Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications
Divide-and-conquer recurrences of the form <i>f</i>(<i>n</i>) = <i>f</i> (⌊ &fracn2;⌋ ) + <i>f</i> ( ⌈ &fracn2;⌉ ) + <i>g</i>(<i>n</i>) (<i>n</i>⩾ 2), with <i>g</i>(<i>n</i>) and <i>f</i>(1) given, appear very frequently in the analysis of computer algorithms and related areas. While most previous methods and results focus on simpler crude approximation to the solution, we show that the solution always satisfies the simple <i>identity</i> <i>f</i>(<i>n</i>) = <i>n</i> <i>P</i>(log<sub>2</sub><i>n</i>) − <i>Q</i>(<i>n</i>) under an optimum (iff) condition on <i>g</i>(<i>n</i>). This form is not only an identity but also an asymptotic expansion because <i>Q</i>(<i>n</i>) is of a smaller order than linearity. Explicit forms for the <i>continuous</i> periodic function <i>P</i> are provided. We show how our results can be easily applied to many dozens of concrete examples collected from the literature and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary but leads to the strongest types of results for all examples to which our theory applies.