Concrete mathematics : a foundation for computer science

8.5 HASHING 399
After a successful search, the desired data D(K) appears in DATA [jl , as in
the previous algorithm. After an unsuccessful search, we can enter K and D(K)
in the table by doing the following operations:
n := n+l;
if j < 0 then FIRSTCil :=n else NEXT[il :=n;
KEYCn.1 := K; DATACnl := D(K); NEXT[n] := 0. (8.83)
Now the table will once again be up to date.
We hope to get lists of roughly equal length, because this will make the
task of searching about m times faster. The value of m is usually much greater
than 4, so a factor of l/m will be a significant improvement.
We don’t know in advance what keys will be present, but it is generally
possible to choose the hash function h so that we can consider h(K) to be a
random variable that is uniformly distributed between 1 and m, independent
of the hash values of other keys that are present. In such cases computing the
hash function is like rolling a die that has m faces. There’s a chance that all
the records will fall into the same list, just as there’s a chance that a die will
always turn up � ; but probability theory tells us that the lists will almost
always be pretty evenly balanced.
Analysis of Hashing: Introduction.
“Algorithmic analysis” is a branch of computer science that derives quantitative
information about the efficiency of computer methods. “Probabilistic
analysis of an algorithm” is the study of an algorithm’s running time, considered
as a random variable that depends on assumed characteristics of the
input data. Hashing is an especially good candidate for probabilistic analysis,
because it is an extremely efficient method on the average, even though its
worst case is too horrible to contemplate. (The worst case occurs when all
keys have the same hash value.) Indeed, a computer programmer who uses
hashing had better be a believer in probability theory.
Let P be the number of times step H3 is performed when the algorithm
above is used to carry out a search. (Each execution of H3 is called a “probe”
in the table.) If we know P, we know how often each step is performed,
depending on whether the search is successful or unsuccessful:
Step Unsuccessful search
Hl 1 time
H2 P + 1 times
H3 P times
H4 P times
Successful search
1 time
P times
P times
P - 1 times