Numerous and narrow absorption lines in quasar spectra, the so-called Lyman alpha forest, can be a unique probe to explore the early universe. In addition, several detections of an absorption trough in the spectra have placed stringent constraints on the physical properties of the intergalactic medium at high redshifts. They would play an important role in relation to the evolution of the UV background radiation field. Focusing on the mutual relations between the quasar absorption system and the UV background radiation field, we examine the evolution of minihalos, gas clouds confined by the gravity of cold dark matter, and explore reasonable possibilities for minihalos to reproduce the redshift distribution of Lyalpha clouds at 0 <~ z <~ 4. We can find that the redshift distribution of minihalos shows good agreement with the observations over the whole range of redshifts 0 <~ z <~ 4 under two alternative cases: first, the UV background radiation has a peak intensity at z ~ 1; second, the Lyalpha forest has a comoving number density that decreases exponentially with a characteristic timescale, tau ~ a few Gyr.