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This paper presents novel, highly efficient approaches for updating Euclidean distance maps and Voronoi diagrams represented on grid maps. Our methods employ a dynamic variant of the brushfire algorithm to update only those cells that are actually affected by changes in the environment. In experiments in different environments we show that our update strategies for distance maps and Voronoi diagrams require substantially fewer cell visits and significantly less computation time compared to previous approaches. Furthermore, the dynamic Voronoi diagram also improves on previous work by correctly dealing with non-convex obstacles such as building walls. We also present a dynamic variant of a skeletonization-based approach to Voronoi diagrams that is especially robust to noise. All of our algorithms consider actual Euclidean distances rather than grid steps. An open source implementation is available online.