Parallel Algorithms. Parallel Algorithms

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1 Parallel Algorithms Parallel Algorithms Goals: Speedup and Efficiency Speedup, in general, is limited linearly to the number of processors (P) used. When can we epect to have linear speedup? Fully (embarrassingly) parallel programs where calculation at each data point is independent of calculations at other data points. Not likely 1

2 Parallel Algorithms Problem Domain: N, the (number of) data elements to which the algorithm is being applied Problem Decomposition: Assignment of data elements to each processor The Problem Height K() or Ψ(.0) Ψ(.t) String anchored at both ends Initial plucking of String Shape of String at time t The problem: Solve the 1-D Wave Equation Fo etal.: Solving Problems on Concurrent Processors, page

3 The Wave Equation Height K() or Ψ(,0) Ψ(.t) String anchored at both ends Initial plucking of string Shape of string at time t In general, modeled with the hyperbolic differential equation: 1 δ 2 ψ(,t) δ 2 ψ(,t) - = 0 c 2 δ 2 t 2 δ 2 2 where Ψ(.t) is the vibration amplitude, representing the activity in the system and c is a material constant. The Wave Equation Height K() or Ψ(,0) Ψ(.t) String anchored at both ends Initial plucking of string Shape of string at time t Initial Conditions: Ψ(,0) = K(), the function describing the plucking of the string at time 0. Boundary Conditions: Ψ(0,t) = Ψ(L,t) = 0, i.e., the string is anchored at each end. 3

4 A Numerical Solution Note: we can construct a numerical solution using difference equations to solve the previous equation when y is sufficiently small df dy = limit y 0 f(y + y) - f(y) y d 2 f dy = limit 2 y 0 f(y - y) - 2 f(y) + f(y+ y) ( y) 2 The Numerical Solution Ψ(.t) L = N-1 = 0 +(N-1)*d Assume discrete steps in both and t (, t) Construct the difference equation: 1 ψ( i, t- t) - 2 ψ( i, t) + ψ( i, t+ t) ψ( i-1, t) - 2 ψ( i, t) + ψ( i+1, t) - = 0 c 2 ( t) 2 ( ) 2 4

5 The Numerical Solution If all ψ s are known for all i up to current time t, can solve the previous equation for ψ( i, t+ t): t ψ( i, t+ t)=2(1- c 2 ( ) 2 ) ψ( i, t) - ψ( i, t- t) +c 2 ( ) 2 (ψ( i-1, t) + ψ( i+1, t)) Using appropriate substitutions, this simplifies to ψ( i, t+ t) = A ψ( i, t) + B ψ( i, t- t) + C (ψ( i-1, t) + ψ( i+1, t)) t Algorithm P = N Ψ(.t) L = N-1 = 0 +(N-1)*d The algorithm for finding i is Echange info with neighboring processors Calculate ψ( i, t+ t) The problem becomes more interesting if N >> P 5

6 The Wave Equation Problem Domain: The physical etent of the string and the displacement Ψ(.t) which is a function of the distance along the string and the time t. Problem Decomposition: The choice of grid points ( s) assigned to the various processors Decomposition of string into contiguous sections The Algorithm when P << N? i*k P i (i+1)*k-1 How should we decompose the problem? What is the algorithm for each processor? How long does it take to calculate Ψ(.t) for all i, where (0 <= i <= N-1) each time step? What part does communication play? Fractional communication overhead (fc) = communication time/calculation time What is the speedup? What assumptions? How would a 2D version work? 6

