Distributional Analysis

TL;DR

Distributional analysis evaluates algorithm performance based on average-case scenarios with respect to specific input distributions, highlighting its benefits and limitations for designing robust algorithms.

Contribution

This chapter discusses the advantages and drawbacks of distributional analysis, illustrating its use and limitations through various examples and setting the stage for hybrid analysis methods.

Findings

Distributional analysis can optimize algorithms for specific input distributions.

Pure distributional analysis may lead to overfitting solutions to assumptions.

Hybrid worst- and average-case analysis offers a more robust framework.

Abstract

In distributional or average-case analysis, the goal is to design an algorithm with good-on-average performance with respect to a specific probability distribution. Distributional analysis can be useful for the study of general-purpose algorithms on "non-pathological" inputs, and for the design of specialized algorithms in applications in which there is detailed understanding of the relevant input distribution. For some problems, however, pure distributional analysis encourages "overfitting" an algorithmic solution to a particular distributional assumption and a more robust analysis framework is called for. This chapter presents numerous examples of the pros and cons of distributional analysis, highlighting some of its greatest hits while also setting the stage for the hybrids of worst- and average-case analysis studied in later chapters.