Graph Sketches: Sparsification, Spanners, and Subgraphs

When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., those based on linear projections of the data. These are applicable in many models including various parallel, stream, and compressed sensing settings. A rich body of analytic and empirical work exists for sketching numerical data such as the frequencies of a set of entities. Our work investigates graph sketching where the graphs of interest encode the relationships between these entities. The main challenge is to capture this richer structure and build the necessary synopses with only linear measurements. In this paper we consider properties of graphs including the size of the cuts, the distances between nodes, and the prevalence of dense sub-graphs. Our main result is a sketch-based sparsifier construction: we show that O̅(nε-2) random linear projections of a graph on n nodes suffice to (1 + ε) approximate all cut values. Similarly, we show that O(ε-2) linear projections suffice for (additively) approximating the fraction of induced sub-graphs that match a given pattern such as a small clique. Finally, for distance estimation we present sketch-based spanner constructions. In this last result the sketches are adaptive, i.e., the linear projections are performed in a small number of batches where each projection may be chosen dependent on the outcome of earlier sketches. All of the above results immediately give rise to data stream algorithms that also apply to dynamic graph streams where edges are both inserted and deleted. The non-adaptive sketches, such as those for sparsification and subgraphs, give us single-pass algorithms for distributed data streams with insertion and deletions. The adaptive sketches can be used to analyze MapReduce algorithms that use a small number of rounds

A taxonomy framework for unsupervised outlier detection techniques for multi-type data sets

The term "outlier" can generally be defined as an observation that is significantly different from the other values in a data set. The outliers may be instances of error or indicate events. The task of outlier detection aims at identifying such outliers in order to improve the analysis of data and further discover interesting and useful knowledge about unusual events within numerous applications domains. In this paper, we report on contemporary unsupervised outlier detection techniques for multiple types of data sets and provide a comprehensive taxonomy framework and two decision trees to select the most suitable technique based on data set. Furthermore, we highlight the advantages, disadvantages and performance issues of each class of outlier detection techniques under this taxonomy framework

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph $G=(V, E)$ in the streaming model of computation. The goal is to find a partitioning $\mathcal{C}$ of vertices $V$ such that the subgraphs of $G$ induced by the clusters $C \in \mathcal{C}$ are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of $V$) to within a $(\delta, \epsilon)$-multiplicative/additive error with high probability. The power cut sparsifier uses $\tilde{O}(n/\epsilon\delta)$ space and edges, which we show is asymptotically tight up to polylogarithmic factors in $n$ for constant $\delta$.Comment: 31 pages, 0 figures, to appear in ITCS 202

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?

On Constructing Spanners from Random Gaussian Projections

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work