Concrete mathematics : a foundation for computer science

398 DISCRETE PROBABILITY
For example, suppose ,the keys are names, and suppose that there are
m = 4 lists based on the first letter of a name:
1, for ,4-F;
2, for G-L;
h(name) =
3, for M-R;
1 4, for !3-Z.
We start with four empty lists and with n = 0. If, say, the first record has
Nora as its key, we have h(Nora) = 3, so Nora becomes the key of the first
item in list 3. If the next two names are Glenn and Jim, they both go into
list 2. Now the tables in memory look like this:
FIRST[l] = -1, FIRST[2] = 2, FIRST [31 = 1, FIRST [41 = -1
KEY Cl1 = Nora, NEXT[l1 = 0;
KEY [21 = Glenn, NEXTC21 = 3;
KEY [31 = Jim, NEXTC31 = 0; n = 3.
(The values of DATA [ll , DATA[21, and DATAC31 are confidential and will not
be shown.) After 18 records have been inserted, the lists might contain the Let’s hear it for
names
the Concrete Math
students who sat in
list 1 list 2 list 3 list 4
the front rows and
lent their names to
Dianne Glenn Nora Scott
this experiment.
Ari Jim Mike Tina
Brian Jennifer Michael
Fran Joan Ray
Doug Jerry
Jean
Paula
and these names would appear intermixed in the KEY array with NEXT entries
to keep the lists effectively separate. If we now want to search for John, we
have to scan through the six names in list 2 (which happens to be the longest
list); but that’s not nearly as bad as looking at all 18 names.
Here’s a precise specification of the algorithm that searches for key K in
accordance with this scheme:
Hl Set i := h(K) and j := FIRSTCil.
H2 If j 6 0, stop. (The search was unsuccessful.)
H3 If KEY Cjl = K, stop. (The search was successful.)
H4 Set i := j, then set j := NEXTCi] and return to step H2. (We’ll try again.)
For example, to search for Jennifer in the example given, step Hl would set
i := 2 and j := 2; step H3 ,would find that Glenn # Jennifer; step H4 would
set j := 3; and step H3 would find Jim # Jennifer.
1 bet their parents
are glad about that.