Property testing is concerned with the design of superfast algorithms for structural analysis of huge amounts of data, where by structural analysis we mean an analysis aimed at unveiling global features of the data. Examples include determining whether the data as a whole have some property or estimating some global parameter of the data. The focus is on properties and parameters that go beyond simple statistics of the type that refers to the frequency of occurrence of various local patterns. The algorithms are given direct access to items of a huge data set, and determine whether this data set has some predetermined (global) property or is far from having this property. Remarkably, this decision is made by accessing only a small portion of the data set.

In other words, property testing is concerned with the design of superfast algorithms for approximate decision making, where the decision refers to properties or parameters of huge objects. In particular, we seek algorithms that inspect only relatively small portions of the huge object. Such algorithms must be randomized and can provide only approximate answers. Indeed, two salient aspects of property testing are that (1) it studies algorithms that can read only parts of the input, and (2) it focuses on algorithms that solve “approximate decision” problems. Both aspects are quite puzzling: What can one do without even reading the entire input? What does approximate decision mean?

The answer is that these two aspects are indeed linked: Approximate decision means distinguishing objects that have some predetermined property (i.e., reside in some predetermined set) from objects that are “far” from having the property (i.e., are far from any object having the property), where the notion of distance employed here is the relative number of different symbols in the descriptions of the objects. Such approximate decisions may be valuable in settings in which an exact decision is infeasible or very expensive or just considerably more expensive than obtaining an approximate decision.

The point is that, in many cases, an approximate decision can be achieved by means of superfast randomized algorithms. One well-known example is the common practice of estimating various statistics by sampling, which can be cast as a small collection of approximate decision problems (with respect to some threshold values).