## Convex recoloring of strings and trees (2003)

Citations: | 2 - 2 self |

### BibTeX

@TECHREPORT{Moran03convexrecoloring,

author = {Shlomo Moran and Sagi Snir},

title = {Convex recoloring of strings and trees},

institution = {},

year = {2003}

}

### OpenURL

### 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. eg, 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. One common measure for this is based on the parsimony score, which is the number of edges whose endpoints have different colors. In this paper we study another natural measure for this distance: 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. We also study a similar measure which aims at minimizing the number of “exceptional edges ” w.r.t. a closest convex coloring. We show that finding each of these distances is NP-hard even for strings. We then focus on the first measure and generalize it to weighted trees, and then to non-uniform coloring costs. In the positive side we present few algorithms for convex recoloring of strings of trees: First we present algorithms for optimal convex recolorings of strings and trees with non-uniform coloring costs, which for any fixed number of colors are linear in the input size. Then we present fixed parameter tractable algorithms and approximation algorithms for convex recolorings of weighted strings and trees.

### Citations

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- 1999
(Show Context)
Citation Context ... algorithm which receives as an input a colored tree on n vertices, (T, C), and an integer k, and decides whether OP T (T, C) ≤ k in p(n)f(k) time, for some polynomial p and arbitrary function f (see =-=[6]-=-). It is easy to see that an FPT recoloring algorithm exists iff there is an algorithm which finds an optimal recoloring of the input coloring C in p(n) · f(k) time for p and f as above, where k is th... |

165 |
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Citation Context ...ithm makes use of the local ratio technique, which is useful for approximating optimization covering problems such as vertex cover, dominating set, minimum spanning tree, feedback vertex set and more =-=[4, 2, 3]-=-. We hereafter describe it briefly: The input to the problem is a triplet (V, f : 2V → {0, 1}, w : V → R + ), and the goal is to find a subset X ⊆ V such that f(X) = 1 and w(X) is minimized, i.e. w(X)... |

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(Show Context)
Citation Context ...parsimony. The input is a set of characters while the tree sought is homoplasy free for each of the characters. The general perfect phylogeny problem was shown to be NP-Complete by Bodlaender et. al. =-=[5]-=- and independently by Steel [19]. Nevertheless, when restricting the number of possible states for each character, r, or considering r as a parameter, there are some positive results: For binary chara... |

74 | A 2-approximation algorithm for the undirected feedback vertex set problem
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Citation Context ...ithm makes use of the local ratio technique, which is useful for approximating optimization covering problems such as vertex cover, dominating set, minimum spanning tree, feedback vertex set and more =-=[4, 2, 3]-=-. We hereafter describe it briefly: The input to the problem is a triplet (V, f : 2V → {0, 1}, w : V → R + ), and the goal is to find a subset X ⊆ V such that f(X) = 1 and w(X) is minimized, i.e. w(X)... |

60 | One for the price of two: A unified approach for approximating covering problems
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Citation Context ...smaller support (set of vertices of positive weight); depending on the specific instance, the reduction can be either of “local ratio” type, which keeps the topology but decreases the weight function =-=[3]-=-, or of “combinatorial” type, which changes the topology of the input in a local manner. The rest of the paper is organized as follows. The next section presents the notations used and define the unwe... |

25 |
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(Show Context)
Citation Context ...x iff the total number of violations of C is zero. In fact, some authors use the above sum, taken over all characters, as a measure of the distance of a given phylogenetic tree from perfect phylogeny =-=[8]-=-. The definition of convex coloring is extended to partially colored trees, in which the input coloring C assigns colors to some subset of the vertices. A partial coloring is said to be convex if it c... |

25 |
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Citation Context ...03 Figure 1: The primates tree, taken from “The Tree of Life” project. One of the best known and most widely used character-based methods for constructing phylogenetic trees is maximum parsimony (MP) =-=[9]-=-. This is a combinatorial method, which attempts to construct trees which minimize the number of violations of the convexity property. Specifically, Fitch algorithm for ”small parsimony” [9] finds, fo... |

24 |
Convex tree realizations of partitions
- Dress, Steel
- 1992
(Show Context)
Citation Context ...ositive results: For binary character, i.e. each state is either 0 or 1, Gusfield [11] gave an O(nk) time algorithm where n is the number of species and k is the number of characters. Dress and Steel =-=[7]-=- devised an O(nk 2 ) for r ≤ 3 and Kannan and Warnow [13] gave a O(n 2 k) algorithm for r ≤ 4. when considering r as a parameter, Agarwala and Fernandez-Baca [1] showed a fixed parameter tractable alg... |

22 | Simple Algorithms for Perfect Phylogeny and Triangulating Colored Graphs
- Agarwala, Fernandez-Baca
- 1996
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Citation Context ... number of characters. Dress and Steel [7] devised an O(nk 2 ) for r ≤ 3 and Kannan and Warnow [13] gave a O(n 2 k) algorithm for r ≤ 4. when considering r as a parameter, Agarwala and Fernandez-Baca =-=[1]-=- showed a fixed parameter tractable algorithm that runs in time O(2 3r (nk 3 + k 4 )). This work was later improved by Kannan and Warnow [14] to O(2rnk 2 ) time. In this work we deal with following as... |