一遍动态流上保持密集子图的空间和时间效率算法

Sayan Bhattacharya, M. Henzinger, Danupon Nanongkai, Charalampos E. Tsourakakis
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引用次数: 127

摘要

虽然在许多图挖掘应用程序中,从时间和空间的角度有效地处理更新流是至关重要的,但人们对实现这种类型的算法知之甚少。在本文中,我们研究了许多图挖掘应用的核心问题——最密集子图问题。我们开发了一种同时实现时间和空间效率的算法。据我们所知,这是图形问题的第一种方法。给定一个输入图,密度最大的子图是使边数与节点数之比最大化的子图。对于任何ε>0,我们的算法在插入和删除边缘的情况下,使用~O(n)空间和~O(1)平摊时间,有很高的概率保持一个(4+ε)-近似解;这里,$n$是图中的节点数,~O隐藏了O(polylog_{1+ε} n)项。随着时间的延长,近似比可以提高到(2+ε)。它可以推广为(2+ε)近似亚线性时间算法和分布式流算法。我们的算法是第一个可以一次维护最密集子图的流算法。在此之前,即使在增量流的特殊情况下,即使没有时间限制,也没有算法可以做到这一点。在此设置中,之前的最佳算法需要通过O(log n)次[BahmaniKV12]。我们的算法所需的空间紧到一个多对数因子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams
While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of 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 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. Given an input graph, the densest subgraph is the subgraph that maximizes the ratio between the number of edges and the number of nodes. For any ε>0, our algorithm can, with high probability, maintain a (4+ε)-approximate solution under edge insertions and deletions using ~O(n) space and ~O(1) amortized time per update; here, $n$ is the number of nodes in the graph and ~O hides the O(polylog_{1+ε} n) term. The approximation ratio can be improved to (2+ε) with more time. 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 one pass. Prior to this, no algorithm could do so even in the special case of an incremental stream and even when there is no time restriction. The previously best algorithm in this setting required O(log n) passes [BahmaniKV12]. The space required by our algorithm is tight up to a polylogarithmic factor.
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