## Abstract

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 Ending 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.

Original language | English |
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Pages (from-to) | 192-208 |

Number of pages | 17 |

Journal | Lecture Notes in Computer Science |

Volume | 3624 |

DOIs | |

State | Published - 2005 |

Externally published | Yes |

Event | 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005 - Berkeley, CA, United States Duration: 22 Aug 2005 → 24 Aug 2005 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science