Core Concepts

Computing a sequence of expander decompositions, where each decomposition is applied to the inter-cluster edges of the previous one, requires space that depends on the sparsity parameter φ, even in insertion-only streams.

Abstract

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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Key Insights Distilled From

by Yu Chen,Mich... at **arxiv.org** 04-26-2024

Deeper Inquiries

The lower bound for the space complexity of computing a sequence of expander decompositions, as shown in the context, states that any streaming algorithm that computes at least two levels of an (ǫ, φ)-RED requires e Ω(n/φ) bits of space. This lower bound implies that there is a fundamental limitation in terms of space complexity when dealing with repeated expander decompositions in streaming algorithms. The lower bound suggests that achieving a space complexity better than e Ω(n/φ) bits for computing a sequence of expander decompositions is not possible, at least based on the current understanding of the problem.
To improve or match this lower bound with an algorithm, one would need to devise a fundamentally different approach that circumvents the inherent limitations imposed by the sparsity parameter φ. This could involve exploring novel data structures, algorithmic techniques, or theoretical breakthroughs that allow for more efficient space utilization while maintaining the quality of the expander decompositions. However, as of now, there is no known algorithm that can achieve a space complexity better than e Ω(n/φ) bits for computing a sequence of expander decompositions in streaming settings.

Expander decompositions have a wide range of applications beyond the specific context discussed. One potential area where the dependence on the sparsity parameter φ can be avoided in streaming algorithms is in the field of network analysis and anomaly detection. In network traffic analysis, expander decompositions can be used to identify clusters of network nodes that exhibit anomalous behavior or patterns. By partitioning the network into expanders without being constrained by φ-dependent space complexities, streaming algorithms can efficiently detect and analyze network anomalies in real-time.
Another application where the dependence on φ can be mitigated is in social network analysis. Expander decompositions can help identify cohesive communities within social networks, aiding in targeted marketing, recommendation systems, and community detection. By developing streaming algorithms that can construct expander decompositions without space constraints tied to φ, researchers can gain deeper insights into the structure and dynamics of social networks in a more efficient manner.

The inherent sparsity dependence in the design of streaming algorithms that rely on repeated applications of expander decompositions has significant implications for various algorithmic applications, such as the maximum flow algorithm mentioned in the context.
One implication is that the space complexity of these algorithms will be influenced by the sparsity parameter φ, potentially limiting their scalability and efficiency. This dependence can pose challenges in scenarios where the sparsity of the input graph varies widely, impacting the performance of the algorithm.
Furthermore, the sparsity dependence can affect the applicability of these algorithms in dynamic environments where the graph structure evolves over time. Adapting streaming algorithms to handle dynamic graphs while maintaining efficient expander decompositions may require innovative approaches to address the inherent limitations imposed by φ.
Overall, the sparsity dependence in streaming algorithms utilizing expander decompositions underscores the importance of developing robust and adaptive algorithms that can effectively balance space efficiency with computational accuracy in dynamic and evolving graph settings.

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