  The S=1/2 and S=1 two-dimensional quantum Heisenberg antiferromagnets on the
anisotropic dimerized square lattice are investigated by the quantum Monte
Carlo method. By finite-size-scaling analyses on the correlation lengths, the
ground-state phase diagram parametrized by strengths of the dimerization and of
the spatial anisotropy is determined much more accurately than the previous
works. It is confirmed that the quantum critical phenomena on the phase
boundaries belong to the same universality class as that of the classical
three-dimensional Heisenberg model. Furthermore, for S=1, we show that all the
spin-gapped phases, such as the Haldane and dimer phases, are adiabatically
connected in the extended parameter space, though they are classified into
different classes in terms of the string order parameter in the
one-dimensional, i.e., the zero-interchain-coupling, case.
