Computer Science 320

Measuring Speedup

What Is Running Time?

• T(N, K) says that the running time T is a function of the problem size N and the number of processors K

• To determine speedup, we’ll always determine both Tseq(N, K) and Tpar(N, K), where K = 1 and K > 1, respectively

What Is Speed?

• Speed is the rate at which program runs can be done

• S(N, K) = 1 / T(N, K)

• Measured in program runs per second, rather than seconds per run

What Is Speedup?

• Speedup is the speed of a parallel version running on K processors relative to a sequential version running on one processor

• Speedup(N, K) = Spar(N, K) / Sseq(N, 1)

• Why seq in the denominator?

Speedup in Terms of Running Time

Speedup(N, K) = Tseq(N, K) / Tpar(N, K)

Ideally, the speedup should equal K, or a linear speedup

The real speedup of most algorithms is sublinear

What Is Efficiency?

Eff(N, K) = Speedup (N, K) / K

Usually a fraction < 1

Amdahl’s Law

• The sequential portion of a parallel program puts an upper bound on the efficiency it can achieve

• The sequential parts are usually run at startup and cleanup

What Is the Sequential Fraction?

• The sequential fraction F is the fraction of the code that must be run sequentially

• F * T(N, 1) is the running time of the sequential part on a single processor

• (1 – F) * T(N, 1) is the running time of the parallel part on a single processor

Amdahl’s Law

T(N, K) = F * T(N, K) + 1/K * (1 – F) * T(N, K)

Running time when the parallel part is equally divided among K processors

Speedup and Efficiency in Terms of Sequential Fraction

Speedup(N, K) = 1 / (F + (1 – F) / K)

Eff(N, K) = 1 / (K * F + 1 – F)

Consequences of Amdahl

• As K increases, the speedup approaches 1 / F

• As K goes to infinity, the efficiency approaches 0

Predicted Speedups

Predicted Efficiencies

A Normal Plot for up to 8 Processors

Experimentally Determining F

To calculate the sequntial fraction F from the running time measurements, we rearrange Amdahl’s Law to get

F = (K * T(N, K) - T(N, 1)) / (K * T(N, 1) - T(N, 1))

Called EDSF

If F is not a constant or EDSF versus K is not a horizontal line, something’s not right with the program

Measuring Running Times

• Not the same on each run with the same data set

• Take the minimum time of several runs, not the average time. Why?

Experimental Situation

• Close all unnecessary apps and prevent remote logins

• Tseq(N, 1) must be at least 60 seconds

• Run the sequential program 7 times on each data size N, and take the minimum time

Experimental Situation

• For each data size N and for each K from 1 up to the number of available processors, run Tpar(N, K) 7 times and take each minimum time

Running Time of Key Search

Speedup of Key Search

Efficiency of Key Search

EDSF of Key Search