7 1D Wave Equation in Parallais (* This program solves the wave equation for the case where there are many more points than processors. *) MODULE WaveEquation; CONST NumberOfSecs = 2; NumberOfStepsPerSec = 100; NumberOfPoints = 80; NumberOfProcessors = 5; PrintOutsPerSec = 10; NumberOfSteps = NumberOfSecs * NumberOfStepsPerSec; PointsPerProcessor = NumberOfPoints DIV NumberOfProcessors; (* Assume divides evenly *) c = 1.0; d = 1.0 / float( NumberOfPoints ); dt = 1.0 / float( NumberOfStepsPerSec ); 1D Wave Equation in Parallais(2) CONFIGURATION Processor[ 1..NumberOfProcessors ]; CONNECTION right: Processor[j] <-> Processor[j+1].left; VAR step, count, p : INTEGER; ScalarAmplitude : ARRAY[1..PointsPerProcessor],[1..NumberOfProcessors] OF REAL; : REAL; previous, current, net : Processor OF INTEGER; i : Processor OF INTEGER; amplitude : Processor OF ARRAY[0..2], [0..PointsPerProcessor+1] OF REAL; a, b, d, j : Processor OF REAL; 7

8 1D Wave Equation in Parallais(3) BEGIN previous := 0; current := 1; net := 2; a := (2.0 * (c * dt / d) * (c * dt / d)); b := -1.0; d := (c * dt / d) * (c * dt / d); 1D Wave Equation in Parallais(4) FOR i := 0 TO (PointsPerProcessor+1) DO j := float(((dim1-1) * PointsPerProcessor) + i) / float( NumberOfPoints); IF j <= 0.5 THEN amplitude[0][i] := j; ELSE amplitude[0][i] := j; END; (* IF *) amplitude[1][i] := amplitude[0][i]; END; (* FOR *) IF dim1 = 1 THEN amplitude[0][0] := 0.0; amplitude[1][0] := 0.0; amplitude[2][0] := 0.0 ELSIF dim1 = NumberOfProcessors THEN amplitude[0][pointsperprocessor+1] := 0.0; amplitude[1][pointsperprocessor+1] := 0.0; amplitude[2][pointsperprocessor+1] := 0.0; END; (* IF *) 8

9 1D Wave Equation in Parallais(5) FOR count := 1 TO PointsPerProcessor DO STORE( amplitude[current][count], ScalarAmplitude[count] ); END; (* FOR *) FOR p := 1 TO NumberOfProcessors DO FOR count := 1 TO PointsPerProcessor DO WriteReal( ScalarAmplitude[count][p], 5 ); Write( ' '); END; (* FOR count *) END; (* FOR p *) WriteLn; 1D Wave Equation in Parallais(6) FOR Step := 1 TO NumberOfSteps DO SEND.right( amplitude[current][pointsperprocessor], amplitude[current][0] ); SEND.left( amplitude[current][1], amplitude[current][pointsperprocessor+1] ); IF dim1 = 1 THEN amplitude[current][0] := 0.0; ELSIF dim1 = NumberOfProcessors THEN amplitude[current][pointsperprocessor+1] := 0.0; END; (* IF *) FOR i := 1 TO PointsPerProcessor DO amplitude[net][i] := a * amplitude[current][i] + b * amplitude[previous][i] + d * (amplitude[current][i-1] + amplitude[current][i+1]); END; (* FOR i *) previous := current; current := net; net := (net + 1) MOD 3; 9

10 1D Wave Equation in Parallais(7) IF (Step MOD(NumberOfStepsPerSec DIV PrintOutsPerSec)) = 0 THEN FOR count := 1 TO PointsPerProcessor DO STORE( amplitude[current][count], ScalarAmplitude[count] ); END; (* FOR *) FOR p := 1 TO NumberOfProcessors DO FOR count := 1 TO PointsPerProcessor DO WriteReal( ScalarAmplitude[count][p], 5 ); Write( ' '); END; (* FOR count *) END; (* FOR p *) WriteLn; WriteLn; END; (* IF *) END; (* FOR Step *) END WaveEquation. 10

The Vibrating String

CS 789 Multiprocessor Programming The Vibrating String School of Computer Science Howard Hughes College of Engineering University of Nevada, Las Vegas (c) Matt Pedersen, 200 P P2 P3 P4 The One-Dimensional