Map Labeling Heuristics: Provably Good and Practically Useful

Frank Wagner, Alexander Wolff Institut für Informatik Freie Universität Berlin email:, Report B 95-04 April 1995

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Map Labeling Heuristics: Provably Good and Practically Useful The lettering of maps is a classical problem of cartography that consists of placing names, symbols, or other data near to spe\-ci\-fied sites on a map. Certain design rules have to be obeyed. A practically interesting special case, the {\em Map Labeling Problem}, consists of placing axis parallel rectangular labels of common size so that one of its corners is the site, no two labels overlap, and the labels are of maximum size in order to have legible inscriptions. The problem is \NP-hard; it is even \NP-hard to approximate the solution with quality guaranty better than 50 percent. There is an approximation algorithm $A$ with a quality guaranty of 50 percent and running time $\bigO (n \log n)$. So $A$ is the best possible algorithm from a theoretical point of view. This is even true for the running time, since there is a lower bound on the running time of any such approximation algorithm of $\Omega (n \log n)$. Unfortunately $A$ is useless in practice as it typically produces results that are intolerably far off the maximum size. The main contribution of this paper is the presentation of a heuristical approach that has $A$'s advantages while avoiding its disadvantages:\\ 1. It uses $A$'s result in order to guaranty the same optimal running time efficiency; a method which is new as far as we know.\\ 2. Its practical results are close to the optimum. The practical quality is analysed by comparing our results to the exact optimum, where this is known; and to lower and upper bounds on the optimum otherwise. The sample data consists of three different classes of random problems and a selection of problems arising in the production of groundwater quality maps by the authorities of the City of M\"unchen.