On the Maximum Quartet Distance between Phylogenetic Trees

Alon, Noga; Naves, Humberto; Sudakov, Benny

doi:10.1137/15m1041754

Cited by 11 publications

(30 citation statements)

References 25 publications

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“…So, a quartet is a caterpillar of order 4 where every edge except the central edge is an endpoint edge and every tree of order 5 is isomorphic to CAT 5 , but already for n = 6 not every tree is a caterpillar. The result of [1] also proves that M CATn = (1 + o(1)) 2 3 n 4 so at least for caterpillars, the Bandelt and Dress conjecture holds.…”

Section: Problems Definitions and Statements Of Main Resultsmentioning

confidence: 74%

“…for every tree T of order n. The result of [1] proves that M n ≤ (0.69 + o(1)) n 4 . The caterpillar CAT n of order n ≥ 4 is the tree having only two internal vertices x, y adjacent to two leaves each.…”

Section: Problems Definitions and Statements Of Main Resultsmentioning

confidence: 84%

“…Alon, Snir, and Yuster [2] improved the upper bound on the maximum quartet distance proving it is asymptotically smaller than 9 10 n 4 . Alon, Naves, and Sudakov [1] further improved the upper bound to at most (0.69 + o(1)) n 4 . In fact, for the particular important case where both T 1 and T 2 are the caterpillar tree they established the validity of Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 94%

“…Type-0 variables: In this case E[X T (q)] = 2 3 since each of the three possible quartets on the elements of q is equally likely to be in Q(T * ), so q is satisfied by T * with probability 1 3 . Type-1 variables: Suppose q = [ax|yz] where a ∈ [k].…”

Section: A Formula For E[x T (S) ]mentioning

confidence: 99%

“…This raises the problem of finding a tree maximizing the number of compatible quartets -maximum quartet compatibility (MQC) [15]. As MQC is NP-hard, several approximation algorithms have been designed for it [4,13,16,17], but the best approximation to the general problem is still obtained by a naive "random tree" with expected approximation ratio of 1 3 . Related to the problem of compatibility is the concept of quartet distance [9,19].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the quartet distance given partial information

Snir¹,

Weissberg²,

Yuster³

2021

Preprint

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Let T be an arbitrary phylogenetic tree with n leaves. It is well-known that the average quartet distance between two assignments of taxa to the leaves of T is 2 3 n 4 . However, a longstanding conjecture of Bandelt and Dress asserts that ( 2 3 + o(1)) n 4 is also the maximum quartet distance between two assignments. While Alon, Naves, and Sudakov have shown this indeed holds for caterpillar trees, the general case of the conjecture is still unresolved. A natural extension is when partial information is given: the two assignments are known to coincide on a given subset of taxa. The partial information setting is biologically relevant as the location of some taxa (species) in the phylogenetic tree may be known, and for other taxa it might not be known. What can we then say about the average and maximum quartet distance in this more general setting? Surprisingly, even determining the average quartet distance becomes a nontrivial task in the partial information setting and determining the maximum quartet distance is even more challenging, as these turn out to be dependent of the structure of T . In this paper we prove nontrivial asymptotic bounds that are sometimes tight for the average quartet distance in the partial information setting. We also show that the Bandelt and Dress conjecture does not generally hold under the partial information setting. Specifically, we prove that there are cases where the average and maximum quartet distance substantially differ.

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Section: Problems Definitions and Statements Of Main Resultsmentioning

confidence: 74%

Section: Problems Definitions and Statements Of Main Resultsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 94%

Section: A Formula For E[x T (S) ]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the quartet distance given partial information

Snir¹,

Weissberg²,

Yuster³

2021

Preprint

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On the quartet distance given partial information

Snir

Weissberg

Yuster

2021

Journal of Graph Theory

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A High Quartet Distance Construction

2019

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Given two binary trees on N labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is 2 3 N 4 . However, no strongly explicit construction reaching this bound asymptotically was known.We consider complete, balanced binary trees on N = 2 n leaves, labeled by n long bit sequences. Ordering the leaves in one tree by the prefix order, and in the other tree by the suffix order, we show that the resulting quartet distance is 2 3 + o(1) N 4 , and it always exceeds the 2 3 N 4 bound.

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On the Maximum Quartet Distance between Phylogenetic Trees

Cited by 11 publications

References 25 publications

On the quartet distance given partial information

On the quartet distance given partial information

On the quartet distance given partial information

A High Quartet Distance Construction

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