Sorry, I noticed, I made 2 small errors: 1. search inside array: log[(n/2)!] << log[(n/4)^(n/2)] = n/2 * log(n/4) !!! 2. move array: log[(n/4)!] << log[(n/8)^(n/4)] = n/4 * log(n/8) !!!
This is because (n-k) * k is always < (n/2)^2 and is equal only and only when k = n/2!!! (where k = 0::n) In this way: O(...) << 5/4 * n * log(n) -5/4 * n -1 Please note, that the approximations presented at 1 and 2 are very conservative: 10! / 5^10 = 0.4 12! / 6^12 = 0.2 14! / 7^14 = 0.13 So basically, the terms: log[(n/2)!] are always much less than n/2 * log(n/4); [<< means much less] For really big numbers, even my previous formula was correct, with n/4 instead of n/2. Andreas J. Guelzow wrote: > On Sat, 2007-10-02 at 19:49 +0200, Leonard Mada wrote: > > > >> - move array (with worst algorithm): log[(n/2)!] << log[(n/4)^(n/4)] = >> n/4 * log(n/4) >> > > for n=100 log[(n/2)!] > 140 > but n/4 * log(n/4) < 81 > > What exactly is << supposed to mean? > > >> so, even under the worst assumption, it will be less than n * log(n), >> > > assuming this is true > > >> and because log[(n/2-1)!] << log[(n/4)^(n/4)], there will indeed be a >> significant difference. I presume, that more realistically, this >> algorithm will be 2 times faster than the corresponding sort and uses >> only half the memory. >> > > "2 times faster" means nothing. You are counting artifical actions > assuming that a comparison is the same as moving a memory block. When > you write the real code a small constant factor can easily disappear. > And typically with complex algorithms it will disappear. > > Andreas > _______________________________________________ gnumeric-list mailing list [email protected] http://mail.gnome.org/mailman/listinfo/gnumeric-list
