Adjacency queries in dynamic sparse graphs

Abstract

We deal with the problem of maintaining a dynamic graph so that queries of the form “is there an edge between u and v?” are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like, for example, planar graphs. Brodal and Fagerberg [G.S. Brodal, R. Fagerberg, Dynamic representations of sparse graphs, in: Proc. 6th Internat. Workshop on Algorithms and Data Structures (WADS'99), in: Lecture Notes in Comput. Sci., vol. 1663, Springer, Berlin, 1999, pp. 342–351] described a very simple linear-size data structure which processes queries in constant worst-case time and performs insertions and deletions in O ( 1 ) and O ( log n ) amortized time, respectively. We show a complementary result that their data structure can be used to get O ( log n ) worst-case time for query, O ( 1 ) amortized time for insertions and O ( 1 ) worst-case time for deletions. Moreover, our analysis shows that by combining the data structure of Brodal and Fagerberg with efficient dictionaries one gets O ( log log log n ) worst-case time bound for queries and deletions and O ( log log log n ) amortized time for insertions, with size of the data structure still linear. This last result holds even for graphs of arboricity bounded by O ( log k n ) , for some constant k.