Approximate map labeling is in Ω ( n log n)

Abstract

Given n real numbers, the ε-CLOSENESS problem consists in deciding whether any two of them are within ε of each other, where ε is a fixed parameter of the problem. This is a basic problems with an Ω ( n log n) lower bound on the running time in the algebraic computation tree model. In this paper we show that for a natural approximation version thereof the same lower bound holds. The main tool used is a lower bound theorem of Ben-Or. We introduce a new interpolation method relating the approximation version of the problem to two corresponding exact versions. Using this result, we are able to prove the optimality of the running time of a cartographic algorithm, that determines an approximate solution to the so-called MAP LABELING problem. MAP LABELING was shown to be NP-complete. The approximation algorithm discussed here is of provably optimal approximation quality. Our result is the first n log n lower bound for an approximate problem. The proof method is general enough to be potentially helpful for further results of this type.