A Greedoid and a Matroid Inspired by Bhargava's $p$-Orderings

Consider a finite set $E$. Assume that each $e \in E$ has a "weight" $w \left(e\right) \in \mathbb{R}$ assigned to it, and any two distinct $e, f \in E$ have a "distance" $d \left(e, f\right) = d \left(f, e\right) \in \mathbb{R}$ assigned to them, such that the distances satisfy the ultrametric triangle inequality $d(a,b)\leqslant \max \left\{d(a,c),d(b,c)\right\}$. We look for a subset of $E$ of given size with maximum perimeter (where the perimeter is defined by summing the weights of all elements and their pairwise distances). We show that any such subset can be found by a greedy algorithm (which starts with the empty set, and then adds new elements one by one, maximizing the perimeter at each step). We use this to define numerical invariants, and also to show that the maximum-perimeter subsets of all sizes form a strong greedoid, and the maximum-perimeter subsets of any given size are the bases of a matroid. This essentially generalizes the "$P$-orderings" constructed by Bhargava in order to define his generalized factorials, and is also similar to the strong greedoid of maximum diversity subsets in phylogenetic trees studied by Moulton, Semple and Steel. We further discuss some numerical invariants of $E, w, d$ stemming from this construction, as well as an analogue where maximum-perimeter subsets are replaced by maximum-perimeter tuples (i.e., elements can appear multiple times).

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Cladistic hypotheses as degree of equivalence relational structures: implications for three-item statements

10.1101/2021.01.14.426769 ◽

2021 ◽

Author(s):

Valentin Rineau ◽

Stéphane Prin

Keyword(s):

Phylogenetic Trees ◽

Phylogenetic Reconstruction ◽

Equivalence Relations ◽

Strong Connection ◽

Relational Structures ◽

Degree Of Equivalence ◽

Axiomatic Definition ◽

Finite Set ◽

Definition Of ◽

Supertree Methods

AbstractThree-item statements, as minimal informative rooted binary phylogenetic trees on three items, are the minimal units of cladistic information. Their importance for phylogenetic reconstruction, consensus and supertree methods relies on both (i) the fact that any cladistic tree can always be decomposed into a set of three-item statements, and (ii) the possibility, at least under some conditions, to build a new cladistic tree by combining all or part of the three-item statements deduced from several prior cladistic trees. In order to formalise such procedures, several k-adic rules of inference, i.e., rules that allow us to deduce at least one new three-item statement from exactly k other ones, have been identified. However, no axiomatic background has been proposed, and it remains unknown if a particular k-adic rule of inference can be reduced to more basic rules. In order to solve this problem, we propose here to define three-item statements in terms of degree of equivalence relations. Given both the axiomatic definition of the latter and their strong connection to hierarchical classifications, we establish a list of the most basic properties for three-item statements. With such an approach, we show that it is possible to combine five three-item statements from basic rules although they are not combinable only from dyadic rules. Such a result suggests that all higher k-adic rules are well reducible to a finite set of simpler rules.

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Tree-Based Unrooted Phylogenetic Networks

Bulletin of Mathematical Biology ◽

10.1007/s11538-017-0381-3 ◽

2017 ◽

Vol 80 (2) ◽

pp. 404-416 ◽

Cited By ~ 10

Author(s):

A. Francis ◽

K. T. Huber ◽

V. Moulton

Keyword(s):

Gene Transfer ◽

Horizontal Gene Transfer ◽

Phylogenetic Tree ◽

Phylogenetic Trees ◽

Phylogenetic Network ◽

Simple Graph ◽

Phylogenetic Networks ◽

Underlying Graph ◽

Finite Set ◽

Computational Properties

Abstract Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An unrooted phylogenetic network on a non-empty, finite set X of taxa, or network, is a connected, simple graph in which every vertex has degree 1 or 3 and whose leaf set is X. It is called a phylogenetic tree if the underlying graph is a tree. In this paper we consider properties of tree-based networks, that is, networks that can be constructed by adding edges into a phylogenetic tree. We show that although they have some properties in common with their rooted analogues which have recently drawn much attention in the literature, they have some striking differences in terms of both their structural and computational properties. We expect that our results could eventually have applications to, for example, detecting horizontal gene transfer or hybridization which are important factors in the evolution of many organisms.

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POSI-MODULAR SYSTEMS WITH MODULOTONE REQUIREMENTS UNDER PERMUTATION CONSTRAINTS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830910000474 ◽

2010 ◽

Vol 02 (01) ◽

pp. 61-76 ◽

Cited By ~ 1

Author(s):

TOSHIMASA ISHII ◽

KAZUHISA MAKINO

Keyword(s):

