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Given a set [Formula: see text] of n points and a set [Formula: see text] of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in [Formula: see text] is covered by at least one disk in [Formula: see text] or not and (ii) if so, then find a minimum cardinality subset [Formula: see text] such that the unit disks in [Formula: see text] cover all the points in [Formula: see text]. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within [Formula: see text], for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is [Formula: see text]. The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time [Formula: see text].

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An automated method is presented for the generalization of linear features that are already in a topologically structured computer-readable form. The method consists of three stages: feature elimination; feature simplification; and interference detection and resolution. The first stage is implemented using English-like rules to specify which map features are to be eliminated at the generalized map scale. The second stage consists of the simplification of linear map features, specifically in the form of polyline simplification and riverbank combination. The third stage deals with interference detection and resolution among linear map features. Due to the width of the symbol representing a feature, interference may occur when a simplified feature is added to the generalized map and it overlaps another feature already on the generalized map. If that occurs, one or both of the interfering features have to be displaced so as to eliminate the interference between them. This displacement may cause other features to be subsequently displaced in a process called feature displacement propagation. The algorithms used to detect feature interference and to perform the subsequent feature displacement propagation are described. A FORTRAN 77 computer program called MAPEX was written to perform automated generalization of linear map features. The intended scale range for MAPEX is 1:24 000 to 1: 250 000. The results of running MAPEX on two topologically structured test data files are given, along with some of the algorithmic details used in the implementation of the program.

Given a set [Formula: see text] of m unit disks and a set [Formula: see text] of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset [Formula: see text] to cover [Formula: see text]. This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by Carmi et al. [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n( log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of Carmi et al. [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem.