The $k$-Equal Problem

  • Authors:

  • Martin Aigner

  • Affiliations:

  • Freie Universität Berlin, Fachbereich Mathematik und Informatik Arnimallee 14, D-14195 Berlin (e-mail: aigner@math.fu-berlin.de)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2005

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Abstract

Suppose we are given $n$ coloured balls and an integer $k$ between 2 and $n$. How many colour-comparisons $Q(n,k)$ are needed to decide whether $k$ balls have the same colour? The corresponding problem when there is an (unknown) linear order with repetitions on the balls was solved asymptotically by Björner, Lovász and Yao, the complexity being \smash{$\theta (n\log\frac{2n}{k})$}. Here we give the exact answer for \smash{$k\frac{n}{2}: Q(n,k)=2n-k-1$}, and the order of magnitude for arbitrary \smash{$k:Q(n,k)=\theta(\frac{n^2}{k})$}.