  (abridged) We study high redshift structure formation and reionization in a
LCDM universe under the assumption that the spectral power index of primordial
density fluctuations is a function of length scale. We adopt a particular
formulation of the running spectral index (RSI) model as suggested by the
recent WMAP data. While early structure forms hierarchically in the RSI model,
the reduced power on small scales causes a considerable delay in the formation
epoch of low mass (~ 10^6 Msun) ``mini-halos'' compared to the LCDM model. The
extremely small number of gas clouds in the RSI model indicates that
reionization is initiated later than z<15, generally resulting in a smaller
total Thomson optical depth than in the LCDM model. By carrying out radiative
transfer calculations, we also study reionization by stellar populations formed
in galaxies. Even with a top-heavy intial mass function representing an early
population of massive stars and/or an extraordinarily high photon emission rate
from galaxies, the total optical depth can only be as large as tau ~ 0.1 for
reasonable models of early star-formation. The RSI model is thus in conflict
with the large Thomson optical depth inferred by the WMAP satellite.
