Implicit representation of graphs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On the complexity of a game related to the dictionary problem
SIAM Journal on Computing
Planar orientations with low out-degree and compaction of adjacency matrices
Theoretical Computer Science
Surpassing the information theoretic bound with fusion trees
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Dynamic Perfect Hashing: Upper and Lower Bounds
SIAM Journal on Computing
Optimal bounds for the predecessor problem and related problems
Journal of Computer and System Sciences - STOC 1999
Faster deterministic sorting and searching in linear space
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Oracles for bounded-length shortest paths in planar graphs
ACM Transactions on Algorithms (TALG)
A dynamic implicit adjacency labelling scheme for line graphs
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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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(logn) amortized time, respectively. We show a complementary result that their data structure can be used to get O(logn) 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(logloglogn) worst-case time bound for queries and deletions and O(logloglogn) 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^kn), for some constant k.