Dynamic Graph Stream Algorithms in o(n)o(n) Space

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 $\tilde{O}(n)$ 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 $\tilde{O}(n)$ 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 $\tilde{O}(n^{1-\varepsilon})$ space, which is $o(n)$ for any constant $\varepsilon$. To complement our algorithms, we present $\Omega(n^{1-O(\varepsilon)})$ space lower bounds for these problems, which show that such a dependence on $\varepsilon$ is necessary.Comment: ICALP 201

Testing Uniformity of Stationary Distribution

A random walk on a directed graph gives a Markov chain on the vertices of the graph. An important question that arises often in the context of Markov chain is whether the uniform distribution on the vertices of the graph is a stationary distribution of the Markov chain. Stationary distribution of a Markov chain is a global property of the graph. In this paper, we prove that for a regular directed graph whether the uniform distribution on the vertices of the graph is a stationary distribution, depends on a local property of the graph, namely if (u,v) is an directed edge then outdegree(u) is equal to indegree(v). This result also has an application to the problem of testing whether a given distribution is uniform or "far" from being uniform. This is a well studied problem in property testing and statistics. If the distribution is the stationary distribution of the lazy random walk on a directed graph and the graph is given as an input, then how many bits of the input graph do one need to query in order to decide whether the distribution is uniform or "far" from it? This is a problem of graph property testing and we consider this problem in the orientation model (introduced by Halevy et al.). We reduce this problem to test (in the orientation model) whether a directed graph is Eulerian. And using result of Fischer et al. on query complexity of testing (in the orientation model) whether a graph is Eulerian, we obtain bounds on the query complexity for testing whether the stationary distribution is uniform

An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs

We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by d, any property that can be tested with query complexity O_{?,d}(1) in the bidirectional model can be tested with n^{1-?_{?,d}(1)} queries in the unidirectional model. In particular, {if the proximity parameter ? approaches 0, then the query complexity of the transformed tester in the unidirectional model approaches n}. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation. We prove that testing subgraph-freeness in which the subgraph contains k source components, requires ?(n^{1-1/k}) queries in the unidirectional model. This directly gives the first explicit properties that exhibit an O_{?,d}(1) vs ?(n^{1-f(?,d)}) separation of the query complexities between the bidirectional model and unidirectional model, where f(?,d) is a function that approaches 0 as ? approaches 0. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing k-star-freeness

Dynamic graph stream algorithms in o(n) space

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 Ω(n) space, where n is the number of vertices, existing works mainly focused on designing 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 εn of a general graph and (1+ε) -approximating the weight of the minimum spanning tree of a connected graph with bounded edge weights, for any small constant ε>0 . The latter improves upon the previous 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 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(n1−ε⋅polylogn) space, which is o(n) for any constant ε . To complement our algorithms, we present Ω(n1−O(ε)) space lower bounds for these problems, which show that such a dependence on ε is necessary

Relating two property testing models for bounded degree directed graphs

We study property testing algorithms in directed graphs (digraphs) with maximum indegree and maximum outdegree upper bounded by d. For directed graphs with bounded degree, there are two different models in property testing introduced by Bender and Ron (2002). In the bidirectional model, one can access both incoming and outgoing edges while in the unidirectional model one can only access outgoing edges. In our paper we provide a new relation between the two models: we prove that if a property can be tested with constant query complexity in the bidirectional model, then it can be tested with sublinear query complexity in the unidirectional model. A corollary of this result is that in the unidirectional model (the model allowing only queries to the outgoing neighbors), every property in hyperfinite digraphs is testable with sublinear query complexity

The Subgraph Testing Model

We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base graph G=([n],E), and oracle access to a function f:E -> {0,1} that represents a subgraph of G. The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property. We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model. We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models. Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds