Number of Comparisons finding the median of 7 numbers [closed]

Question

I can find the median with 12 comparisons. But I want to know the minimum number of comparisons and how to do it.

Answer 1

Donald Knuth has a chapter on "Minimum-Comparison Selection" in Volume III of The Art of Computer Programming.

Knuth says, "no general method [of selection in the minimum number of comparisons] is evident as yet" but he gives some general methods that come close to the minimum.

Looking at Table 5.3.3–1, we can see that V₄(7) = 10 (that is, you can find the 4th largest of 7 items using at most 10 comparisons), and the algorithm ("found manually by trial and error") is given in the solution to exercise 5.3.3–10.

Answer 2

If you allow comparisons in parallel (a modern CPU will probably do this for you), you can use a sorting network to solve the problem in six steps.

Answer 3

For (7s) you know you're on the right track when you spot the flying robots (Hasse Diagram). There are two main cases to verify for 10 comps; other cases are not close to the limit. (7s) is close to a perfect puzzle, like E's Zebra Puzzle. Not too hard, but hard enough so only the disciplined will crack it on their first go.

Now (4s, 3), also with 7 numbers, is the equal of (7s). Here we assume that one of the numbers repeats three times (has multiplicity 3). I won't tell you what the answer (height of the optimal ternary dec. tree) is. Go find it fellas!