Almost Optimal Distance Oracles for Planar Graphs

Abstract: We present new tradeoffs between space and query-time for exact distance oracles in directed weighted planar graphs. These tradeoffs are almost optimal in the sense that they are within polylogarithmic, sub-polynomial or arbitrarily small polynomial factors from the na\"{\i}ve linear space, constant query-time lower bound. These tradeoffs include: (i) an oracle with space $\tilde{O}(n<sup>{1+\epsilon})$ and query-time $\tilde{O}(1)$ for any constant $\epsilon&gt;0$, (ii) an oracle with space $\tilde{O}(n)$ and query-time $\tilde{O}(n<sup>{\epsilon})$ for any constant $\epsilon&gt;0$, and (iii) an oracle with space $n<sup>{1+o(1)}$ and query-time $n<sup>{o(1)}$.