Gossip is synteny: incomplete gossip and an exact algorithm for syntenic distance

The <i>syntenic distance</i> between two genomes is given by the minimum number of fusions, fissions, and translocations required to transform one into the other, ignoring the order of genes within chromosomes. The problem of computing this distance is NP-complete. In this paper, we give an <i>&Ogr;</i>(2<sup><i>&Ogr;</i>(<i>n</i> log <i>n</i>)</sup>) algorithm to exactly compute the syntenic distance between two genomes that contain at most <i>n</i> chromosomes. Our algorithm requires <i>&Ogr;</i>(2<sup><i>&Ogr;</i>(<i>d</i> log <i>d</i>)</sup>) time when this distance is <i>d</i>, improving the <i>&Ogr;</i>(2<sup><i>&Ogr;</i>(<i>d</i><sup>2</sup>)</sup>) running time of the beat previous exact algorithm. Our result is based upon a tight connection between syntenic distance and a novel generalization of the classical <i>gossip problem</i>. We define the <i>incomplete gossip problem</i>, in which there are <i>n</i> gossipers who each have a unique piece of initial information. They communicate by phone calls in which the participants exchange all their information, and the goal is to minimize the total number of phone calls necessary to inform each gossiper of his set of <i>relevant gossip</i> which he desires to learn.

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