Question

One of the things we've already had to do is pick out electives for sixth grade. Basically, students rank a (long) set of electives in order of preference. But suppose for a moment that preferences are instead only provided in pairs. For instance, a student can specify that he or she "prefers" one elective to another. A given elective can be part of multiple pairwise preferences, but only two electives can be mentioned in any pair. So my kid might have specified the following: · I prefer dance over astronom » I prefer photography over dance » I prefer band over astronom o I prefer photography over band Assuming there are n electives, a student can specify up to n(n-1) such preferences. And a student can specify as few as zero preferences The school wants to convert the pairwise preferences into a partially ordered list. The most preferred electives should be at the top of the list. More concretely, an elective i should not be below some other elective j if the student has a preference for i over j. (For example. in my daughter's list, dance should never be below astronomy.) Your goal for this problem is to write an efficient algorithm that either (a) generates a partially ordered elective list from a set of preferences or (b) outputs that such an ordering is not possible. When more than one partially ordered list is possible, your algorithm can output any such list that respects the student's preferences. Your algorithm should be as efficient as possible. Give a tight asymptotic upper bound on the running time of your algorithm, and provide a brief justification for it. Full credit will be given for the asymptotically most efficient algorithm. Partial credit will be given for less efficient algorithms

Answer 1

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