Dynamic Tree Cross Products

Range searching over tree cross products – a variant of classic range searching – recently has been introduced by Buchsbaum et al (Proc 8th ESA, vol 1879 of LNCS, pp 120–131, 2000) A tree cross product consist of hyperedges connecting the nodes of trees T1,...,Td In this context, range searching means to determine all hyperedges connecting a given set of tree nodes Buchsbaum et al describe a data structure which supports, besides queries, adding and removing of edges; the tree nodes remain fixed In this paper we present a new data structure, which additionally provides insertion and deletion of leaves of T1,...,Td; it combines the former approach with a novel technique of using search trees superimposed over ordered list maintenance structures The extra cost for this dynamization is roughly a factor of ${\mathcal O}({\rm log} {\it n}/{\rm log log} {\it n})$ The trees being dynamic is especially important for maintaining hierarchical graph views, a problem that can be modeled as tree cross product Such views evolve from a large base graph by the contraction of subgraphs defined recursively by an associated hierarchy The graph view maintenance problem is to provide methods for refining and coarsening a view In previous solutions only the edges of the underlying graph were dynamic; with the application of our data structure, the node set becomes dynamic as well.