Concrete mathematics : a foundation for computer science

Somehow the verb
“to hash” magically
became standard
terminology for key
transformation doring
the mid-l 96Os,
yet nobody was rash
enough to use such
an undignified word
publicly until 1967.
-D. E. Knoth [I 75)
8.5 HASHING
8.5 HASHING 397
Let’s conclude this chapter by applying probability theory to computer
programming. Several important algorithms for storing and retrieving
information inside a computer are based on a technique called “hashing!’
The general problem is to maintain a set of records that each contain a “key”
value, K, and some data D(K) about that key; we want to be able to find
D(K) quickly when K is given. For example, each key might be the name of
a student, and the associated data might be that student’s homework grades.
In practice, computers don’t have enough capacity to set aside one memory
cell for every possible key; billions of keys are possible, but comparatively
few keys are actually present in any one application. One solution to the
problem is to maintain two tables KEY [jl and DATACjl for 1 6 j 6 N, where
N is the total number of records that can be accommodated; another variable
n tells how many records are actually present. Then we can search for a
given key K by going through the table sequentially in an obvious way:
Sl Set j := 1. (We’ve searched through all positions < j.)
S2 If j > n, stop. (The search was unsuccessful.)
S3 If KEY Cjl = K, stop. (The search was successful.)
S4 Increase j by 1 and return to step S2. (We’ll try again.)
After a successful search, the desired data entry D(K) appears in DATACjl.
After an unsuccessful search, we can insert K and D(K) into the table by
setting
n := j, KEY Cnl := K, DATACnl := D(K),
assuming that the table was not already filled to capacity.
This method works, but it can be dreadfully slow; we need to repeat
step S2 a total of n + 1 times whenever an unsuccessful search is made, and
n can be quite large.
Hashing was invented to speed things up. The basic idea, in one of its
popular forms, is to use m separate lists instead of one giant list. A “hash
function” transforms every possible key K into a list number h(K) between 1
and m. An auxiliary table FIRSTCil for 1 6 i 6 m points to the first record
in list i; another auxiliary table NEXTCjl for 1 < j 6 N points to the record
following record j in its list. We assume that
FIRSTCi] = -1, if list i is empty;
NEXT[jl = 0, if record j is the last in its list.
As before, there’s a variable n that tells how many records have been stored
altogether.