Gomory–Hu Trees

Keywords and Synonyms

Cut trees  

Problem Definition

Let ${ G = (V,E) }$ be an undirected graph with ${ |V| = n }$ and ${ |E| = m }$. The edge connectivity of two vertices ${ s,t \in V }$, denoted by ${ \lambda(s,t) }$, is defined as the size of the smallest cut that separates s and t; such a cut is called a minimum s–t cut. Clearly, one can represent the ${ \lambda(s, t) }$ values for all pairs of vertices s and t in a table of size O(n ²). However, for reasons of efficiency, one would like to represent all the ${ \lambda(s, t) }$ values in a more succinct manner. Gomory–Hu trees (also known as cut trees) offer one such succinct representation of linear (i. e., O(n)) space and constant (i. e., O(1)) lookup time. It has the additional advantage that apart from representing all the ${ \lambda(s, t) }$ values, it also contains structural information from which a minimum s–t cut can be retrieved easily for any pair of vertices s and t.

Formally, a Gomory–Hu...