Efficient visual navigation of hierarchically structured graphs

Visual navigation of hierarchically structured graphs is a technique for interactively exploring large graphs that possess an additional hierarchical structure. This structure is expressed in form of a recursive clustering of the nodes: in call graphs of telephone networks, for instance, the nodes are identified with phone numbers; they are clustered recursively through the implicit structure of the numbers, e.g., nodes with the same area code belong to a cluster. In order to reduce the complexity and the size of the graph, only those subgraphs that are currently needed are shown in detail, while the others are collapsed, i. e., represented by meta nodes. In such a graph view the subgraphs in the areas of interest are expanded furthest, whereas those on the periphery are abstracted. As the areas of interest change over time, clusters in a view need to be expanded or contracted. First and foremost, there is need for an efficient data structure for this graph view maintenance problem. Depending on the admissible modifications of the graph and its hierarchical clustering, three variants have been discussed in the literature: in the static case, everything is fixed; in the dynamic graph variant, only edges of the graph can be inserted and deleted; finally, in the dynamic graph and tree variant the graph additionally is subject to node insertions and deletions and the clustering may change through splitting and merging of clusters. We introduce a new variant, dynamic leaves, which is based on the dynamic graph variant, but additionally allows insertion and deletion of graph nodes, i. e., leaves of the hierarchy. So far efficient data structures were known only for the static and the dynamic graph variant, i. e., neither the nodes of the graph nor the clustering could be modified. As this is unsatisfactory in an interactive editor for hierarchically structured graphs,we first generalize the approach of Buchsbaum et. al (Proc. 8th ESA, vol. 1879 of LNCS, pp. 120–131, 2000), in which graph view maintenance is formulated as a special case of range searching over tree cross products, to the new dynamic leaves variant. This generalization builds on a novel technique of superimposing a search tree over an ordered list maintenance structure. With an additional factor of roughly O(log n/log log n), this is the first data structure for the problem of graph view maintenance where the node set is dynamic.