Greedy Algorithm ◽

Location Problem ◽

Monotone Function ◽

Modular Function ◽

Discrete Math ◽

Natural Generalization ◽

Source Location ◽

Alternative Proof ◽

Modular Systems ◽

Finite Set

Given a system (V, f, r) on a finite set V consisting of a posi-modular function f : 2V → ℝ and a modulotone function r : 2V → ℝ, we consider the problem of finding a minimum set R ⊆ V such that f(X) ≥ r(X) for all X ⊆ V - R. The problem, called the transversal problem, was introduced in [M. Sakashita, K. Makino, H. Nagamochi and S. Fujishige, Minimum transversals in posi-modular systems, SIAM J. Discrete Math.23 (2009) 858–871] as a natural generalization of the source location problem and external network problem with edge-connectivity requirements in undirected graphs and hypergraphs. By generalizing [H. Tamura, H. Sugawara, M. Sengoku and S. Shinoda, Plural cover problem on undirected flow networks, IEICE Trans.J81-A (1998) 863–869] for the source location problem, we show that the transversal problem can be solved by a simple greedy algorithm if r is π-monotone, where a modulotone function r is π-monotone if there exists a permutation π of V such that the function pr: V × 2V → ℝ associated with r satisfies pr(u, W) ≥ pr(v, W) for all W ⊆ V and u, v ∈ V with π(u) ≥ π(v). Here we show that any modulotone function r can be characterized by pr as r(X) = max {pr(v, W) | v ∈ X ⊆ V - W}. We also show the structural properties on the minimal deficient sets [Formula: see text] for the transversal problem for π-monotone function r, i.e., there exists a basic tree T for [Formula: see text] such that π(u) ≤ π(v) for all arcs (u,v) in T, which, as a corollary, gives an alternative proof for the correctness of the greedy algorithm for the source location problem. Furthermore, we show that a fractional version of the transversal problem can be solved by the algorithm similar to the one for the transversal problem.

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Similarity, separability, and the triangle inequality.

Psychological Review ◽

10.1037/0033-295x.89.2.123 ◽

1982 ◽

Vol 89 (2) ◽

pp. 123-154 ◽

Cited By ~ 236

Author(s):

Amos Tversky ◽

Itamar Gati

Keyword(s):

Triangle Inequality

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DEVELOPMENT OF A HIGH DENSITY FINITE SET OF UNIFORM FIELD EMITTERS ON A THIN FILM GLASS SUBSTRATE

Le Journal de Physique Colloques ◽

10.1051/jphyscol:1986212 ◽

1986 ◽

Vol 47 (C2) ◽

pp. C2-79-C2-83

Author(s):

G. A. KITZMANN

Keyword(s):

Thin Film ◽

Glass Substrate ◽

High Density ◽

Uniform Field ◽

Field Emitters ◽

Finite Set

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Implementasi Algoritma Dijkstra Pada Game Pacman

CCIT Journal ◽

10.33050/ccit.v12i2.687 ◽

2019 ◽

Vol 12 (2) ◽

pp. 170-176

Author(s):

Anggit Dwi Hartanto ◽

Aji Surya Mandala ◽

Dimas Rio P.L. ◽

Sidiq Aminudin ◽

Andika Yudirianto

Keyword(s):

Artificial Intelligence ◽

Greedy Algorithm ◽

Dijkstra Algorithm ◽

Testing Phase ◽

A Value ◽

Route Problem ◽

Starting Point ◽

The Greedy Algorithm

Pacman is one of the labyrinth-shaped games where this game has used artificial intelligence, artificial intelligence is composed of several algorithms that are inserted in the program and Implementation of the dijkstra algorithm as a method of solving problems that is a minimum route problem on ghost pacman, where ghost plays a role chase player. The dijkstra algorithm uses a principle similar to the greedy algorithm where it starts from the first point and the next point is connected to get to the destination, how to compare numbers starting from the starting point and then see the next node if connected then matches one path with the path). From the results of the testing phase, it was found that the dijkstra algorithm is quite good at solving the minimum route solution to pursue the player, namely by getting a value of 13 according to manual calculations

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Darwin's two competing phylogenetic trees: marsupials as ancestors or sister taxa?

Archives of Natural History ◽

10.3366/anh.2012.0091 ◽

2012 ◽

Vol 39 (2) ◽

pp. 217-233 ◽

Cited By ~ 4

Author(s):

J. David Archibald

Keyword(s):

Fossil Record ◽

Evolutionary Biology ◽

Phylogenetic Trees ◽

Charles Darwin ◽

The Other ◽

Charles Lyell ◽

Single Origin ◽

Molecular Studies ◽

Richard Owen ◽

First Time

Studies of the origin and diversification of major groups of plants and animals are contentious topics in current evolutionary biology. This includes the study of the timing and relationships of the two major clades of extant mammals – marsupials and placentals. Molecular studies concerned with marsupial and placental origin and diversification can be at odds with the fossil record. Such studies are, however, not a recent phenomenon. Over 150 years ago Charles Darwin weighed two alternative views on the origin of marsupials and placentals. Less than a year after the publication of On the origin of species, Darwin outlined these in a letter to Charles Lyell dated 23 September 1860. The letter concluded with two competing phylogenetic diagrams. One showed marsupials as ancestral to both living marsupials and placentals, whereas the other showed a non-marsupial, non-placental as being ancestral to both living marsupials and placentals. These two diagrams are published here for the first time. These are the only such competing phylogenetic diagrams that Darwin is known to have produced. In addition to examining the question of mammalian origins in this letter and in other manuscript notes discussed here, Darwin confronted the broader issue as to whether major groups of animals had a single origin (monophyly) or were the result of “continuous creation” as advocated for some groups by Richard Owen. Charles Lyell had held similar views to those of Owen, but it is clear from correspondence with Darwin that he was beginning to accept the idea of monophyly of major groups.

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