Constructing a Gene Team Tree in Almost $O$$(n\; {\rm lg}\; n)$ Time

An important model of a conserved gene cluster is called the gene team model, in which a chromosome is defined to be a permutation of distinct genes and a gene team is defined to be a set of genes that appear in two or more species, with the distance between adjacent genes in the team for each chromosome always no more than a certain threshold
$\delta$
. A gene team tree is a succinct way to represent all gene teams for every possible value of
$\delta$
. The previous fastest algorithm for constructing a gene team tree of two chromosomes requires
$O(n\ {\rm lg}\ n \;{\rm lglg}\ n)$
time, which was given by Wang and Lin. Its bottleneck is a problem called the maximum-gap problem. In this paper, by presenting an improved algorithm for the maximum-gap problem, we reduce the upper bound of the gene team tree problem to
$O(n\ {\rm lg}\ n\ \alpha (n))$
. Since
$\alpha$
grows extremely slowly, this result is almost as efficient as the current best upper bound,
$O(n\ {\rm lg}\ n)$
, for finding the gene teams of a fixed
$\delta$
value. Our new algorithm is very efficient from both the theoretical and practical points of view. Wang and Lin's gene-team-tree algorithm can be extended to
$k$
chromosomes with complexity
$O(kn\ {\rm lg}\ n\ {\rm lglg}\ n)$
. Similarly, our improved algorithm for the maximum-gap problem reduces this running time to
$O(kn\ {\rm lg}\ n\ \alpha (n))$
. In addition, it also provides new upper bounds for the gene team tree problem on general sequences, in which multiple copies of the same gene are allowed.