Concrete mathematics : a foundation for computer science

400 DISCRETE PROBABILITY
Thus the main quantity that governs the running time of the search procedure
is the number of probes, P.
We can get a good mental picture of the algorithm by imagining that we
are keeping an address book that is organized in a special way, with room for
only one entry per page. On the cover of the book we note down the page
number for the first entry in each of m lists; each name K determines the list
h(K) that it belongs to. Every page inside the book refers to the successor
page in its list. The number of probes needed to find an address in such a
book is the number of pages we must consult.
If n items have been inserted, their positions in the table depend only
on their respective hash val.ues, (h’ , hz, . . . , &). Each of the m” possible
sequences (h’ , h2, . . . , &) is considered to be equally likely, and P is a random
variable depending on such a sequence.
Case 1: The key is not present. Check under the
Let’s consider first the behavior of P in an unsuccessful search, assuming doormat.
that n records have previously been inserted into the hash table. In this case
the relevant probability spac:e consists of mn+’ elementary events
w = (h’,hz,...,h,;hT,+‘)
where b is the hash value of the jth key inserted, and where &+’ is the hash
value of the key for which the search is unsuccessful. We assume that the
hash function h has been chosen properly so that Pr(w) = 1 /mnf’ for every
such CU.
For example, if m = n == 2, there are eight equally likely possibilities:
hl h2 h3: P
1 1 1:2
11 2:o
1 2 1:l
1 2 2: 1
2 1 1:l
2 1 2:l
2 2 1:o
2 2 212
If h’ = h2 = h3 we make two unsuccessful probes before concluding that the
new key K is not present; if h’ = h2 # h3 we make none; and so on. This list
of all possibilities shows that P has a probability distribution given by the pgf
(f + $2 + $2’) = (i + iz)‘, when m = n = 2.
An unsuccessful search makes one probe for every item in list number
hn+‘, so we have the general formula
P = [h, =hm+,l + [hz=hn+,l + ... + [h,,=hn+ll. (8.84)