Multidimensional Divide-and-Conquer Maximin Recurrences

Abstract

Bounds are obtained for the solution to the divide-and-conquer recurrence \[ M( n ) = \max \limits_{k_1 + \cdots + k_p = n} ( M( k_1 ) + M( k_2 ) + \cdots + M( k_p )+ \min ( f ( k_1 ), \cdots ,f ( k_p ) ) ), \] for nondecreasing functions f. Similar bounds are found for the recurrence with “min” replaced by “sum-of-all-but-the-max.” Such recurrences appear in the analysis of various algorithms. The bounds follow from analyses of partition trees.