On the Loop Switching Addressing Problem

The following graph addressing problem was studied by Graham and Pollak in devising a routing scheme for Pierce’s loop switching network. Let G be a graph with n vertices. It is desired to assign to each vertex $v_i $ an address in $\{ 0,1, * \} ^l $, such that the Hamming distance between the addresses of any two vertices agrees with their distance in G. Let $N(G)$ be the minimum length l for which an assignment is possible. It was shown by Graham and Pollak that $N(G) \leqq m_G (n - 1)$, where $m_G $ is the diameter of G. In the present paper, we shall prove that $N(G) \leqq 1.09(\lg m_G )n + 8n$ by an explicit construction. This shows in particular that any graph has an addressing scheme of length $O(n\log n)$.