# Dynamic Graph Stream Algorithms in o(n) Space

arXiv: https://arxiv.org/abs/1605.00089 · HTML: https://mlsanity.com/paper/1605.00089/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2016.

## Built on by (most cited first, 1 of 1)

- [Continual Release of Densest Subgraphs: Privacy Amplification & Sublinear Space via Subsampling](https://mlsanity.com/paper/2510.11640/) (2025)

## Abstract (arXiv)

In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require Omega(n) space, where n is the number of vertices, existing works mainly focused on designing tildeO(n) space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. n is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present o(n) space algorithms for estimating the number of connected components with additive error varepsilon n and (1+varepsilon)-approximating the weight of minimum spanning tree, for any small constant varepsilon>0. The latter improves previous tildeO(n) space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are varepsilon-far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly tildeO(n1-varepsilon) space, which is o(n) for any constant varepsilon. To complement our algorithms, we present Omega(n1-O(varepsilon)) space lower bounds for these problems, which show that such a dependence on varepsilon is necessary.