Conclusion
Over the next few articles, I'll discuss ways to apply O(N) and O(K) techniques, and ways to not only cheat Amdahl, but beat him. Shameless teaser: How can you use N cores to get an answer more than N times faster? Normally, getting just N-fold speedups is considered the Holy Grail. Hint: Think about ways you might leverage data locality and/or perform speculative and cancellable execution to set yourself up for such superlinear speedups of p. Then read next month's column.
Cheat Amdahl by doing more of the same: Solve bigger cases of the same problem. As Gustafson concluded: "Speedup should be measured by scaling the problem to the number of processors, not fixing problem size. We expect to extend our success to a broader range of applications and even larger values for N." (To put "even larger" in context, when Gustafson wrote that he was already reporting repeatable successes with N = 1024 processors.)
Cheat Amdahl by finding new O(N) work to do that was once unthinkable. Cheat him by O(K)-parallelizing the apparently "sequential" parts of your code.
To paraphrase Churchill: We shall cheat him in the data sizes, we shall cheat him in the loops, we shall cheat him in the pipelines and in the caches, we shall cheat him in the unimagined new features. We shall never surrender.
Notes
[1] For more on O(1), O(K), and O(N) concurrency, see my column "How Much Scalability Do You Have or Need?" (DDJ, September 2007), available online at http://www.ddj.com/hpc-high-performance- computing/201202924. Briefly, O(x) means that the code has about x things it could be running concurrently at a given time, where O(1) is sequential, O(K) is code that hardwires a predetermined fixed amount of concurrency that prefers K cores and can't use >K cores, and O(N) is scalable across variable numbers of cores.
[2] Gene Amdahl. "Validity of the Single Processor Approach to Achieving Large-Scale Computing Capabilities" (AFIPS Conference Proceedings, Vol. 30, AFIPS Press, April 1967).
[3] John Gustafson. "Reevaluating Amdahl's Law" (Communications of the ACM, 31(5), 1988).
Herb is a software architect at Microsoft and chair of the ISO C++ Standards committee. He can be contacted at www.gotw.ca.
