The Asymptotic Behavior of Diameters in the Average

Abstract

In 1975 R. Ahlswede and G. Katona posed the following average distance problem ( Discrete Math. 17 (1977), 10): For every cardinality a ∈ {1, ..., 2 n } determine subsets A of {0, 1} n with # A = a, which have minimal average inner Hamming distance. Recently I. Althöfer and T. Sillke ( J. Combin. Theory Ser. B 56 (1992), 296-301) gave an exact solution of this problem for the central value a = 2 n − 1 . Here we present nearly optimal solutions for a = 2 λ n with 0 < λ < 1: Asymptotically it is not possible to do better than choosing A n = {( x 1, ..., x n )|∑ n t = 1 x t = ⌊α n⌋}, where λ = −αlog α − (1 − α) log(1 − α). Next we investigate the following more general problem, which occurs, for instance, in the construction of good write-efficient-memories (WEMs). Given any finite set M with an arbitrary cost function d: M × M → R , the corresponding sum type cost function d n : M n × M n → R is defined by d n (( x 1, ..., x n , ( y 1, ..., y n ) = ∑ n t = 1 d( x t , y t ). The task is to find sets A n , of a given cardinality, which minimize the average inner cost (1/(# A n ) 2)∑ a∈ A n ∑ a′∈ A n d n ( a, a′). We prove that asymptotically optimal sets can be constructed by using "mixed typical sequences" with at most two different local configurations. As a non-trivial example we look at the Hamming distance for M = {1, ..., m} with m ≥ 3.