In this paper we study graph problems in the dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require $$\varOmega (n)$$Ω(n) space, where n is the number of vertices, existing works mainly focused on designing $${O}(n\cdot \mathrm {poly}\log n)$$O(n·polylogn) 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 $$\varepsilon n$$εn of a general graph and $$(1+\varepsilon )$$(1+ε)-approximating the weight of the minimum spanning tree of a connected graph with bounded edge weights, for any small constant $$\varepsilon >0$$ε>0. The latter improves upon the previous $$O(n\cdot \mathrm {poly}\log n)$$O(n·polylogn) space algorithm given by Ahn et al. (SODA 2012) for the same class of graphs. 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 $$\varepsilon $$ε-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 $${O}(n^{1-\varepsilon }\cdot \mathrm {poly}\log n)$$O(n1-ε·polylogn) space, which is o(n) for any constant $$\varepsilon $$ε. To complement our algorithms, we present $$\varOmega (n^{1-O(\varepsilon )})$$Ω(n1-O(ε)) space lower bounds for these problems, which show that such a dependence on $$\varepsilon $$ε is necessary.

中文翻译：

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o(n) 空间中的动态图流算法

在本文中，我们研究动态流模型中的图问题，其中输入由一系列边插入和删除定义。由于许多自然问题需要 $$\varOmega (n)$$Ω(n) 空间，其中 n 是顶点数，现有工作主要集中在设计 $${O}(n\cdot \mathrm {poly}\log n)$$O(n·polylogn) 空间算法。尽管密集图的边数是次线性的，但对于许多应用程序来说它仍然可能太大（例如，n 很大或图很稀疏）。在这项工作中，我们为两类问题提供了克服这个空间障碍的单遍算法。我们提出了 o(n) 空间算法，用于估计具有一般图的加性误差 $$\varepsilon n$$εn 和 $$(1+\varepsilon )$$(1+ε)-近似权重的连通分量的数量对于任何小常数 $$\varepsilon >0$$ε>0，具有有界边权重的连通图的最小生成树的计算。后者改进了之前 Ahn 等人给出的 $$O(n\cdot\mathrm {poly}\log n)$$O(n·polylogn) 空间算法。（SODA 2012）用于同一类图。我们开始研究动态流模型中的近似图属性测试，我们希望将满足该属性的图与 $$\varepsilon $$ε-远不具有该属性的图区分开来。我们考虑测试 k 边连通性、k 顶点连通性、循环自由度和二部性（平面图的）的问题，为此，我们提供大约使用 $${O}(n^{1-\varepsilon }\cdot \mathrm {poly}\log n)$$O(n1-ε·polylogn) 空间的算法，对于任何常数 $$\varepsilon $$ε。为了补充我们的算法，我们提出了这些问题的 $$\varOmega (n^{1-O(\varepsilon )})$$Ω(n1-O(ε)) 空间下界，这表明对 $ $\varepsilon $$ε 是必要的。