I was wondering if problems for which sublinear time (in the input size) algorithms exist can be characterized as possessing specific properties. This includes sublinear time (e.g. property testing, an alternative notion of approximation for decision problems), sublinear space (e.g. sketching/streaming algorithms in which the Turing machine has a read-only tape, a sublinear working space, and a write-only output tape) and sublinear measurements (e.g. sparse recovery/compressive sensing). In particular, I am interested to such a characterization for both the framework of property testing algorithms and in the classical model of randomized and approximation algorithms.
For instance, the problems for which a dynamic programming solution exist exhibit optimal substructure and overlapping subproblems; those for which a greedy solution exists exhibit optimal substructure and the structure of a matroid. And so on. Any reference dealing with this topic is welcome.
With the exception of a few problems that admit a deterministic sub linear algorithm, almost all of the sublinear algorithms I have seen are randomized. Is there any specific complexity class related to problems admitting sublinear time algorithms ? If yes, is this class included in BPP or PCP?