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<sup>1+1/k</sup>) for graphs of n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degree Ω(n<sup>1/k</sup>).

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