How do we spot interesting events from e-mail or transportation logs? How can we detect port scan or denial of service attacks from IP-IP communication data? In general, given a sequence of weighted, directed or bipartite graphs, each summarizing a snapshot of activity in a time window, how can we spot anomalous graphs containing the sudden appearance or disappearance of large dense subgraphs (e.g., near bicliques) in near real-time using sublinear memory? To this end, we propose a randomized sketching-based approach called SpotLight, which guarantees that an anomalous graph is mapped 'far' away from 'normal' instances in the sketch space with high probability for appropriate choice of parameters. Extensive experiments on real-world datasets show that SpotLight (a) improves accuracy by at least 8.4% compared to prior approaches, (b) is fast and can process millions of edges within a few minutes, (c) scales linearly with the number of edges and sketching dimensions and (d) leads to interesting discoveries in practice.

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SpotLight: Detecting Anomalies in Streaming Graphs

We study the problem of finding an approximate maximum matching in two closely related computational models, namely, the dynamic graph streaming model and the simultaneous multi-party communication model. In the dynamic graph streaming model, the input graph is revealed as a stream of edge insertions and deletions, and the goal is to design a small space algorithm to approximate the maximum matching. In the simultaneous model, the input graph is partitioned across k players, and the goal is to design a protocol where the k players simultaneously send a small-size message to a coordinator, and the coordinator computes an approximate matching.Dynamic graph streams. We resolve the space complexity of single-pass turnstile streaming algorithms for approximating matchings by showing that for any e > 0, Θ(n2-3e) space is both sufficient and necessary (up to polylogarithmic factors) to compute an ne-approximate matching; here n denotes the number of vertices in the input graph.The simultaneous communication model. Our results for dynamic graph streams also resolve the (per-player) simultaneous communication complexity for approximating matchings in the edge partition model. For the vertex partition model, we design new randomized and deterministic protocols for k players to achieve an α-approximation. Specifically, for α ≥ [EQUATION], we provide a randomized protocol with total communication of O(nk/α2) and a deterministic protocol with total communication of O(nk/α). Both these bounds are tight. Our work generalizes the results established by Dobzinski et al. (STOC 2014) for the special case of k = n. Finally, for the case of α = o([EQUATION]), we establish a new lower bound on the simultaneous communication complexity which is super-linear in n.

/pdf/maximum-matchings-in-dynamic-graph-streams-and-the-2deriiurnl.pdf

Maximum matchings in dynamic graph streams and the simultaneous communication model

While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em 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 {\em 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. 
In a graph $G = (V, E)$, the "density" of a subgraph induced by a subset of nodes $S \subseteq V$ is defined as $|E(S)|/|S|$, where $E(S)$ is the set of edges in $E$ with both endpoints in $S$. In the densest subgraph problem, the goal is to find a subset of nodes that maximizes the density of the corresponding induced subgraph. For any $\epsilon>0$, we present a dynamic algorithm that, with high probability, maintains a $(4+\epsilon)$-approximation to the densest subgraph problem under a sequence of edge insertions and deletions in a graph with $n$ nodes. It uses $\tilde O(n)$ space, and has an amortized update time of $\tilde O(1)$ and a query time of $\tilde O(1)$. Here, $\tilde O$ hides a $O(\poly\log_{1+\epsilon} n)$ term. The approximation ratio can be improved to $(2+\epsilon)$ at the cost of increasing the query time to $\tilde O(n)$. It can be extended to a $(2+\epsilon)$-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in {\em one pass}. The previously best algorithm in this setting required $O(\log n)$ passes [Bahmani, Kumar and Vassilvitskii, VLDB'12]. The space required by our algorithm is tight up to a polylogarithmic factor.

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results:• Matching: Our main result for matchings is that there exists an O(k2) space algorithm that returns the edges of a maximum matching on the assumption the cardinality is at most k. The best previous algorithm used O(kn) space where n is the number of vertices in the graph and we prove our result is optimal up to logarithmic factors. Our algorithm has O(1) update time. We also show that there exists an O(n2/α3) space algorithm that returns an α-approximation for matchings of arbitrary size. In independent work, Assadi et al. (SODA 2016) proved this approximation algorithm is optimal and provided an alternative algorithm. We generalize our exact and approximate algorithms to weighted matching. For graphs with low arboricity such as planar graphs, the space required for constant approximation can be further reduced. While there has been a substantial amount of work on approximate matching in insert-only graph streams, these are the first non-trivial results in the dynamic setting.• Vertex Cover and Hitting Set: There exists an O(kd) space algorithm that solves the minimum hitting set problem where d is the cardinality of the input sets and k is an upper bound on the size of the minimum hitting set. We prove this is optimal up to logarithmic factors. Our algorithm has O(1) update time. The case d = 2 corresponds to minimum vertex cover.Finally, we consider a larger family of parameterized problems (including b-matching, disjoint paths, vertex coloring among others) for which our subgraph sampling primitive yields fast, small-space dynamic graph stream algorithms. We then show lower bounds for natural problems outside this family.

