Analysis of a Recurrence Arising from a Construction for Nonblocking Networks

Define f on the integers n > 1 by the recurrence f(n) mAn{n, minm[ 2f(m) + 3f(n/m)}. The function f has f(n) n as its upper envelope, attained for all prime n. The goal of this paper is to determine the corresponding lower envelope. It is shown that this has the form f(n) C(logn)1+1/ for certain constants and C, in the sense that for any > 0, the inequality f(n) <_ (C+)(logn) 1+1/’ holds for infinitely many n, while f(n) _ (C-)(logn)1+1/ holds for only finitely many. In fact, -y 0.7878... is the unique real solution of the equation 2+ 31, and C 1.5595... is given by the expression C ((2log2 + 3log3)1/)/(( + 1)(15log+1 5 log+l kl 1/) 3-Eh 1 with the reals x > 1 in the above recurrence: fo(x) min{x, infl<y<x 2f0(y) + 3fo(x/y)}. The author shows that fo(x) Co(logx) 1+1/, where Co 1.5586... is given by Co 6e (2log2+ 3-’ log 3-)1/ (-/(--t1)) 1+1/ and is smaller than C by a factor of 0.9994 Key words, asymptotic analysis, recurrence relation AMS subject classification. 26A12