Concrete mathematics : a foundation for computer science

[The value of s1
cancels out, so it
can be anything
but zero.)
2.2 SUMS AND RECURRENCES 27
(Notice that we’ve left the term for k = 0 out of this sum.) The sum of the
geometricseries2~‘+2~2+~~~+2~“=(~)’+(~)2+~~~+(~)nwillbederived
later in this chapter; it turns out to be 1 - (i )“. Hence T,, = 2”S, = 2” - 1.
We have converted T, to S, in this derivation by noticing that the recurrence
could be divided by 2n. This trick is a special case of a general
technique that can reduce virtually any recurrence of the form
a,T,, = bnTn-1 + cn (2.9)
to a sum. The idea is to multiply both sides by a summation factor, s,:
s,a,T,, = s,,bnTn-1 + snc,, .
This factor s, is cleverly chosen to make
s b
n n =h-1 an-l s
Then if we write S, = s,a,T,, we have a sum-recurrence,
Hence
Sn = Sn-1 +SnCn.
%I = socuT + t skck = s.lblTo + c skck ,
k=l k=l
and the solution to the original recurrence (2.9) is
1
T, = -
ha,
n
s,b,To + &Ck
k=l
(2.10)
For example, when n = 1 we get T, = (s~b,To +slcl)/slal = (b,To +cl)/al.
But how can we be clever enough to find the right s,? No problem: The
relation s,, = snPl anPI /b, can be unfolded to tell us that the fraction
a,- 1 a,-2.. . al
S
n = b,bnp,...bz ’
(2.11)
or any convenient constant multiple of this value, will be a suitable summation
factor. For example, the Tower of Hanoi recurrence has a,, = 1 and b, = 2;
the general method we’ve just derived says that sn = 2-” is a good thing to
multiply by, if we want to reduce the recurrence to a sum. We don’t need a
brilliant flash of inspiration to discover this multiplier.
We must be careful, as always, not to divide by zero. The summationfactor
method works whenever all the a’s and all the b’s are nonzero.