Pipelined Computations

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1 Chapter 5 Slide 155 Pipelined Computations

2 Pipelined Computations Slide 156 Problem divided into a series of tasks that have to be completed one after the other (the basis of sequential programming). Each task executed by a separate process or processor. P 0 P 1 P 2 P 3 P 4 P 5

3 Slide 157 Example Add all the elements of array a to an accumulating sum: for (i = 0; i < n; i++) sum = sum + a[i]; The loop could be unfolded to yield sum = sum + a[0]; sum = sum + a[1]; sum = sum + a[2]; sum = sum + a[3]; sum = sum + a[4];.

4 Slide 158 Pipeline for an unfolded loop a[0] a[1] a[2] a[3] a[4] sum s in a s out s in a s out s in a s out s in a s out s in a s out

5 Another Example Slide 159 Frequency filter - Objective to remove specific frequencies (f 0, f 1, f 2, f 3, etc.) from a digitized signal, f(t). Signal enters pipeline from left: Signal without Signal without frequency f frequency f 1Signal 3 Signal without without frequency f 0 frequency f 2 f(t) f 1 f in f out f in f out f 0 f 4 f 2 f in f out f 3 f in f out f in f out Filtered signal

6 Where pipelining can be used to good effect Slide 160 Assuming problem can be divided into a series of sequential tasks, pipelined approach can provide increased execution speed under the following three types of computations: 1. If more than one instance of the complete problem is to be executed 2. If a series of data items must be processed, each requiring multiple operations 3. If information to start the next process can be passed forward before the process has completed all its internal operations

7 Type 1 Pipeline Space-Time Diagram Slide 161 p - 1 m P 5 P 4 P 3 P 2 P 1 P 0 Instance 1 Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Instance Time Execution time = m + p - 1 cycles for a p-stage pipeline and m instances.

8 Slide 162 Alternative space-time diagram Instance 0 Instance 1 Instance 2 Instance 3 Instance 4 P 0 P 1 P 2 P 3 P 4 P 5 P 0 P 1 P 2 P 3 P 4 P 5 P 0 P 1 P 2 P 3 P 4 P 5 P 0 P 1 P 2 P 3 P 4 P 5 P 0 P 1 P 2 P 3 P 4 P 5 Time

9 Type 2 Pipeline Space-Time Diagram Input sequence d 9 d 8 d 7 d 6 d 5 d 4 d 3 d 2 d 1 d 0 P 0 P 1 P 2 P 3 P 4 P 5 (a) Pipeline structure P 6 P 7 P 8 P 9 Slide 163 p - 1 n P 9 P 8 P 7 P 6 P 5 P 4 P 3 P 2 P 1 P 0 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 Time (b) Timing diagram

10 Type 3 Pipeline Space-Time Diagram Slide 164 Information transfer sufficient to start next process P 0 P 1 P 2 P 3 P 4 P 5 Information passed to next stage P 5 P 4 P 3 P 2 P 1 P 0 Time (a) Processes with the same execution time Time (b) Processes not with the same execution time Pipeline processing where information passes to next stage before end of process.

11 Slide 165 If the number of stages is larger than the number of processors in any pipeline, a group of stages can be assigned to each processor: Processor 0 Processor 1 Processor 2 P 0 P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11

12 Slide 166 Computing Platform for Pipelined Applications Multiprocessor system with a line configuration. Host computer Multiprocessor Processors Strictly speaking pipeline may not be the best structure for a cluster - however a cluster with switched direct connections, as most have, can support simultaneous message passing.

13 Slide 167 Example Pipelined Solutions (Examples of each type of computation)

14 Slide 168 Pipeline Program Examples Adding Numbers Σ 1 i Σ 1 2 i Σ 1 3 i Σ 1 4 Σ 1 P 0 P 1 P 2 P 3 P 4 i 5 i Type 1 pipeline computation

15 Basic code for process P i : Slide 169 recv(&accumulation, P i-1 ); accumulation = accumulation + number; send(&accumulation, P i+1 ); except for the first process, P 0, which is send(&number, P 1 ); and the last process, P n 1, which is recv(&number, P n-2 ); accumulation = accumulation + number;

16 Slide 170 SPMD program if (process > 0) { recv(&accumulation, P i-1 ); accumulation = accumulation + number; } if (process < n-1) send(&accumulation, P i+1 ); The final result is in the last process. Instead of addition, other arithmetic operations could be done.

