Approximate Distance Oracles for Planar Graphs with Subpolynomial Error Dependency
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Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there exists a (1+ϵ)-approximate distance oracle for planar graphs with O(n (log n)ϵ^-1) space and O(ϵ^-1) query time. While the dependency on n is nearly linear, the space-query product of their oracles depend quadratically on 1/ϵ. Many follow-up results either improved the space or the query time of the oracles while having the same, sometimes worst, dependency on 1/ϵ. Kawarabayashi, Sommer, and Thorup [SODA'13] were the first to improve the dependency on 1/ϵ from quadratic to nearly linear (at the cost of log^*(n) factors). It is plausible to conjecture that the linear dependency on 1/ϵ is optimal: for many known distance-related problems in planar graphs, it was proved that the dependency on 1/ϵ is at least linear. In this work, we disprove this conjecture by reducing the dependency of the space-query product on 1/ϵ from linear all the way down to subpolynomial (1/ϵ)^o(1). More precisely, we construct an oracle with O(nlog(n)(ϵ^-o(1) + log^*n)) space and log^2+o(1)(1/ϵ) query time. Our construction is the culmination of several different ideas developed over the past two decades.
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