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 δ. A gene team tree is a succinct way to represent all gene teams for every possible value of δ. The previous fastest algorithm for constructing a gene team tree of two chromosomes requires O(n lg n 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 lg n α(n)). Since α grows extremely slowly, this result is almost as efficient as the current best upper bound, O(n lg n), for finding the gene teams of a fixed δ 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 lg n lglg n). Similarly, our improved algorithm for the maximum-gap problem reduces this running time to O(kn lg n α(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.