Analyzing massive data sets has been one of the key motivations for studying streaming algorithms. In recent years, there has been significant progress in analysing distributions in a streaming setting, but the progress on graph problems has been limited. A main reason for this has been the existence of linear space lower bounds for even simple problems such as determining the connectedness of a graph. However, in many new scenarios that arise from social and other interaction networks, the number of vertices is significantly less than the number of edges. This has led to the formulation of the semi-streaming model where we assume that the space is (near) linear in the number of vertices (but not necessarily the edges), and the edges appear in an arbitrary (and possibly adversarial) order.
In this paper we focus on graph sparsification, which is one of the major building blocks in a variety of graph algorithms. There has been a long history of (non-streaming) sampling algorithms that provide sparse graph approximations and it a natural question to ask if the sparsification can be achieved using a small space, and in addition using a single pass over the data? The question is interesting from the standpoint of both theory and practice and we answer the question in the affirmative, by providing a one pass $\tilde{O}(n/ε^{2})$ space algorithm that produces a sparsification that approximates each cut to a $(1+ε)$ factor. We also show that $Ω(n \log \frac1ε)$ space is necessary for a one pass streaming algorithm to approximate the min-cut, improving upon the $Ω(n)$ lower bound that arises from lower bounds for testing connectivity.
In this paper we give a construction of cut sparsifiers of Benczur and Karger in the {\em dynamic} streaming setting in a single pass over the data stream. Previous constructions either required multiple passes or were unable to handle edge deletions. We use $\tilde{O}(1/\e^2)$ time for each stream update and $\tilde{O}(n/\e^2)$ time to construct a sparsifier. Our $\e$-sparsifiers have $O(n\log^3 n/\e^2)$ edges. The main tools behind our result are an application of sketching techniques of Ahn et al.[SODA'12] to estimate edge connectivity together with a novel application of sampling with limi…
A graph G'(V,E') is an \eps-sparsification of G for some \eps>0, if every (weighted) cut in G' is within (1\pm \eps) of the corresponding cut in G. A celebrated result of Benczur and Karger shows that for every undirected graph G, an \eps-sparsification with O(n\log n/\e^2) edges can be constructed in O(m\log^2n) time. Applications to modern massive data sets often constrain algorithms to use computation models that restrict random access to the input. The semi-streaming model, in which the algorithm is constrained to use \tilde O(n) space, has been shown to be a good abstraction for analyzing…
We derive a new proof to show that the incremental resparsification algorithm proposed by Kelner and Levin (2013) produces a spectral sparsifier in high probability. We rigorously take into account the dependencies across subsequent resparsifications using martingale inequalities, fixing a flaw in the original analysis.
In this paper we consider the problem of computing spectral approximations to graphs in the single pass dynamic streaming model. We provide a linear sketching based solution that given a stream of edge insertions and deletions to a $n$-node undirected graph, uses $\tilde O(n)$ space, processes each update in $\tilde O(1)$ time, and with high probability recovers a spectral sparsifier in $\tilde O(n)$ time. Prior to our work, state of the art results either used near optimal $\tilde O(n)$ space complexity, but brute-force $Ω(n^2)$ recovery time [Kapralov et al.'14], or with subquadratic runtime…
A spectral sparsifier of a graph $G$ is a sparser graph $H$ that approximately preserves the quadratic form of $G$, i.e. for all vectors $x$, $x^T L_G x \approx x^T L_H x$, where $L_G$ and $L_H$ denote the respective graph Laplacians. Spectral sparsifiers generalize cut sparsifiers, and have found many applications in designing graph algorithms. In recent years, there has been interest in computing spectral sparsifiers in semi-streaming and dynamic settings. Natural algorithms in these settings often involve repeated sparsification of a graph, and accumulation of errors across these steps. We …
A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $Δ$, sampling $O(\log{n})$ colors per each vertex independently from $Δ+1$ colors almost certainly allows for proper coloring of $G$ from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown to have various applications to design of algorithms for $(Δ+1)$ coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms. In this paper, we further study…
In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use $2^{\frac1ε}$ passes or $n^{\frac1ε}$ time to compute a $(1-ε)$ approximation. In this paper we provide the first FPTAS (polynomial in $n,\frac1ε$) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the …
We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph G, our algorithm maintains a randomized linear sketch of the incidence matrix of G into dimension O((1/epsilon^2) n polylog(n)). Using this sketch, at any point, the algorithm can output a (1 +/- epsilon) spectral sparsifier for G with high probability. While O((1/epsilon^2) n polylog(n)) space algorithms are known for computing "cut sparsifiers" in dynamic streams [AGM12b, GKP12] a…
We present a simple deterministic single-pass $(2+ε)$-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of $(4+ε)$. Our algorithm uses $O(n\log^2 n)$ bits of space for constant values of $ε$. It relies on a variation of the local-ratio theorem, which may be of use for other algorithms in the semi-streaming model as well.
