II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 119
shift the required value into a standard position within the word. The
unwanted values of neighbouring elements of the array can then be masked
off. The method of selectively updating an element of a packed array is
even more laborious, since the new value must be inserted at the right position
within the word, without disturbing the values of the neighbouring elements.
The efficiency of both operations may be slightly increased if the number of
elements per word is an exact power of two, since then the integer division
of the subscript may be replaced by a shift to find the quotient, and a mask
to find the remainder. On some machines, further efficiency may be gained
if each element is stored in a single character position.
The minimal representation for an array is similar to that for a Cartesian
product, except that the multiplier of each element value is equal to the
cardinality of range type, raised to the power of the subscript value. The
process of selecting or updating a value of an element of an array stored in
minimal representation is even more laborious than that described above,
unless the cardinality of the range type is an exact power of two. It would
be prohibitive if the array were to stretch over more than one normal com-
puter word. For this reason, the minimal representation for arrays is of
mainly academic interest.
Standard Loose Packed Tight Pecked
'r ~ I~I~!
.
A [o] A~ !A[o] AB] A[2] ]
A [ ~ ]
A [ 2 ]
i A[2] A[32
iA[4] A[5] ~I IA[6]IA[711 )
A [3] iAE6] A[7] A[~]
A E4]
. . . . .
~,
A [5]
A [6]
A [7]
padding
(a) (b) (c)
FIG. 3. Representation of A" array 0.. 7 of T
When the domain of a finite mapping is itself a data structure, for example,
a Cartesian product, it is usual to represent this domain in the minimal
representation, so as to avoid allocation of unused storage space. For
example, the display page has a domain which is the Cartesian product of
the integer ranges I to 40 and 1 to 27. In the minimal representation, this
gives a range of integers between 0 and 40x 27 -1 = 1079. Consequently
1080 consecutive words are allocated to hold values of elements of the array.
In order to access any element in a given row and character position, it is
necessary first to construct a minimal representation for the subscript, in
the manner described in Section 4.2.
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