  We carefully examine common measures of dynamical heterogeneity for a model
polymer melt and test how these scales compare with those hypothesized by the
Adam and Gibbs (AG) and random first-order transition (RFOT) theories of
relaxation in glass-forming liquids. To this end, we first analyze clusters of
highly mobile particles, the string-like collective motion of these mobile
particles, and clusters of relative low mobility. We show that the time scale
of the high-mobility clusters and strings is associated with a diffusive time
scale, while the low-mobility particles' time scale relates to a structural
relaxation time. We find a high degree of similarity in the geometrical
structure of these particle clusters. In particular, we show that the fractal
dimensions of these clusters are consistent with those of swollen branched
polymers or branched polymers with screened excluded-volume interactions. In
contrast, the fractal dimension of the strings crosses over from that of
self-avoiding walks for small strings, to simple random walks for longer, {more
strongly interacting, strings, corresponding to flexible polymers with screened
excluded-volume interactions. We examine the appropriateness of identifying the
size scales of either mobile particle clusters or strings with the size of
cooperatively rearranging regions (CRR) in the AG and RFOT theories. We find
that the string size appears to be the most consistent measure of CRR for both
the AG and RFOT models. Identifying strings or clusters with the "mosaic"
length of the RFOT model relaxes the conventional assumption that the "entropic
droplets" are compact. We also confirm the validity of the entropy formulation
of the AG theory, constraining the exponent values of the RFOT theory. This
constraint, together with the analysis of size scales, enables us to estimate
the characteristic exponents of RFOT.
