Concrete mathematics : a foundation for computer science

8.5 HASHING 409
When such “coincidences” occur, we suspect that there’s a mathematical
reason; there might be another way to attack the problem, explaining why
the answer has such a simple form. And indeed, there is another approach (in
exercise 60), which shows that the variance of the average successful search
has the general form
VA =
k=l
(8.103)
when sk is the probability that the kth-inserted element is being sought.
Equation (8.102) is the special case sk = l/n for 1 < k 6 n.
Besides the variance of the average, we might also consider the average of
the variance. In other words, each sequence (hl , . . . , hn) that defines a hash
table also defines a probability distribution for successful searching, and the
variance of this probability distribution tells how spread out the number of
probes will be in different successful searches. For example, let’s go back to
Where have 1 seen the case where we inserted n = 16 things into m = 10 lists:
that pattern before?
Where have 1 seen h , . . . ,h16)=3 141592653589793
that grafito before?
IqvP, .
Pl, * *.,P,6)=1112111122312133
A successful search in the resulting hash table has the pgf
G(3,1,4,1,... ,3) = f SkZP(3,1,4,1,...,3;k)
k=l
= SlZ+S2Z+S3Z+S4Z2+...+S~~Z3.
We have just considered the average number of probes in a successful search
of this table, namely A(3,1,4,1,. . . ,3) = Mean(G(3,1,4,1,. . . ,3)). We can
also consider the variance,
This variance is a random variable, depending on (hl , . . . , h,), so it is natural
to consider its average value.
In other words, there are three natural kinds of variance that we may
wish to know, in order to understand the behavior of a successful search: The
overuZZ variance of the number of probes, taken over all (h1,. , . , h,,) and k;
the variance of the average number of probes, where the average is taken
over all k and the variance is then taken over all (h, , . . . , h,,); and the average
of the variance of the number of the probes, where the variance is taken over