Efficient union-find for planar graphs and other sparse graph classes

Abstract

We solve the Union-Find problem (UF) efficiently for the case the input is restricted to several graph classes, namely partial k-trees for any fixed k, d-dimensional grids for any fixed dimension d and for planar graphs. For the later we develop a technique of decomposing such a graph into small subgraphs, patching, that might be useful for other algorithmic problems on planar graphs, too. By efficiency we do not only mean "linear time" in a theoretical setting but also a practical reorganization of memory such that a dynamic data structures for UF is allocated consecutively and thus to reduce the amount of page fault produced by UF implementations drastically.