Efficient Approximation of Convex Recolorings
A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex coloring of trees arises in areas such as phylogenetics, linguistics, etc. e.g., a perfect phylogenetic tree is one in which the states of each character induce a convex coloring of the tree. Research on perfect phylogeny is usually focused on finding a tree so that few predetermined partial colorings of its vertices are convex.
When a coloring of a tree is not convex, it is desirable to know “how far” it is from a convex one. In [MS05], a natural measure for this distance, called the recoloring distance was defined: the minimal number of color changes at the vertices needed to make the coloring convex. This can be viewed as minimizing the number of “exceptional vertices” w.r.t. to a closest convex coloring. The problem was proved to be NP-hard even for colored strings.
In this paper we continue the work of [MS05], and present a 2-approximation algorithm of convex recoloring of strings whose running time O(cn), where c is the number of colors and n is the size of the input, and an O(cn 2) 3-approximation algorithm for convex recoloring of trees.
KeywordsCovered Vertex Partial Coloring Colored Tree Perfect Phylogeny Versus Support
Unable to display preview. Download preview PDF.
- [BDFY01]Ben-Dor, A., Friedman, N., Yakhini, Z.: Class discovery in gene expression data. In: RECOMB, pp. 31–38 (2001)Google Scholar
- [BFW92]Bodlaender, H.L., Fellows, M.R., Warnow, T.: Two strikes against perfect phylogeny. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 273–283. Springer, Heidelberg (1992)Google Scholar
- [Hoc97]Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problem. PWS Publishing Company (1997)Google Scholar
- [MS03]Moran, S., Snir, S.: Convex recoloring of strings and trees. Technical Report CS-2003-13, Technion (November 2003)Google Scholar
- [Vaz01]Vazirani, V.: Approximation Algorithms. Springer, Berlin (2001)Google Scholar