Concrete mathematics : a foundation for computer science

28 SUMS
Let’s apply these ideas to a recurrence that arises in the study of “quicksort,”
one of the most important methods for sorting data inside a computer. (Quicksort was
The average number of comparison steps made by quicksort when it is applied
to n items in random order satisfies the recurrence
invented bY H0arc
in 1962 [158].)
k=O
for n > 0.
(2.12)
Hmmm. This looks much scarier than the recurrences we’ve seen before; it
includes a sum over all previous values, and a division by n. Trying small
cases gives us some data (Cl = 2, Cl = 5, CX = T) but doesn’t do anything
to quell our fears.
We can, however, reduce the complexity of (2.12) systematically, by first
getting rid of the division and then getting rid of the 1 sign. The idea is to
multiply both sides by n, obtaining the relation
n-1
nC, =n2+n+2xCk, for n > 0;
k=O
hence, if we replace n by n - 1,
n-2
(n-l)cnpj = (n-1)2+(n-1)+2xck, forn-1 >O.
We can now subtract the second equation from the first, and the 1 sign
disappears:
k=O
nC, - (n - 1)&l = 2n + 2C,-1 , for n > 1.
It turns out that this relation also holds when n = 1, because Cl = 2. Therefore
the original recurrence for C, reduces to a much simpler one:
co = 0;
nC, = (n + 1 )C,-I + 2n, for n > 0.
Progress. We’re now in a position to apply a summation factor, since this
recurrence has the form of (2.9) with a, = n, b, = n + 1, and c, = 2n.
The general method described on the preceding page tells us to multiply the
recurrence through by some multiple of
a,._1 an-l. . . a1 (n-l).(n-2).....1 2
S n = b,b,-, . . b2 = (n+l).n...:3 = (n+l)n