[Paper Review] Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

Sayan Bhattacharya, Monika Henzinger+2 more|arXiv (Cornell University)|Apr 9, 2015

Complexity and Algorithms in Graphs28 references4 citations

TL;DR

This paper presents the first one-pass dynamic streaming algorithm for maintaining a densest subgraph with optimal space and near-constant update time. It achieves a $(4+\epsilon)$-approximation using $\tilde{O}(n)$ space and $\tilde{O}(1)$ amortized update and query time, significantly improving upon prior multi-pass approaches and setting a new standard for time- and space-efficient graph stream processing.

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 $ε>0$, we present a dynamic algorithm that, with high probability, maintains a $(4+ε)$-approximation to the densest subgraph problem under a sequence of edge insertions and deletions in a graph with $n$ nodes. It uses $ ilde O(n)$ space, and has an amortized update time of $ ilde O(1)$ and a query time of $ ilde O(1)$. Here, $ ilde O$ hides a $O(\poly\log_{1+ε} n)$ term. The approximation ratio can be improved to $(2+ε)$ at the cost of increasing the query time to $ ilde O(n)$. It can be extended to a $(2+ε)$-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.

Motivation & Objective

• To address the challenge of maintaining dense subgraphs in dynamic graph streams with minimal space and time overhead.
• To design a streaming algorithm that supports both edge insertions and deletions efficiently in a single pass.
• To achieve near-constant update and query times while maintaining strong approximation guarantees.
• To close the gap between theoretical streaming models and practical dynamic graph workloads by enabling real-time analysis.

Proposed method

• The algorithm uses a novel $(\alpha,d,L)$-decomposition to maintain structural properties of dense subgraphs under dynamic updates.
• It employs $\ell_0$-sampling and uniform hashing to efficiently sample edges and maintain a compact sketch of the graph.
• A potential function-based analysis ensures amortized $\tilde{O}(1)$ update time by bounding the number of structural changes per update.
• The algorithm dynamically partitions time into sparse and dense intervals, applying different optimization strategies based on edge density.
• For dense intervals, it uses a specialized subroutine called Dynamic-Stream that maintains a sampled edge set to enable fast approximation.
• It leverages concentration bounds and polylogarithmic factors to ensure high-probability correctness under $\tilde{O}(n)$ space.

Experimental results

Research questions

• RQ1 Can a one-pass dynamic streaming algorithm maintain a $(4+\epsilon)$-approximate densest subgraph with $\tilde{O}(n)$ space and $\tilde{O}(1)$ amortized update time?
• RQ2 Is it possible to achieve sublinear query time while maintaining a strong approximation ratio in dynamic graph streams?
• RQ3 Can the algorithm be extended to support sublinear-time and distributed streaming settings with minimal communication?
• RQ4 What is the theoretical limit of approximation ratio for space- and time-efficient dynamic densest subgraph algorithms?
• RQ5 Can similar techniques be applied to other fundamental graph problems like maximum matching or shortest paths in dynamic streams?

Key findings

• The algorithm achieves a $(4+\epsilon)$-approximation for the densest subgraph problem with $\tilde{O}(n)$ space and $\tilde{O}(1)$ amortized update and query time.
• The approximation ratio can be improved to $(2+\epsilon)$ at the cost of $\tilde{O}(n)$ query time, enabling a sublinear-time algorithm.
• The algorithm is the first to maintain a densest subgraph in a single pass over dynamic streams, improving upon the prior $O(\log n)$-pass requirement.
• It supports extensions to directed graphs, sublinear-time algorithms, and distributed streaming models with $\tilde{O}(k+n)$ communication.
• The space complexity is tight up to polylogarithmic factors, as shown by a lower bound of $\tilde{\Omega}(n/k^2)$ in a multi-party communication model.
• A hardness result for $\lambda$-approximation algorithms implies that $\tilde{\Omega}(n/\lambda^2)$ queries are necessary, supporting the optimality of the space bound.