Question

If you have 5 distinct numbers, how many comparisons at most do you need to sort this using merge sort?

Answer 1

I find the question interesting, so I decided to explore it thoroughly (with a little experimentation in Python).

I downloaded mergesort.py from here and modified it to add a cmp argument for a comparator function. Then:

import collections
import itertools
import mergesort
import sys

class CountingComparator(object):
  def __init__(self):
    self.count = 0
  def __call__(self, a, b):
    self.count += 1
    return cmp(a, b)

ms_histo = collections.defaultdict(int)

for perm in itertools.permutations(range(int(sys.argv[1]))):
  cc = CountingComparator()
  lperm = list(perm)
  mergesort.mergesort(lperm, cmp=cc)
  ms_histo[cc.count] += 1

for c in sorted(ms_histo):
  print "%d %2d" % (c, ms_histo[c])

The resulting simple histogram (starting with a length of 4, as I did for developing and debugging this) is:

4  8
5 16

For the problem as posted, with a length of 5 instead of 4, I get:

5  4
6 20
7 48
8 48

and with a length of 6 (and a wider format;-):

7    8
8   56
9  176
10 288
11 192

Finally, with a length of 7 (and even wider format;-):

 9   16
10  128
11  480
12 1216
13 1920
14 1280

Surely some perfectly regular combinatorial formula lurks here, but I’m finding it difficult to gauge what it might be, either analytically or by poring over the numbers. Anybody’s got suggestions?