Approximate distance queries and compact routing in sparse graphs

An approximate distance query data structure is a compact representation of a graph, and can be queried to approximate shortest paths between any pair of vertices. Any such data structure that retrieves stretch 2k Ω 1 paths must require space Ω(n
^1+1/k
) for graphs of n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degree Ω(n
^1/k
).