ramblings • Due to their regular structure, parallel computations involving matrices and vectors readily lend themselves to data-decomposition. • Typical algorithms rely on input, output, or intermediate data decomposition. • Discuss one-and two-dimensional block, cyclic, and block-cyclic partitionings. • Use one task per process

4. Matrix-Vector Multiplication • Multiply a dense n xn matrix A with an n x 1vector x to yield an n x 1 vector y. • The serial algorithm requires n2multiplications and additions.

5. Rowwise 1-D Partitioning

6. One row per process • Each process starts with only one element of x , need all-to-all broadcast to distribute all the elements of x to all of the processes. • Process Pi then computes • The all-to-all broadcast and the computation of y[i] both take time Θ(n) . Therefore, the parallel time is Θ(n) .

7. P<N • Use block 1D partitioning. • Each process initially stores n/p complete rows of the matrix and a portion of the vector of size n/p. • all-to-all broadcast takes place among p processes and involves messages of size n/p takes time tslog p + tw(n/p)(p-1)~tslog p+twn for large p • This is followed by n/p local dot products • The parallel runtime is TP=n2/p + ts log p +twn • pTP=n2 + pts log p + ptwn => • Cost optimal(pTp) if p=O(n)

8. Scalability Analysis • We know that T0 = pTP - W, therefore, we have, • For isoefficiency, we have W = KT0, where K = E/(1 – E) for desired efficiency E. • TO=tsplog p + twnp • W=n2=Ktwnp from tw term alone =>W=n2=K2tw2p2 • From this, we have W = O(p2) from the tw term • There is also a bound on isoefficiency because of concurrency. In this case, p < n, therefore, W = n2 = Ω(p2). • From these 2 bounds on W, the overall isoefficiency isW = (p2).

9. 2-D Partitioning (naïve version) • Begin with one element per process partitioning • The n x n matrix is partitioned among n2 processors such that each processor owns a single element. • The n x 1 vector x is distributed in the last column of n processors.Each processor has one element.

10. 2-D Partitioning

11. 2-D Partitioning • We must first align the vector with the matrix appropriately. • The first communication step for the 2-D partitioning aligns the vector x along the principal diagonal of the matrix. • The second step copies the vector elements from each diagonal process to all the processes in the corresponding column using n simultaneous broadcasts among all processors in the column. • Finally, the result vector is computed by performing an all-to-one reduction along the columns.

12. 2-D Partitioning • Three basic communication operations are used in this algorithm: • one-to-one communication to align the vector along the main diagonal, • one-to-all broadcast of each vector element among the n processes of each column, and • all-to-one reduction in each row. • Each of these operations takes Θ(log n) time and the parallel time is Θ(log n) . • There are n2 processes,so the cost (process-time product) is Θ(n2 log n) ; hence, the algorithm is not cost-optimal.

13. The less naïve version-fewer than n2 processes • When using fewer than n2 processors, each process owns an (n/√ p) x (n/ √ p) block of the matrix. • The vector is distributed in portions of (n/√ p) elements in the last process-column only. • In this case, the message sizes for the alignment, broadcast, and reduction are all (n/√ p) • The computation is a product of an (n/√ p) x (n/ √ p) submatrix with a vector of length (n/√ p) .

14. Parallel Run Time • Sending message of size n/√p to diagonal takes time ts + twn/√p • Column-wise one to all broadcast takes (ts + twn/√p)log √p using hypercube algorithm • All to one reduction takes same amount of time • Assuming a multiplication and addition takes unit time,each process spends n2/p computing • TP on next page

15. Next page • TP= • {computation} TP= n2/p + • {aligning vector} ts +twn/√p+ • {columwise one to all broadcast }(ts + twn/ √ p)log√ p + • {all to one reduction} ts + twn/ √ p)log √ p • TP ~ n2/p + ts log p + tw (n/ √ p) log p

16. Scalability Analysis • From W=n2, expression for TP, and TO=pTP-W, TO=tsp log p + twn p log p • As before, find out what each term contributes to W • W=Ktsp log p • W=n2=Ktwn√plog p=>n=Ktw √p log p=> n2=K2tw2 p log 2p=> • W=K2tw2 p log2 p (**) • Concurrency is n2=>p=O(n2)=>n2= Ω(p) and • W= Ω(p) • The tw term dominates (**) everything => • W= (p log2 p )

17. Scalability Analysis • Maximum number of processes which can be used cost-optimally for a problem of size W is determined by p log2 p= O(n2) • After some manipulation, p=O(n2/log2n), • Asymptotic upper bound on the number of processes which can be used for cost-otpimal solution • Bottom line:2-D partitioning is better than 1-D because: • It is faster! • It has a smaller isoefficiency function-get the same efficiency on more processes!

