Examples of One-Dimensional Systolic Arrays

1. Examples of One-Dimensional Systolic Arrays

2. Motivation & Introduction • We need a high-performance , special-purpose computer • system to meet specific application. • I/O and computation imbalance is a notable problem. • The concept of Systolic architecture can map high-level • computation into hardware structures. • Systolic system works like an automobile assembly line. • Systolic system is easy to implement because of its • regularity and easy to reconfigure. • Systolic architecture can result in cost-effective , high- • performance special-purpose systems for a wide range • of problems.

3. Pipelined Computations

4. Pipelined Computations P1 P2 P3 P4 P5 f, e, d, c, b, a • Pipelined program divided into a series of tasks that have to be completed one after the other. • Each task executed by a separate pipeline stage • Data streamedfrom stage to stageto form computation

5. Pipelined Computations P5 P4 P3 P2 P1 a b c d e f a b c d e f a b c d e f P1 P2 P3 P4 P5 f, e, d, c, b, a a b c d e f a b c d e f time • Computation consists of data streaming through pipeline stages • Execution Time = Time to fill pipeline (P-1) + Time to run in steady state (N-P+1) + Time to empty pipeline (P-1) P = # of processors N = # of data items (assume P < N) This slide must be explained in all detail. It is very important

6. Pipelined Example: Sieve of Eratosthenes • Goal is to take a list of integers greater than 1 and produce a list of primes • E.g. For input 2 3 4 5 6 7 8 9 10, output is 2 3 5 7 • A pipelined approach: • Processor P_i divides each input by the i-th prime • If the input is divisible (and not equal to the divisor), it is marked (with a negative sign) and forwarded • If the input is not divisible, it is forwarded • Last processor only forwards unmarked (positive) data [primes]

7. Sieve of Eratosthenes Pseudo-Code Code for processor Pi (and prime p_i): x=recv(data,P_(i-1)) If (x>0) then If (p_i divides x and p_i = x ) then send(-x,P_(i+1) If (p_i does not divide x or p_i = x) then send(x, P_(i+1)) Else Send(x,P_(i+1)) Code for last processor x=recv(data,P_(i-1)) If x>0 then send(x,OUTPUT) P2 P3 P5 P7 out / Processor P_i divides each input by the i-th prime

8. Programming Issues P13 P17 P2 P3 P5 P7 P11 • Algorithm will take N+P-1 to run where N is the number of data items and P is the number of processors. • Can also consider just the odd bnys or do some initial part separately • In given implementation, number of processors must store all primes which will appear in sequence • Not a scalable approach • Can fix this by having each processor do the job of multiple primes, i.e. mapping logical “processors” in the pipeline to each physical processor • What is the impact of this on performance? processor does the job of three primes

9. Processors for such operation • In pipelined algorithm, flow of data moves through processors in lockstep. • The design attempts to balance the work so that there is no bottleneck at any processor • In mid-80’s, processors were developed to support in hardware this kind of parallel pipelined computation • Two commercial products from Intel: • Warp (1D array) • iWarp (components for 2D array) • Warp and iWarp were meant to operate synchronously Wavefront Array Processor (S.Y. Kung) was meant to operate asynchronously, • i.e. arrival of data would signal that it was time to execute

10. Systolic Arrays

11. Example 1:“pipelined” polynomial evaluation

12. Example 1: “pipelined” polynomial evaluation • Polynomial Evaluation is done by using a Linear array with 2D. • Expression: Y = ((((anx+an-1)*x+an-2)*x+an-3)*x……a1)*x + a0 • Function of PEs in pairs • 1. Multiply input by x • 2. Pass result to right. • 3. Add aj to result from left. • 4. Pass result to right.

13. Example 1: polynomial evaluation Y = ((((anx+an-1)*x+an-2)*x+an-3)*x……a1)*x + a0 Multiplying processor X is broadcasted • Using systolic array for polynomial evaluation. • This pipelined array can produce a polynomial on new X value on every cycle - after 2n stages. • Another variant: you can also calculate various polynomials on the same X. • This is an example of a deeply pipelined computation- • The pipeline has 2n stages. Adding processor x an-1 an-2 an x x a0 x ………. X + X + X + X +

14. For you to think about • Pipelined Graph Coloring • Pipelined Satisfiability • Pipelined sorting/absorbing • Pipelined decision function like Petrick Function. • Pipelined multiplication. • Pipelined calculation of (A + B) * (C – D) on vectors A, B, C, D.

15. Example 2:Matrix Vector Multiplication

16. Example 2:Matrix Vector Multiplication • There are many ways to solve a matrix problems using systolic arrays, some of the methods are: • Triangular Array performing gaussian elimination with neighbor pivoting. • Triangular Array performing orthogonal triangularization. • Simple matrix multiplication methods are shown in next slides.

17. Example 2:Matrix Vector Multiplication - - n PE1 PE3 q r p • Matrix Vector Multiplication: • Each cell’s function is: • 1. To multiply the top and bottom inputs. • 2. Add the left input to the product just obtained. • 3. Output the final result to the right. • Each cell consists of an adder and a few registers. (Booth Algorithm for mul). • Or, a cell can include a hardware multiplier.

18. Matrix Multiplication Example 2:Matrix Vector Multiplication - -i - h f g ec d b - a - - n m l PE1 PE2 PE3 z y x q r p • At time t0 the array receives 1, a, p, q, and r ( The other inputs are all zero). • At time t1, the array receive m, d, b, p, q, and r ….e.t.c • The results emerge after 5 steps.

19. - -i - h f g ec d b - a - - n m l z y x PE1 PE2 PE3 q r p • Explain how to multiply the first row of the matrix by the vector, • how data are shifted from left to right in the architecture To visualize how it works it is good to do a snapshot animation

20. Systolic Algorithms and Architectures

21. Systolic Algorithms • Systolic arrays were built to support systolic algorithms, a hot area of research in the early 80’s • Systolic algorithms used pipelining through various kinds of arrays to accomplish computational goals: • Some of the data streaming and applications were very creative and quite complex • CMU a hotbed of systolic algorithm and array research (especially H.T. Kung and his group)

22. Systolic Arrays from Intel • Warp and iWarp were examples of systolic arrays • Systolic means regular and rhythmic, • data was supposed to move through pipelined computational units in a regular and rhythmic fashion • Systolic arrays meant to be special-purpose processors or co-processors. • They were very fine-grained • Processors implement a limited and very simple computation, usually called cells • Communication is very fast, granularity meant to be around one operation/communication!

23. Systolic Processors, versus Cellular Automata versus Regular Networks of Automata Data Path Block Data Path Block Data Path Block Data Path Block Systolic processor Control Block Control Block Control Block Control Block These slides are for one-dimensional only Cellular Automaton

24. Systolic Processors, versus Cellular Automata versus Regular Networks of Automata Control Block Control Block Control Block Control Block General and Soldiers, Symmetric Function Evaluator Cellular Automaton Control Block Control Block Control Block Control Block Data Path Block Data Path Block Data Path Block Data Path Block Regular Network of Automata