Parallel Algorithms and Performance

Parallel algorithm design seeks computations that expose abundant parallelism with low communication, and performance analysis quantifies how well they exploit additional processors.

Definition

A parallel algorithm specifies how a problem is solved by multiple processors cooperating; its quality is judged by work (total operations) and depth (longest dependency chain), and by performance metrics such as speedup and efficiency that measure the benefit of using p processors.

Scope

This topic covers models for designing and analyzing parallel algorithms—most notably the PRAM and its work-depth analysis—the design of parallel primitives such as prefix sums (scan), reduction, and sorting, and the performance framework of speedup, efficiency, cost-optimality, isoefficiency, and the scaling laws of Amdahl and Gustafson. It provides the analytical foundation for reasoning about parallel programs across all the programming models.

Core questions

• How is the inherent parallelism of a problem characterized by its work and depth?
• How are speedup, efficiency, and scalability defined and measured?
• How does a problem's serial fraction or growth determine its parallel scalability?

Key theories

PRAM and work-depth analysis

The PRAM model idealizes a parallel machine with shared memory and synchronous steps, supporting analysis of an algorithm's work (total operations) and depth (critical-path length), which together bound its achievable parallel time.

Speedup, efficiency, and isoefficiency

Speedup measures how much faster p processors solve a problem, efficiency normalizes it per processor, and isoefficiency expresses how the problem size must grow with processor count to keep efficiency constant, capturing scalability.

Amdahl's and Gustafson's laws

Amdahl's law bounds fixed-size speedup by the serial fraction, while Gustafson's law observes that scaling the problem with the machine allows near-linear speedup, jointly framing strong versus weak scaling.

Clinical relevance

Work-depth analysis and scaling laws guide whether and how a computation should be parallelized and predict its behavior at scale, informing the design of everything from sorting and graph kernels to large simulations and machine-learning training.

History

Amdahl's 1967 argument and Gustafson's 1988 rebuttal framed the speedup debate; the PRAM model and the work-depth framework matured through the 1980s and were codified in JaJa's 1992 text and Grama and colleagues' analysis of scalability, giving parallel computing its theoretical core.

Debates

Strong scaling versus weak scaling

Amdahl's strong-scaling view (fixed problem, more processors) predicts diminishing returns, while Gustafson's weak-scaling view (grow the problem with the processors) predicts sustained speedup; which is the right measure depends on whether problem size or time-to-solution is fixed.

Key figures

• Joseph JaJa
• Gene Amdahl
• John Gustafson
• Vipin Kumar

Seminal works

• jaja1992
• amdahl1967
• gustafson1988

Frequently asked questions

What is the difference between the work and the depth of a parallel algorithm?

Work is the total number of operations performed across all processors, while depth (or span) is the length of the longest chain of dependent operations. Depth bounds the best possible parallel running time, and the work-to-depth ratio indicates how much parallelism is available.