II. Notes on Data Structuring * - Cornell University

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162 C. A. R. HOARE hallsize, k'integer, initially given; load:array student of powerset exam, initially given; type session = powerset exam; timetable:powerset session, initially { }; examcount: array exam of integer, initially all (0); incompat:array exam of powerset exam, initially constant ( { } ); function sessioncount (s: session): integer; begin sum: integer, initially 0; end; for e in s do sum: + examcount (e); sessioncount: -- sum remainin/~: powerset exam, initially exam. all; function suitable: session; begin bestsofar, trial: session; untried: powerset exam; end; e: exam; e from remainder; bestsofar: = {e }; trial: = {e}; untried: = remainder - trial - incompat (e); gensupersets; suitable: = bestsofar The following two procedures are local to suitable: procedure record; if sessioncount (bestsofar) < sessioncount (trial) then bestsofar: = trial; procedure gensupersets; begin e" exam; save 1, save 2"powerset exam; record; save 1 : = untried; if size (trial) < k then while untried ~ { } do begin e from untried; save 2: = untried A incompat (e); untried" - save; trial: v {e}; if sessioncount (trial) < hallsize then gensupersets;
end; end gensupersets; NOTES ON DATA STRUCTURING 163 untried: v save 2; trial: - (e) untried: = save 1 The main program is as follows: for st: student do for e in load (st) do begin examcount (e): + 1; incompat (e): v (load (st) - (e)) end; while remaining -¢ ( ) do end; print timetable begin s" session; s: = suitable; timetable: v {s); remaining: __ s Before spending any more efforl~ on developing this program, it would be advisable to subject it to a critical examination, to ensure that it will be successful. Now the most obvious reasons why the program might fail are: (1) The size of the timetable turns out to be unacceptably large; we have agreed that nothing can be done about this, until we know more about the data. (2) The amount of time taken to generate all trials at each step is excessive. This will be particularly serious when the remainder is still large at the beginning of the program, and if the untried set remains large on every recursion of gensupersets. The main way in which the untried set is reduced is by removing all exams incompatible with the trial. This suggests that we should always prefer to add first to the trial those exams which have the largest incompatible sets, so that untried is reduced as quickly as possible. Among sets equal according to this criterion, the exam with the largest examcount would be selected first. The exact weighting between these criteria . may have to be adjusted later in the light of experience; meanwhile, the simplest implementation of this policy is to presort the exams in accordance with the criterion, and implement e from untried by selecting the first member. If it turns out that this elementary strategy is insufficient we may have to artificially curtail the number of iterations of the loop in gensupersets. But we would probably need some practical experience in order to select a suitable strategy; and for the time being, let us hope it will not be necessary. i
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II. Notes on Data Structuring * - Cornell University

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