Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 951 — #959
22.4. Divide-and-Conquer Recurrences 951
In particular, T .n/ is not a linear combination of a fixed number of immediately
preceding terms; rather, T .n/ is a function of T .n=2/, a term halfway back in the
sequence.
Merge Sort is an example of a divide-and-conquer algorithm: it divides the input,
“conquers” the pieces, and combines the results. Analysis of such algorithms
commonly leads to divide-and-conquer recurrences, which have this form:
T .n/ D
kX
a i T .b i n/ C g.n/
iD1
Here a 1 ; : : : a k are positive constants, b 1 ; : : : ; b k are constants between 0 and 1,
and g.n/ is a nonnegative function. For example, setting a 1 D 2, b 1 D 1=2 and
g.n/ D n 1 gives the Merge Sort recurrence.
22.4.1 The Akra-Bazzi Formula
The solution to virtually all divide and conquer solutions is given by the amazing
Akra-Bazzi formula. Quite simply, the asymptotic solution to the general divideand-conquer
recurrence
is
where p satisfies
T .n/ D
kX
a i T .b i n/ C g.n/
iD1
Z n
T .n/ D ‚
n p g.u/
1 C
1 u pC1 du
(22.2)
kX
a i b p i
D 1: (22.3)
iD1
A rarely-troublesome requirement is that the function g.n/ must not grow or
oscillate too quickly. Specifically, jg 0 .n/j must be bounded by some polynomial.
So, for example, the Akra-Bazzi formula is valid when g.n/ D x 2 log n, but not
when g.n/ D 2 n .
Let’s solve the Merge Sort recurrence again, using the Akra-Bazzi formula instead
of plug-and-chug. First, we find the value p that satisfies
2 .1=2/ p D 1: