[Paper Review] Densest Subgraph in Dynamic Graph Streams

Andrew McGregor, David Tench+2 more|arXiv (Cornell University)|Jun 14, 2015

Complexity and Algorithms in Graphs16 references4 citations

TL;DR

This paper presents a single-pass dynamic graph stream algorithm that approximates the densest subgraph within a (1+ε) factor using O(ε⁻²n polylog n) space and polylog(n) per-update time. By leveraging pairwise independent hashing and ℓ₀-sampling primitives, it efficiently samples edges and computes a sparse proxy graph to estimate maximum subgraph density with high probability.

ABSTRACT

In this paper, we consider the problem of approximating the densest subgraph in the dynamic graph stream model. In this model of computation, the input graph is defined by an arbitrary sequence of edge insertions and deletions and the goal is to analyze properties of the resulting graph given memory that is sub-linear in the size of the stream. We present a single-pass algorithm that returns a $(1+ε)$ approximation of the maximum density with high probability; the algorithm uses $O(ε^{-2} n \polylog n)$ space, processes each stream update in $\polylog (n)$ time, and uses $\poly(n)$ post-processing time where $n$ is the number of nodes. The space used by our algorithm matches the lower bound of Bahmani et al.~(PVLDB 2012) up to a poly-logarithmic factor for constant $ε$. The best existing results for this problem were established recently by Bhattacharya et al.~(STOC 2015). They presented a $(2+ε)$ approximation algorithm using similar space and another algorithm that both processed each update and maintained a $(4+ε)$ approximation of the current maximum density in $\polylog (n)$ time per-update.

Motivation & Objective

• Address the challenge of approximating the densest subgraph in dynamic graph streams with sub-linear memory.
• Achieve a (1+ε)-approximation of the maximum subgraph density with high probability.
• Maintain polylogarithmic update time per edge insertion/deletion while using space close to the information-theoretic lower bound.
• Enable efficient post-processing to compute the final estimate using exact densest subgraph algorithms on the sampled graph.
• Overcome limitations of prior work that achieved only (2+ε) or (4+ε) approximation with similar resource constraints.

Proposed method

• Use pairwise independent hash functions to partition edges into groups E_{i,j} based on hash values.
• Sample τ = 24c log n edges uniformly without replacement from each E_{i,j} using ℓ₀-sampling primitives.
• Leverage linear sketches from Jowhari et al. to sample distinct edges by subtracting previously sampled edge indicators from sketches.
• Maintain counters for |E_{i,j}| and total edge count m to compute sampling probability p = (ε² m)/(n log n).
• Construct a sparse proxy graph G' from sampled edges, ensuring that subgraph densities in G' approximate those in the original graph.
• Compute the maximum density in G' using Charikar's exact algorithm, then scale by 1/p to estimate the original graph's densest subgraph density.

Experimental results

Research questions

• RQ1 Can a (1+ε)-approximation of the densest subgraph be maintained in a dynamic graph stream with polylogarithmic update time and near-optimal space?
• RQ2 Is it possible to achieve high-probability (1+ε)-approximation using only a single pass over the stream?
• RQ3 How can ℓ₀-sampling be adapted to sample without replacement efficiently in a dynamic setting?
• RQ4 Can the space usage be made asymptotically tight relative to the known lower bound for this problem?
• RQ5 What is the trade-off between approximation quality and update efficiency in fully dynamic graph stream models?

Key findings

• The algorithm achieves a (1+ε)-approximation of the densest subgraph with high probability, improving upon the (2+ε) and (4+ε) bounds of prior work.
• Space usage is O(ε⁻²n polylog n), matching the lower bound of Bahmani et al. up to a polylogarithmic factor for constant ε.
• Each edge update is processed in polylog(n) time, achieving worst-case rather than amortized efficiency.
• The post-processing step runs in poly(n) time, which is feasible due to the sparsity of the sampled graph.
• The use of multiple hash partitions and ℓ₀-sampling with subtraction enables distinct, uniform sampling without replacement in polylog(n) time.
• The analysis shows that with high probability, the number of sampled edges per group E_{i,j} remains below τ = 24c log n, ensuring correctness and efficiency.