II. Notes on Data Structuring * - Cornell University

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164 c.A.R. HOARE 11.2. DATA REPRESENTATION In order to design a successful data representation, it is necessary to know something of the likely size of the problem. In this example, we will make the following assumptions: (1) There are not more than 500 exams, each taken by less than 1000 students (typically 50). (2) There are about 5000 students. (3) Each student takes less than ten exams, and typically five. (4) The examination hall will take about 1000 students. (5) An acceptable limit on the number of concurrent exams is 30, and the typical number is 10. (6) Manual timetabling methods have succeeded in constructing timetables with not more than 50 sessions. We will consider the individual items of data. (1) type exam The obvious representation is as an integer subrange:0.. 500. (2) type session There is obviously a choice between a bitpattern representation (500 bits), and an array of 30 nine-bit elements (+ pointer) (270 bits + one word). The number of sessions to be stored is not great, so considerations of storage economy are not significant. The main operations on the session are the insertion of an exam which is known not to be in it already, and the removal of an exam, which is the most recently inserted. Thus the array method would be the best, since the insertion and removal of members can be accomplished by stack methods. Since we frequently wish to know the session-count, it would pay to record this together with the session, and keep it up to date as members are inserted and removed. This representation is used for trial and bestsofar. (3) t#netable The only operation on the timetable is the insertion of new sessions. Since sessions are of variable length, the timetable could be organised as a sequence of variable-length sequences. Since each exam occurs exactly once in the timetable, the maximum size of the timetable is 500 x nine bits, plus perhaps sixty words to indicate the separation of the sessions (if there are more than sixty sessions, the program will have failed anyway).
NOTES ON DATA STRUCTURING 165 An alternative and much simpler representation is simply to record for each exam which session it occurs in. This requires only array exam of 1.. 60 This representation is made possible only by the fact that the sessions of the timetable are mutually exclusive. (4) examcount : array exam of integer A standard representation is the obvious choice. (5) remaining, untried, save 1, save 2 These variables start rather full, and get emptier as the program progresses. Their average density is therefore about fifty percent, and there is no point in adopting a sparse representation. Furthermore, the frequency of standard set operations applied to them indicate a standard bitpattern representation. (6) incompat The most frequent use of elements of incompat is to subtract them from untried. They should therefore also use the bitpattern representation. This will require 500 x 500 bits, of the order of 10000 words. This is by far the largest data structure required, but its total size is probably fully justified by the extra speed which it imparts to the program, and since it is acceptable on most computers on which this program will run, it does not seem worth while to seek a more compact representation. (7) load The load of each student is the primary input data for the problem; it may also be extremely voluminous. It is therefore doubly fortunate that the program only needs to make a single scan of the data; for not only will this enable the data to be presented as an external sequence; it also means that the representation can be designed to be suitable for human reading, writing, and punching. We therefore allocate one card for each student, and use ten columns of six characters each to hold the examination numbers. To save unnecessary punching, the first blank column will signify the end of the examination set. For identification purposes, each card should also contain the student number; fortunately this can be wholly ignored by the program, though it should probably be checked to avoid duplications or omissions. Exercise Code the abstract program described above using the recommended data representations.
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II. Notes on Data Structuring * - Cornell University

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