We study divide-and-conquer recursion of the form \begin{equation*}.


    = \alphaf(\lfloor \tfrac n2\rfloor)

    + \beta f(\lceil \tfrac n2\rceil)

    + \end{equation*} given g(n) \qquad(n\ge2), $g(n)$ and $f(1)$, where $\alpha,\beta\ge0$ is $ \alpha+\beta>0$; Such recurrences are common in computer algorithms, computational systems, combinatorial sequencing, and related fields of analysis. We show that the solution always satisfies the simple \emph{identity} \begin{equation*}.


    = n^{\log_2(\alpha+\beta)} P(\log_2n) – Q(n) \end{equation*} under optimal (iff) conditions for $g(n)$. Since $Q(n)$ has a small degree, the form is not only an identity, but also an asymptotic expansion. An explicit form of \emph{continuity} for the periodic function $P$ is provided along with some other smoothness properties. We show how our results can be easily applied to dozens of concrete examples gleaned from the literature, and how they can be extended in different directions. Our proof method is surprisingly simple and rudimentary, but leads to the most powerful type of result for all examples to which our theory applies.

    Source link


    Leave A Reply