Streaming Complexity Separations for Dense and Sparse Graphs
For streaming algorithm designers, this work provides tight space bounds for fundamental graph problems, revealing that logarithmic factors can be shaved for dense streams.
This paper identifies a sharp separation in the streaming space complexity of Maximum Cut between dense and sparse graphs, showing that dense graphs require O(n/ε²) space while sparse graphs require Ω(n log(ε² n)/ε²) space. Similar separations are proven for Densest Subgraph and other problems.
We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that $O(n/\varepsilon^2)$ space is sufficient and that $Ω(n)$ space is necessary. In contrast, for graphs with $Θ(n/\varepsilon^2)$ edges, the situation is markedly different: we show that the problem requires $Ω(n \log(\varepsilon^2 n)/\varepsilon^2)$ space for any $\varepsilon=ω(1/\sqrt{n})$, which is tight for the full range of $\varepsilon$. We also give an $Ω(n \log n/\varepsilon^2)$-space lower bound against deterministic algorithms for outputting a $(1-\varepsilon)$ approximation to the value of the maximum cut. Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.