Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams (1504.02268v3)

Abstract: While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em both} time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called {\em densest subgraph problem}. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. In a graph $G = (V, E)$, the "density" of a subgraph induced by a subset of nodes $S \subseteq V$ is defined as $|E(S)|/|S|$, where $E(S)$ is the set of edges in $E$ with both endpoints in $S$. In the densest subgraph problem, the goal is to find a subset of nodes that maximizes the density of the corresponding induced subgraph. For any $\epsilon>0$, we present a dynamic algorithm that, with high probability, maintains a $(4+\epsilon)$-approximation to the densest subgraph problem under a sequence of edge insertions and deletions in a graph with $n$ nodes. It uses $\tilde O(n)$ space, and has an amortized update time of $\tilde O(1)$ and a query time of $\tilde O(1)$. Here, $\tilde O$ hides a $O(\poly\log_{1+\epsilon} n)$ term. The approximation ratio can be improved to $(2+\epsilon)$ at the cost of increasing the query time to $\tilde O(n)$. It can be extended to a $(2+\epsilon)$-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in {\em one pass}. The previously best algorithm in this setting required $O(\log n)$ passes [Bahmani, Kumar and Vassilvitskii, VLDB'12]. The space required by our algorithm is tight up to a polylogarithmic factor.

• The paper introduces a novel algorithm achieving a (4+ε) approximation for densest subgraphs in one-pass dynamic streams.
• It employs sampling and sketching techniques to maintain graph updates with polylogarithmic space and nearly constant amortized times.
• The approach extends to a (2+ε) approximation with added computation, enabling real-time analysis in large-scale streaming applications.

A Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams: An Expert Review

The paper presents an advanced algorithmic solution for maintaining dense subgraphs in dynamic graph streams. This problem is a core component in many graph mining applications with streaming data, where the underlying graphs are subject to frequent updates such as edge insertions and deletions. The proposed algorithm efficiently handles these updates while balancing computational time and space usage, which was an area with few known results for dynamic graph streams prior to this work.

Key Contributions

The paper introduces a dynamic algorithm that provides a $(4+\epsilon)$-approximation for the densest subgraph problem. This algorithm operates with high probability, uses $\tilde{O}(n)$ space (where $\tilde{O}$ hides polylogarithmic factors), and features both amortized update and query times of $\tilde{O}(1)$. The authors also improve the approximation ratio to $(2+\epsilon)$ with increased computational overhead, extending the state of the art by providing an algorithm capable of maintaining such properties in a one-pass streaming context.

Definitions and Theoretical Framework

At the foundation of the authors' approach is the concept of an $(\alpha, d, L)$-decomposition. This structure, an extension of the $d$-core concept, allows for an approximate iterative process of node removal based on degree thresholds, effectively aiding in the approximation of dense subgraphs. The paper thoroughly explores the application of this structure to both build and maintain an approximation to the densest subgraph as the input graph undergoes dynamic changes. These techniques are further extended to directed graphs, showcasing the adaptability of the framework.

Algorithmic Design

The algorithm leverages two main strategies:

1. Sampling and Sketching: By sampling $\tilde{O}(n)$ edges from the dynamic graph stream, the authors develop sketches that permit efficient approximations of dense subgraphs, employing techniques from streaming algorithm literature.
2. Dynamic Maintenance: The novel approach utilized combines space-efficient sampling with dynamic, time-efficient graph updates. This hybrid method bridges the gap between fully dynamic and streaming algorithms, leading to the near-optimal bounds presented.

Results and Implications

The results are significant not only because they provide strong approximation guarantees but also because they achieve low computational overhead. These characteristics make the presented algorithm suitable for large-scale applications that require rapid adaptation to data stream updates. The ability to effectively handle insertions and deletions in a single-pass model is a robust feature with implications for real-time graph analysis systems in various domains including social network analysis, web graph studies, and more.

Future Directions

The paper identifies several avenues for further exploration. Enhancing the existing approximation ratio for maintaining densest subgraphs remains a challenging open problem. Additionally, the utility of similar techniques for other graph problems, such as maximum matching or maintaining shortest paths, suggests a broader applicability of these ideas. Moreover, there is potential for improving worst-case update time bounds, which could lead to more robust algorithms under strict real-time constraints.

Conclusion

Overall, this paper makes a considerable contribution to the field of dynamic graph analysis within streaming contexts, expanding both the theoretical understanding and practical capabilities for handling dense subgraph computations efficiently. The introduction of time- and space-efficient algorithms sets a foundation for continued research into similar challenges in dynamic, streaming data environments.