17 Slide 171 Pipelined addition numbers with a master process and ring configuration Master process Slaves d n-1 d 2 d 1 d 0 P 0 P 1 P 2 P n-1 Sum

18 Sorting Numbers A parallel version of insertion sort. Slide numbers 4, 3, 1, 2, 5 P 0 P 1 P 2 P 3 P Time (cycles) 6 7 4, 3, 1, 2 4, 3, ,

19 Slide 173 Pipeline for sorting using insertion sort Series of numbers x n 1 x 1 x 0 P 0 Smaller numbers P 1 P 2 Compare x max Largest number Next largest number Type 2 pipeline computation

20 Slide 174 The basic algorithm for process P i is recv(&number, P i-1 ); if (number > x) { send(&x, P i+1 ); x = number; } else send(&number, i+1 P ); With n numbers, how many the ith process is to accept is known; it is given by n i. How many to pass onward is also known; it is given by n i 1 since one of the numbers received is not passed onward. Hence, a simple loop could be used.

21 Slide 175 Insertion sort with results returned to the master process using a bidirectional line configuration Master process d n-1 d 2 d 1 d 0 Sorted sequence P 0 P 1 P 2 P n-1

22 Slide 176 Insertion sort with results returned P 4 P 3 Sorting phase 2n - 1 Returning sorted numbers n Shown for n = 5 P 2 P 1 P 0 Time

23 Slide 177 Prime Number Generation Sieve of Eratosthenes Series of all integers is generated from 2. First number, 2, is prime and kept. All multiples of this number are deleted as they cannot be prime. Process repeated with each remaining number. The algorithm removes nonprimes, leaving only primes.

24 Pipeline for Prime Number Generation Slide 178 Not multiples of 1st prime number Series of numbers x n P 0 P 1 P 2 Compare 1st prime 2nd prime 3rd prime multiples number number number Type 2 pipeline computation

25 The code for a process, P i, could be based upon Slide 179 recv(&x, P i-1 ); /* repeat following for each number */ recv(&number, P i-1 ); if ((number % x)!= 0) send(&number, P i+1 ); Each process will not receive the same amount of numbers and the amount is not known beforehand. Use a terminator message, which is sent at the end of the sequence: recv(&x, P i-1 ); for (i = 0; i < n; i++) { recv(&number, P i-1 ); if (number == terminator) break; if (number % x)!= 0) send(&number, i+1 P ); }

26 Slide 180 Solving a System of Linear Equations Upper-triangular form a n-1,0 x 0 + a n-1,1 x 1 + a n-1,2 x 2 + a n-1,n-1 x n-1 = b n-1.. a 2,0 x 0 + a 2,1 x 1 + a 2,2 x 2 = b 2 a 1,0 x 0 + a 1,1 x 1 = b 1 a 0,0 x 0 = b 0 where the a s and b s are constants and the x s are unknowns to be found.

27 Back Substitution Slide 181 First, the unknown x 0 is found from the last equation; i.e., b x 0 = a 0,0 Value obtained for x 0 substituted into next equation to obtain x 1 ; i.e., b x 1 a 1,0 x 1 = a 1,1 Values obtained for x 1 and x 0 substituted into next equation to obtain x 2 : b x 2 a 2,0 x 0 a 2,1 x 2 = a 2,2 and so on until all the unknowns are found.

28 Slide 182 Pipeline Solution First pipeline stage computes x 0 and passes x 0 onto the second stage, which computes x 1 from x 0 and passes both x 0 and x 1 onto the next stage, which computes x 2 from x 0 and x 1, and so on. P 0 P 1 P 2 P 3 x x x 0 x x x 1 x Compute x 1 0 Compute x 1 1 Compute x 2 x Compute x 2 3 x 2 x 3 Type 3 pipeline computation

29 Slide 183 The ith process (0 < i < n) receives the values x 0, x 1, x 2,, x i-1 and computes x i from the equation: i 1 b i a i,j x j j 0 x = i = a i,i

30 Slide 184 Sequential Code Given the constants a i,j and b k stored in arrays a[][] and b[], respectively, and the values for unknowns to be stored in an array, x[], the sequential code could be x[0] = b[0]/a[0][0]; /* computed separately */ for (i = 1; i < n; i++) {/* for remaining unknowns */ sum = 0; for (j = 0; j < i; j++ sum = sum + a[i][j]*x[j]; x[i] = (b[i] - sum)/a[i][i]; }

31 Slide 185 Parallel Code Pseudocode of process P i (1 < i < n) of could be for (j = 0; j < i; j++) { recv(&x[j], P i-1 ); send(&x[j], P i+1 ); } sum = 0; for (j = 0; j < i; j++) sum = sum + a[i][j]*x[j]; x[i] = (b[i] - sum)/a[i][i]; send(&x[i], P i+1 ); Now additional computations after receiving and resending values.

32 Pipeline processing using back substitution Slide 186 P 5 Processes P 4 P 3 P 2 Final computed value P 1 P 0 First value passed onward Time

Overview: Parallelisation via Pipelining

Overview: Parallelisation via Pipelining three type of pipelines adding numbers (type ) performance analysis of pipelines insertion sort (type ) linear system back substitution (type ) Ref: chapter : Wilkinson