FIVE POINT METRIC SPACES: GROMOV PRODUCT STRUCTURES, QUADRANGLE STRUCTURES AND EXPLICIT PARAMETERIZATIONS

Year 2023, Volume: 11 Issue: 2, 167 - 181, 28.08.2023

Ayşe Hümeyra Bilge Derya Çelik Mehmet Şahin Koçak Arash Mohammadıan Rezaeınazhad

https://doi.org/10.20290/estubtdb.1278467

Abstract

Let (X,d) be a finite metric space with elements P_i,i=1,…,n and with distances d_ij:=d(P_i,P_j) for i,j=1,…,n. The “Gromov product” Δ_ijk, is defined as Δ_ijk=1/2(d_ij+d_ik-d_jk). (X,d) is called Δ-generic, if for each fixed i, the set of Gromov products has a unique least element, Δ_(ij_i k_i ). The Gromov product structure on a Δ-generic finite metric space (X,d) is the map that assigns the edge E_(j_i k_i ) to P_i. A finite metric space is called “quadrangle generic”, if for all 4-point subsets {P_i,P_j,P_k,P_l }, the set {d_ij+d_kl,d_ik+d_jl,d_il+d_jk } has a unique maximal element. We define the “quadrangle structure” on a quadrangle generic finite metric space (X,d) as the map that assigns to each 4-point subset of X, the pair of edges corresponding to the maximal element of the sums of the distances. Two metric spaces (X,d) and (X,d') are said to be Δ-equivalent (Q-equivalent), if the corresponding Gromov product (quadrangle) structures are the same, up to a permutation of X.
In this paper, Gromov product structures, quadrangle structures, optimal reductions and explicit parameterizations for 5-point spaces are obtained and compared with previous results in the literature. In the final part of this review paper, we have used the Monte Carlo method to obtain the relative volume of each of the 5-point metric types inside the corresponding metric cone for 5-point spaces, meanwhile 102 different partitions of metric cone for 5-point spaces are derived, considering Gromov product structures. These 102 partitions, come in three symmetric classes forming three types of metrics for 5-point spaces. Thus one can say that all the methods of classification given here or given before in the literature of finite metric spaces, give 3 types of metrics for 5-point spaces.

Keywords

Finite metric spaces, Split metric decompositions, Gromov Products, Quadrangle structures

Supporting Institution

TÜBİTAK

Project Number

118F412

Thanks

This work has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the project number 118F412 titled “Analysis of Finite Metric Spaces via Gromov Products and their Applications to Phylogenetics”.

5-NOKTALI METRİK UZAYLAR: GROMOV ÇARPIM YAPILAR, DÖRTGEN YAPILAR VE AÇIK PARAMETRİZASYONLAR

Abstract

Bu çalışmada, 5 noktalı uzaylar için Gromov çarpım yapıları, dörtgen yapılar, optimal indirgemeler ve açık parametrizasyonlar elde edilmiş ve literatürde var olan sonuçlarla karşılaştırılmıştır.

Keywords

Sonlu Metrik Uzaylar, Split metrik ayrışımlar, Gromov Çarpımlar, Dörtgen Yapılar

Project Number

118F412

References

• [1] Bilge AH, Çelik D, Koçak Ş, Rezaeinazhad, AM. Gromov product structures, quadrangle structures and split metric decompositions for finite metric spaces. Discrete Mathematics 2021; 344(6): 112358.
• [2] Ghys É, Harpe P de la. Sur les groupes hyperboliques d'apres Mikhael Gromov. Progress in Mathematics 83. Springer, 1990. [3] Sturmfels B, Yu J. Classification of Six-Point Metrics. Electronic Journal of Combinatorics 2004; 11 R44.
• [4] Bilge AH, Çelik D, Koçak Ş. An equivalence class decomposition of finite metric spaces via Gromov products. Discrete Mathematics 2017; 340(8): 1928-1932.
• [5] Bilge AH, İncegül M. Gromov Product Decomposition of 7-point Metric Spaces. https://arxiv.org/abs/1804.03051v1.
• [6] Bilge AH, İncegül M. Matrix invariants of finite metric spaces. arXiv:2003.03335.
• [7] Bandelt HJ, Dress AWM. A Canonical Decomposition Theory for Metrics on a Finite Set. Advances in Mathematics 1992; 92: 47-105.
• [8] Koolen J, Lesser A, Moulton V. Optimal realizations of generic five-point metrics. European Journal of Combinatorics 2009; 30(5): 1164-1171. [9] Çelik D, Bilge AH, Koçak Ş. Optimal embeddings of finite metric spaces into graphs. Anadolu Üniversitesi Bilim ve Teknoloji Dergisi-B Teorik Bilimler 2015; 3(2): 133-147.