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.

翻译：本文引入并研究了边权图的**二进制拉伸嵌入**问题。该问题与Graham和Pollak著名的**寻址问题**密切相关。寻址问题要求为输入图$G$的顶点分配字母表$\{0,1,*\}$上尽可能短的字符串（称为“地址”），并满足以下性质：对于任意顶点对$u,v$，它们的地址中一个为$1$、另一个为$0$的位置数量恰好等于$u,v$在图$G$中的距离。当地址不包含符号$*$时，该问题被称为**等距超立方体嵌入**。据我们所知，等距超立方体嵌入由Firsov于1965年首次提出。已知此类地址对一般图并不存在。受寻址问题的启发，本文针对边权无向图引入了**二进制拉伸嵌入问题**（简称BSEP），并论证了该问题与文献中其他图嵌入问题的关联。通过运用Hadamard码、线性规划理论等工具与技术，我们发现了若干类图的上界、下界以及精确解。作为本文结果的应用，我们推导出某些参数下Lee度量码最大尺寸的改进上界或精确值。