  Grover's algorithm provides a quadratic speedup over classical algorithms to
search for marked elements in an unstructured database. The original algorithm
is probabilistic, returning a marked element with bounded error. There are
several schemes to achieve the deterministic version, by using the generalized
Grover's iteration $G(\alpha,\beta):=S_r(\beta)\, S_o(\alpha)$ composed of
phase oracle $S_o(\alpha)$ and phase rotation $S_r(\beta)$. However, in all the
existing schemes the value range of $\alpha$ and $\beta$ is limited; for
instance, in the three early schemes $\alpha$ and $\beta$ are determined by the
proportion of marked states $M/N$. In this paper, we break through this
limitation by presenting a search framework with adjustable parameters, which
allows $\alpha$ or $\beta$ to be arbitrarily given. The significance of the
framework lies not only in the expansion of mathematical form, but also in its
application value, as we present two disparate problems which we are able to
solve deterministically using the proposed framework, whereas previous schemes
are ineffective.
