(1-ε)-approximate fully dynamic densest subgraph: linear space and faster update time
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We consider the problem of maintaining a (1-ϵ)-approximation to the densest subgraph (DSG) in an undirected multigraph as it undergoes edge insertions and deletions (the fully dynamic setting). Sawlani and Wang [SW20] developed a data structure that, for any given ϵ > 0, maintains a (1-ϵ)-approximation with O(log^4 n/ϵ^6) worst-case update time for edge operations, and O(1) query time for reporting the density value. Their data structure was the first to achieve near-optimal approximation, and improved previous work that maintained a (1/4 - ϵ) approximation in amortized polylogarithmic update time [BHNT15]. In this paper we develop a data structure for (1-ϵ)-approximate DSG that improves the one from [SW20] in two aspects. First, the data structure uses linear space improving the space bound in [SW20] by a logarithmic factor. Second, the data structure maintains a (1-ϵ)-approximation in amortized O(log^2 n/ϵ^4) time per update while simultaneously guaranteeing that the worst case update time is O(log^3 n loglog n/ϵ^6). We believe that the space and update time improvements are valuable for current large scale graph data sets. The data structure extends in a natural fashion to hypergraphs and yields improvements in space and update times over recent work [BBCG22] that builds upon [SW20].
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