A Linear-Time Algorithm for Reconciliation of Non-binary Gene Tree and Binary Species Tree

In the reconciliation problem, we are given two phylogenetic trees. A species tree represents the evolutionary history of a group of species, and a gene tree represents the history of a family of related genes within these species. A reconciliation maps nodes of the gene tree to the corresponding points of the species tree, and thus helps to interpret the gene family history. In this paper, we study the case when both trees are unrooted and their edge lengths are known exactly. The goal is to root them and to find a reconciliation that agrees with the edge lengths. We show a linear-time algorithm for finding the set of all possible root locations, which is a significant improvement compared to the previous O(N3logN) algorithm.

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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2021.07.002 ◽

2021 ◽

Author(s):

Zdeněk Dvořák ◽

Daniel Král' ◽

Robin Thomas

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Graphs On Surfaces

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Nearly Linear Time Algorithm for Mean Hitting Times of Random Walks on a Graph

Proceedings of the 13th International Conference on Web Search and Data Mining ◽

10.1145/3336191.3371777 ◽

2020 ◽

Author(s):

Zuobai Zhang ◽

Wanyue Xu ◽

Zhongzhi Zhang

Keyword(s):

Random Walks ◽

Linear Time ◽

Time Algorithm ◽

Hitting Times ◽

Linear Time Algorithm

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A Linear time algorithm for deciding security

10.1109/sfcs.1976.1 ◽

1976 ◽

Cited By ~ 33

Author(s):

A. K. Jones ◽

R. J. Lipton ◽

L. Snyder

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm

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AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

International Journal of Foundations of Computer Science ◽

10.1142/s0129054100000247 ◽

2000 ◽

Vol 11 (03) ◽

pp. 365-371 ◽

Cited By ~ 33

Author(s):

LJUBOMIR PERKOVIĆ ◽

BRUCE REED

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Tree Decomposition ◽

Linear Time Algorithm ◽

Input Graph ◽

Primary Motivation ◽

Small Width ◽

Tree Decompositions ◽

Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

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