Dynamic construction of abstract Voronoi diagrams

The abstract Voronoi diagram (AVD) introduced by R. Klein is a generalization of various concrete Voronoi diagrams-data structures actively used in the last decades for solving theoretical and practical geometric problems. This paper presents a fully dynamic algorithm for AVD construction based on Klein's incremental approach. It needs O(n) worst-case time for a new site insertion in an AVD with n sites. For the first time a possibility of effective site deletion without full reconstruction of AVD is proved. The proposed method for site deletion requires O(mn) expected time, where m is the number of invisible sites, and O(n) if invisible sites are not allowed. The dynamic algorithm consumes O(n) memory at any moment. © 2008 Springer Science+Business Media, Inc.