18. Matrix-Matrix multiplication • Standard serial algorithm involves taking the dot product of each row with each column, has complexity of n3 • Can also use q x q array of blocks, where each block is (n/q x n/q). This yields q3 multiplications and additions of the sub-matrices. Each of the sub-matrices involves (n/q)3 additions and multiplications. • Paralellize the q x q blocks algorithm.

19. Simple Parallel Algorithm • A and B are partitioned into p blocks, i.e. AIJ , BIJ of size (n/√p x n/√p) • They are mapped onto a √p x √p mesh • PI,J stores AI,J and BI,J and computes CI,J • Needs AI,K and BJ,K sub-matrices 0 k< √p • All to all broadcast of A’s blocks done on each row and of B’s blocks on each column • Then multiply A’s and B’s

20. Scalability • 2 all to all broadcasts of process mesh • Messages contain submatrices of n2/p elements • Communication time is 2(ts log (√p) + tw (n2/p)( p-1) {hypercube is assumed} • Each process computes C I,J-takes p multiplications (n/√p x n/√p) submatrices, taking n3/p time. • Parallel time TP= n3/p + ts log p + 2 tw n2/√p • Process time product=n3+tsplog p+2twn2 √p • Cost optimal for p=O(n2)

21. Scalability • The isoefficiency is O(p1.5) due to bandwidth term tw and concurrency • Major drawback-algorithm is not memory optimal-Memory is (n2 √p), or √p times the memory of the sequential algorithm

22. Canon’s algorithm • Idea: schedule the computations of the processes of the ith row such that at any given time each process uses a different block Ai,k. • These blocks can be systematically rotated among the processes after every submatrix multiplication so that every process gets a fresh Ai,k after each rotation • Use same algorithm for columns=>no process holds more then one bock at a time • Memory is (n2)

23. Canon shift

24. Performance • Max shift for a block is √p-1. • 2 shifts (row and column) require 2(ts+twn2/p) • P shifts=>√p2(ts+twn2/p) total comm time • The time for multiplying p matrices of size (n/√p) x (n/√p) is n3/p • TP= n3/p+√p2(ts+twn2/p) • Same cost-optimality condition as simple algorithm and same iso function. • Difference is memory!!

25. DNS Algorithm • Simple and Canon • Use block 2-D partitioning of input and output matrices • Use a max of n2 processes for nxn matrix multiplication • Have Ω(n) run time because of (n3) ops in the serial algorithm • DNS • Uses up to n3 processes • Has a run time of (log n) using Ω(n3/log n) processes

26. DNS Algorithm • Assume an n x n x n mesh of processors. • Move the columns of A and rows of B and perform broadcast. • Each processor computes a single add-multiply. • This is followed by an accumulation along the C dimension. • Addition along C takes time (log n) => • Parallel runtime is (log n) • This is not cost optimal. It can be made cost optimal by using n / log n processors along the direction of accumulation

27. Cost optimal DNS with fewer then n3 • Let p=q3 for q<n • Partition the 2 matrices into blocks of size n/q x n/q • Have a q x q square array of blocks

28. Performance • 1-1 communication takes ts+tw(n/q)2 • 1-all broadcast takes tslog q+tw(n/q)2 for each matrix • Last all-1 reduction takes tslog q+tw(n/q)2log q • Multiplication of n/q x n/q submatrices takes (n/q)3 • TP~(n/q)3 + tslogp+tw(n2/p2/3)logp=>cost is n3+tsplogp+twn2p1/3logp • Isoefficiency function is (p(logp)3) • Algorithm is cost optimal for p=O(n3/(log n)3)

29. Linear Equations

30. Upper Triangular Form • Idea is to convert the equations into this form, • and then back substitute (i.e. go up the chain)

