The paper studies the streaming complexity of computing expander decompositions, particularly in the context of repeated applications where the decomposition is applied to the inter-cluster edges of the previous decomposition.
Key insights:
The authors provide an algorithm that can compute a single-level (O(φ log n), φ)-expander decomposition in dynamic streams using O(n) space, without any dependence on the sparsity parameter φ. This is achieved by introducing the concept of "boundary-linked" expander decompositions and designing a new type of graph sparsifier called a "cluster sparsifier".
However, the authors show that computing a sequence of (O(φ log n), φ)-expander decompositions, where each decomposition is applied to the inter-cluster edges of the previous one, requires Ω(n/φ) bits of space, even in insertion-only streams. This lower bound suggests that the dependence on 1/φ in the space complexity of previous streaming algorithms for expander decompositions is inherent for this repeated application setting.
The key technical challenge in the lower bound proof is to construct a hard distribution of graphs that forces the algorithm to maintain a large amount of information about the structure of the graph, even when the algorithm is only required to output a few levels of the expander decomposition.
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by Yu Chen,Mich... at arxiv.org 04-26-2024
https://arxiv.org/pdf/2404.16701.pdfDeeper Inquiries