$(1-\epsilon)$-approximate fully dynamic densest subgraph: linear space and faster update time

TL;DR

This paper presents a new fully dynamic data structure for maintaining a near-optimal densest subgraph approximation with linear space and faster update times, suitable for large-scale graph data.

Contribution

It improves previous dynamic densest subgraph algorithms by reducing space complexity and enhancing update time bounds, extending naturally to hypergraphs.

Findings

Linear space usage for the data structure.

Amortized update time of O(log^2 n / ε^4).

Worst-case update time of O(log^3 n log log n / ε^6).

Abstract

We consider the problem of maintaining a $(1-\epsilon)$-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 $\epsilon > 0$, maintains a $(1-\epsilon)$-approximation with $O(\log^4 n/\epsilon^6)$ 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 $(1/4 - \epsilon)$ approximation in amortized polylogarithmic update time [BHNT15]. In this paper we develop a data structure for $(1-\epsilon)$-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 $(1-\epsilon)$-approximation in amortized $O(\log^2 n/\epsilon^4)$ time per update while simultaneously guaranteeing that the worst case update time is $O(\log^3 n \log \log n/\epsilon^6)$. 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].