31. Principle behind solution • Can make use of elementary operations on equations to solve them • Elementary operations are • Interchanging two rows • Replace any equation by a linear combination of any other equation and itself

32. Code for Gaussian Elimination

33. What the code is doing

34. Complexity of serial Gaussian • n2/2 divisions (line 6 of code) • n3/3-n2/2 subtractions and multiplications (line 12) • Assuming all ops take unit time, for large enough n have W=2/3 n3

35. Parallel Gaussian • Use 1-D Partitioning • One row per process

36. 1-D Partitioning

37. Parallel 1-D • Assume p = n with each row assigned to a processor. • The first step of the algorithm normalizes the row. This is a serial operation and takes time (n-k) in the kth iteration. • In thesecond step, the normalized row is broadcast to all the processors. This takes time (ts+tw(n-k-1))log n • Each processor can independently eliminate this row from its own. This requires (n-k-1) multiplications and subtractions. • The total parallel time can be computed by summing from k = 1… n-1 as TP=3/2n(n-1)+tsnlog n+1/2twn(n-1)log n. • The formulation is not cost optimal because of the tw term.

38. Parallel 1-D with Pipelining • The (k+1)st iteration starts only after kth iteration completes • In each iteration, all of the active processes collaborate together • This is a synchronous algorithm • Idea: Implement algorithm so that no process has to wait for all of its predecessors to finish their work • The result is an asynchronous algorithm, which makes use of pipelining • Algorithm turns out to be cost-optimal

39. Pipelining • During the kth iteration, P k sends part of the kth row to Pk+1, which forwards it to Pk+1, which….. • P k+1 can perform the elimination step without waiting for the data finish its journey to the bottom of the matrix • Idea is to get the maximum overlap of communication and computation • If a process has data destined for other processes, it sends it right away • If the process can do a computation using the data it has, it does so

40. Pipeline for 1-D, 5x5

41. Pipelining is cost optimal • The total number of steps in the entire pipelined procedure is Θ(n). • In any step, either O(n) elements are communicated between directly-connected processes, or a division step is performed on O(n) elements of a row, or an elimination step is performed on O(n) elements of a row. • The parallel time is therefore O(n2) • Since there are n processes, the cost is O(n3) • Guess what,cost optimal!

42. Pipelining 1-D with p<n • Pipelining algorithm can be easily extended • N x N matrix • n/p processes per processor • Example on next slide • P=4 • 8 x8 matrix

43. Next slide

44. Analysis • In the kth iteration, a processor with all rows belonging to the active part of the matrix performs (n – k -1) / np multiplications and subtractions during elimination step of the kth iteration. • Computation dominates communication at each iteration (n-(k+1)) words are communicated during iteration k (vs (n-(k+1)/np computation ops) • The parallel time is 2(n/p)∑k=0n-1 (n-k-1) ~ n3/p. • The algorithm is cost optimal, but the cost is higher than the sequential run time by a factor of 3/2.

45. Fewer then n processes • The parallel time is 2(n/p)∑k=0n-1 (n-k-1) ~ n3/p. • The algorithm is cost optimal, but the cost is higher than the sequential run time by a factor of 3/2. • Inefficiency due to unbalanced load • In the figure on next slide,1 process is idle, 1 is partially active, 2 are fully active • Use cyclic block distribution to balance load

46. Block and cyclic mappings

47. 2-D Partitioning • A[i,j] is n x n and is mapped to n x n mesh-A[i,j] goes to P I,J • The rest is as before, only the communication of individual elements takes place between processors • Need one to all broadcast of A[i,k] along ith row for k≤ i<n and one to all broadcast of A[k,j] along jth column for k<j<n • Picture on next slide • The result is not cost optimal

48. Picture K=3 for 8 x8 mesh

49. Pipeline • If we use synchronous broadcasts, the results are not cost optimal, so we pipeline the 2-D algorithm • Principle of the pipelining algorithm is the same-if you can compute or communicate, do it now, not later • P k,k+1 can divide A[k,k+1] by A[k,k] before A[k,k+1] reaches P k,n-1 {the end of the row} • After A[k,j] performs the division, it can send the result down column j without waiting • Next slide exhibits algorithm for 2-D pipelining