/pdf/kernelization-via-sampling-with-applications-to-finding-57kfka4vdx.pdf

Kernelization via sampling with applications to finding matchings and related problems in dynamic graph 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 e>0, our algorithm can, with high probability, maintain a (4+e)-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+e} n) term. The approximation ratio can be improved to (2+e) with more time. It can be extended to a (2+e)-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.

/pdf/space-and-time-efficient-algorithm-for-maintaining-dense-io3d17wqs1.pdf

In this paper, we consider the problem of approximating the densest subgraph in the dynamic graph stream model. In this model of computation, the input graph is defined by an arbitrary sequence of edge insertions and deletions and the goal is to analyze properties of the resulting graph given memory that is sub-linear in the size of the stream. We present a single-pass algorithm that returns a $(1+\epsilon )$ approximation of the maximum density with high probability; the algorithm uses $O(\epsilon ^{-2} n {{\mathrm{polylog}}}~n)$ space, processes each stream update in ${{\mathrm{polylog}}}(n)$ time, and uses ${{\mathrm{poly}}}(n)$ post-processing time where n is the number of nodes. The space used by our algorithm matches the lower bound of Bahmani et al. (PVLDB 2012) up to a poly-logarithmic factor for constant $\epsilon $. The best existing results for this problem were established recently by Bhattacharya et al. (STOC 2015). They presented a $(2+\epsilon )$ approximation algorithm using similar space and another algorithm that both processed each update and maintained a $(4+\epsilon )$ approximation of the current maximum density in ${{\mathrm{polylog}}}(n)$ time per-update.

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Densest Subgraph in Dynamic Graph Streams

We present space-efficient data stream algorithms for approximating the number of triangles in a graph up to a factor 1+e. While it can be shown that determining whether a graph is triangle-free is not possible in sub-linear space, a large body of work has focused on minimizing the space required in terms of the number of triangles T (or a lower bound on this quantity) and other parameters including the number of nodes n and the number of edges m. Two models are important in the literature: the arbitrary order model in which the stream consists of the edges of the graph in arbitrary order and the adjacency list order model in which all edges incident to the same node appear consecutively. We improve over the state of the art results in both models. For the adjacency list order model, we show that ~O(e-2m/√T) space is sufficient in one pass and ~O(e-2m3/2/T) space is sufficient in two passes where the ~O(·) notation suppresses log factors. For the arbitrary order model, we show that ~O(e-2m/√T) space suffices given two passes and that ~O(e-2m3/2/T) space suffices given three passes and oracle access to the degrees. Finally, we show how to efficiently implement the "wedge sampling" approach to triangle estimation in the arbitrary order model. To do this, we develop the first algorithm for lp sampling such that multiple independent samples can be generated with O(polylog n) update time; this primitive is widely applicable and this result may be of independent interest.

https://dl.acm.org/doi/pdf/10.1145/2902251.2902283

Better Algorithms for Counting Triangles in Data Streams

The trace reconstruction problem is to reconstruct a string x of length n given m random subsequences where each subsequence is generated by deleting each character of x independently with probability p. Two natural questions are a) how large must m be as a function of n and p such that reconstruction is possible with high probability and b) how can this reconstruction be performed efficiently. Existing work considers the case when x is chosen uniformly at random and when x is arbitrary. In this paper, we relate the complexity of both cases; improve bounds by Holenstein et al. (SODA 2008) on the sufficient value of m in both cases; and present a significantly simpler analysis for some of the results proved by Viswanathan and Swaminathan (SODA 2008), Kannan and McGregor (ISIT 2005), and Batu et al. (SODA 2004). In particular, our work implies the first sub-polynomial upper bound (when the alphabet is polylogn) and super-logarithmic lower bound on the number of traces required when x is random and p is constant.

/pdf/trace-reconstruction-revisited-2yc9dx7a1s.pdf

Trace Reconstruction Revisited

We present data stream algorithms for estimating the size or weight of the maximum matching in low arboricity graphs. A large body of work has focused on improving the constant approximation factor for general graphs when the data stream algorithm is permitted O(n polylog n) space where n is the number of nodes. This space is necessary if the algorithm must return the matching. Recently, Esfandiari et al. (SODA 2015) showed that it was possible to estimate the maximum cardinality of a matching in a planar graph up to a factor of 24+epsilon using O(epsilon^{-2} n^{2/3} polylog n) space. We first present an algorithm (with a simple analysis) that improves this to a factor 5+epsilon using the same space. We also improve upon the previous results for other graphs with bounded arboricity. We then present a factor 12.5 approximation for matching in planar graphs that can be implemented using O(log n) space in the adjacency list data stream model where the stream is a concatenation of the adjacency lists of the graph. The main idea behind our results is finding "local" fractional matchings, i.e., fractional matchings where the value of any edge e is solely determined by the edges sharing an endpoint with e. Our work also improves upon the results for the dynamic data stream model where the stream consists of a sequence of edges being inserted and deleted from the graph. We also extend our results to weighted graphs, improving over the bounds given by Bury and Schwiegelshohn (ESA 2015), via a reduction to the unweighted problem that increases the approximation by at most a factor of two.

Sofya Vorotnikova

Papers

Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams

Densest Subgraph in Dynamic Graph Streams

Better Algorithms for Counting Triangles in Data Streams

Trace Reconstruction Revisited

Planar Matching in Streams Revisited