Approximate distance oracles for geometric spanners

Given an arbitrary real constant ϵ > 0, and a geometric graph <i>G</i> in <i>d</i>-dimensional Euclidean space with <i>n</i> points, <i>O</i>(<i>n</i>) edges, and constant dilation, our main result is a data structure that answers (1 + ϵ)-approximate shortest-path-length queries in constant time. The data structure can be constructed in <i>O</i>(<i>n</i> log <i>n</i>) time using <i>O</i>(<i>n</i> log <i>n</i>) space. This represents the first data structure that answers (1 + ϵ)-approximate shortest-path queries in constant time, and hence functions as an approximate distance oracle. The data structure is also applied to several other problems. In particular, we also show that approximate shortest-path queries between vertices in a planar polygonal domain with “rounded” obstacles can be answered in constant time. Other applications include query versions of <i>closest-pair</i> problems, and the efficient computation of the approximate dilations of geometric graphs. Finally, we show how to extend the main result to answer (1 + ϵ)-approximate shortest-path-length queries in constant time for geometric spanner graphs with <i>m</i> = ω(<i>n</i>) edges. The resulting data structure can be constructed in <i>O</i>(<i>m</i> + <i>n</i> log <i>n</i>) time using <i>O</i>(<i>n</i> log <i>n</i>) space.

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