Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.09311

Binary Stretch Embedding of Weighted Graphs

Javad B. Ebrahimi·Mehri Oghbaei Bonab
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we introduce and study the problem of \textit{binary stretch embedding} of edge-weighted graph. This problem is closely related to the well-known \textit{addressing problem} of Graham and Pollak. Addressing problem is the problem of assigning the shortest possible length strings (called ``addresses") over the alphabet $\{0,1,*\}$ to the vertices of an input graph $G$ with the following property. For every pair $u,v$ of vertices, the number of positions in which one of their addresses is $1$, and the other is $0$ is exactly equal to the distance of $u,v$ in graph $G$. When the addresses do not contain the symbol $*$, the problem is called \textit{isometric hypercube embedding}. As far as we know, the isometric hypercube embedding was introduced by Firsov in 1965. It is known that such addresses do not exist for general graphs. Inspired by the addressing problem, in this paper, we introduce the \textit{binary stretch embedding problem}, or BSEP for short, for the edge-weighted undirected graphs. We also argue how this problem is related to other graph embedding problems in the literature. Using tools and techniques such as Hadamard codes and the theory of linear programming, several upper and lower bounds as well as exact solutions for certain classes of graphs will be discovered. As an application of the results in this paper, we derive improved upper bounds or exact values for the maximum size of Lee metric codes of certain parameters.

Explore related subjects

cs.DMmath.CO

Keep this discovery

BibTeXRIS

Javad B. Ebrahimi, Mehri Oghbaei Bonab. 2024-03-14. Binary Stretch Embedding of Weighted Graphs. https://arxiv.org/abs/2403.09311

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An FPTAS for Two-Machine Open-Shop Scheduling with a Single Unavailability Interval

We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

cs.DM↗
arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗
arXiv2026

The exact asymptotic constant in the metric dimension of Jaccard space

Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $\Theta(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erd\H{o}s--R\'enyi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erd\H{o}s--R\'enyi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindstr\"om and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.

cs.DM↗