II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 153
entry with a different key, some other vacant position in the table must
be found. It is quite usual to search for such a vacant position in the next
following locations of the table; but when the table is nearly full, this may
cause undesirable bunching around an area of the table which happens to
be popular. A solution to this problem is to choose N + 1 as a prime number,
and to use a second hashing function to compute an arbitrary step length
from any given key. The next position to try when any given position is full
is obtained by adding the step length (modulo N + 1) to the previous
position.
10.1.3. lndexed Representation
The tabular method of storage is suitable only when the whole table can be
accommodated in the main store of the computer. In the common case
when this is not possible, a mixture of the tabular and sequential methods is
often used. In this a sparse array is represented as a table, each of whose
entries is a sequence:
(default: R; table:array 1.. N of
(max:D; seq: sequence (key:D; information:R))).
Every entry is placed on that sequence i such that its key falls between
table [i -1]. max (or D.min if i = 1) and table [i].max. The table is sorted
so that the appropriate sequence can be quickly located. This technique
may be likened to the organisation of a multivolume encyclopaedia, in
which the keys of the first and last entries of each volume are indicated on
the spine, so that the right volume can be quickly identified, without extracting
the volumes from the shelf.
When using this representation, it is desirable to ensure that all sequences
are of roughly the same length. Indeed, if disc backing store is used, it is
very advantageous to ensure that each of them is fitted onto a single cylinder,
so that a random access will not involve more than a single head movement.
Thus, when one sequence gets too long, it must exchange material with the
adjacent sequence. This involves extracting the entries with the largest
and/or smallest keys, and is best done when all the sequences are sorted into
order of key-value. The sorting and reshuffling is often carried out as a
separate operation at regular intervals; and the general method of file
organisation is known as "indexed sequential".
Naturally in this method of representation, it is an advantage to keep
the sequences as short as possible, say less than a single track on disk.
Consequently, the table itself may get so large that it will no longer fit in
main store. In this case the table itself is split up into sections, and a second-
level table may be set up to point to its sections, using the same principle
again. Thus at least two accesses to backing store will in general be required
for each access to an element of the array, and it is strongly recommended
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