II. Notes on Data Structuring * - Cornell University

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156 c.A.R. HOARE each student can attend the examination for each course that he has taken. This can always be arranged by allocating a separate session for each examination; but the interests of examiner and student alike dictate that the total examination period be as short as possible. This means that each session should contain as many examinations as possible, subject to some limit k. An additional constraint is imposed by the size of the examination hall, which can only accommodate a certain maximum number of students. Before designing the program, it is desirable to confirm our understanding of the problem by making a more rigorous formalisation in terms of the structure of the various items of data, both given and required. The types "student" and "exam" are obviously unstructured and need no further definition at this stage. The load of exams to be taken by each student is given by a mapping" load: array student of powerset exam. A timetable is a set of sessions, where each session consists of a set of exams: type session = powerset exam; timetable: powerset session. We next attempt to formalise the properties which the input and output data are required to possess. (1) We choose not to formalise the condition that the number of sessions be minimised, since in fact we do not want an absolute minimum if this turns out to be too expensive to compute. (2) Each exam is scheduled for one of the sessions s = exam.all s ia timetable (3) No exam is scheduled for more than one session: s l, s2 in timetable = s l ^ s2 = { } Conditions (2) and (3) effectively state that the timetable is a partitioning of the set of all exams into exhaustive and exclusive subsets. (4) No session includes more than k exams s in timetable ~ size (s) ~< k (5) No session involves more than hallsize students. To formalise this, we need to count the number of students taking each exam: examcount (e:exam) = size (st:studentle in load (st)}. Now the number of students involved in a session is session count (s:session) -- ~ examcount (e) eins
The condition may be formalised: NOTES ON DATA STRUCTURING 157 s in timetable ~ sessioncount (s) ~< hallsize. (6) No student takes more than one exam in a session. To formalise this we introduce the concept of incompatibility of exams: two exams are incompatible if some student is taking both of them. For each exam el there is a set incompat (el) of exams which are incompatible with it: incompat (el) = {e2:exam I e2 -¢- el & 3 st:student (el in load (st) & e2 in load (st))} Now we can define that every pair of exams in a session must be compatible: s in timetable & el, e2 in s D -l el in incompat (e2). These six conditions, defined in terms of load, hallsize, and k, must be possessed by any successful timetable in the real world, and by any successful computer representation of the timetable. They serve to define the objectives and criterion of correctness of our timetabling program. 11.1 THE ABSTRACT PROGRAM Inspection of the conditions reveals that construction of the timetable does not require full knowledge of the load of each student. All that is needed is the examcount of each exam, and for each exam the set of other exams which are incompatible with it: examcount: array exam of integer; incompat: array exam of powerset exam. These two arrays embody an abstraction from the real life data, which concentrate attention on exactly those features which are for the present purpose relevant, and permitting us to ignore for the time being the other features of the situation. It is plain that these two arrays can be readily constructed from a single scan of the student load data: examcount: = all (0); incompat:= all ( { } ); for st: student do for e in load (st) do begin examcount (e)" + 1; end; incompat (e): v (load (st) - {e}) One of the simplifying factors in the search for a solution to the given problem is that the conditions fall readily into two classes: (1) (2) and (3) relate to the timetable as a whole, whereas (4) (5) and (6) relate only to
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II. Notes on Data Structuring * - Cornell University

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