Concrete mathematics : a foundation for computer science

8.5 HASHING 411
possible values of Y, and V(E(XlY)) is th e variance of this random variable
[Now is a good with respect to the probability distribution of Y. Similarly, E(V(XlY)) is the
time to do warmup
exercise 6.)
P is still the number
of probes.
average of the random variables V(Xly) as y varies. On the left of (8.105)
is VX, the unconditional variance of X. Since variances are nonnegative, we
always have
vx 3 V(EW’)) and VX 3 E(V(XlY)). (8.106)
Case 1, again: Unsuccessful search revisited.
Let’s bring our microscopic examination of hashing to a close by doing one
more calculation typical of algorithmic analysis. This time we’ll look more
closely at the total running time associated with an unsuccessful search,
assuming that the computer will insert the previously unknown key into its
memory.
The insertion process in (8.83) has two cases, depending on whether j is
negative or zero. We have j < 0 if and only if P = 0, since a negative value
comes from the FIRST entry of an empty list. Thus, if the list was previously
empty, we have P = 0 and we must set FIRSTC&+,l := n + 1. (The new
record will be inserted into position n + 1.) Otherwise we have P > 0 and we
must set a LINK entry to n + 1. These two cases may take different amounts
of time; therefore the total running time for an unsuccessful search has the
form
T = a+pP$-6[P=O], (8.107)
where OL, fi, and 6 are constants that depend on the computer being used and
on the way in which hashing is encoded in that machine’s internal language.
It would be nice to know the mean and variance of T, since such information
is more relevant in practice than the mean and variance of P.
So far we have used probability generating functions only in connection
with random variables that take nonnegative integer values. But it turns out
that we can deal in essentially the same way with
Gx(z) = t Pr(w)zx(wi
wcn
when X is any real-valued random variable, because the essential characteristics
of X depend only on the behavior of Gx near z = 1, where powers of z are
well defined. For example, the running time (8.107) of an unsuccessful search
is a random variable, defined on the probability space of equally likely hash
values (h1,. . , , h,; h,+l ) with 1 6 hj 6 m; we can consider the series
GT(z) = &i f...f f
h, =l h,=l h,+,=l
Z”+PPlhl ,..., hn;hn+l)+6P(hl a...> hn;hn+l I=01