In this paper we consider the problem of finding a maximum weight set subject to a $k$-extendible constraint in the data stream model. The only non-trivial algorithm known for this problem to date---to the best of our knowledge---is a semi-streaming $k^2(1 + \varepsilon)$-approximation algorithm (Crouch and Stubbs, 2014), but semi-streaming $O(k)$-approximation algorithms are known for many restricted cases of this general problem. In this paper, we close most of this gap by presenting a semi-streaming $O(k \log k)$-approximation algorithm for the general problem, which is almost the best poss…
In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniqu…
Considerable effort has been devoted to the development of streaming algorithms for analyzing massive graphs. Unfortunately, many results have been negative, establishing that a wide variety of problems require $Ω(n^2)$ space to solve. One of the few bright spots has been the development of semi-streaming algorithms for a handful of graph problems -- these algorithms use space $O(n\cdot\text{polylog}(n))$. In the annotated data streaming model of Chakrabarti et al., a computationally limited client wants to compute some property of a massive input, but lacks the resources to store even a small…
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other graph sparsifiers using linear sketching, and obtained near-linear upper bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21]. All these linear sketching algorithms, however, only work on unweighted graphs. In this paper, we initiate the study of weighted…
In the semi-streaming model, an algorithm receives a stream of edges of a graph in arbitrary order and uses a memory of size $O(n \mbox{ polylog } n)$, where $n$ is the number of vertices of a graph. In this work, we present semi-streaming algorithms that perform one or two passes over the input stream for maximum matching with no restrictions on the input graph, and for the important special case of bipartite graphs that we refer to as maximum bipartite matching (MBM). The Greedy matching algorithm performs one pass over the input and outputs a $1/2$ approximation. Whether there is a better o…
Semi-Streaming Algorithms for Submodular Maximization under Random Arrival Order
Tel-Aviv University（特拉维夫大学）；University of Haifa（海法大学）；Queen Mary University of London（伦敦玛丽女王大学）；Tepper School of Business, Carnegie Mellon University (CMU)（卡内基梅隆大学泰珀商学院）；Carnegie Mellon University (CMU)（卡内基梅隆大学）
In a recent breakthrough, Paz and Schwartzman (SODA'17) presented a single-pass ($2+ε$)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. Their algorithm uses $O(n\log^2 n)$ bits of space, for any constant $ε>0$. We present two simplified and more intuitive analyses, for essentially the same algorithm, which also improve the space complexity to the optimal bound of $O(n\log n)$ bits --- this is optimal as the output matching requires $Ω(n\log n)$ bits. Our analyses rely on a simple use of the primal-dual method and a simple accounting method.
The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approxima…
In this paper, we generalize Spencer's hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size $n$, it constructs an expanding Cayley graph of logarithmic degree in near-optimal O(n^2 log^3 n) time. For the latter case, we present a fast …
We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke (Proc. of STACS2008, pages 669-680) by devising a deterministic approach whose performance guarantee is 4.91+epsilon. In addition, we study preemptive online algorithms, a sub-class of one-pass algorithms where we are only allowed to maintain a feasible matching in memory at any point in time. All known results prior to Zelke's belong to this sub-class. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and henc…
In this paper, we study the problem of finding a maximum matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, an algorithm receives a stream of edges and it is allowed to have a memory of $\tilde{O}(n)$ where $n$ is the number of vertices in the graph. A recent inspiring work by Assadi et al. shows that there exists a streaming algorithm with the approximation ratio of $\frac{2}{3}$ that uses $\tilde{O}(n^{1.5})$ memory. However, the memory of their algorithm is much larger than the memory constraint of the semi-streaming algorithms. In this wo…
We present the first semi-streaming PTAS for the minimum feedback arc set problem on directed tournaments in a small number of passes. Namely, we obtain a $(1 + \varepsilon)$-approximation in polynomial time $O \left( \text{poly}(n) 2^{\text{poly}(1/\varepsilon)} \right)$, with $p$ passes in $n^{1+1/p} \cdot \text{poly}\left(\frac{\log n}{\varepsilon}\right)$ space. The only previous algorithm with this pass/space trade-off gave a $3$-approximation (SODA, 2020), and other polynomial-time algorithms which achieved a $(1+\varepsilon)$-approximation did so with quadratic memory or with a linear n…
We study the communication complexity and streaming complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 <= ε<= 1 uses space O(n^{1+ε}) and computes an O(…
We consider the problem of maximizing a non-negative submodular function under the $b$-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of $2+\varepsilon$, $3 + 2 \sqrt{2} \approx 5.828$, and $4 + 2 \sqrt{3} \approx 7.464$, respectively. We also consider a generalized problem, where a $k$-uniform hypergraph is given, along with an extra matroid or a $k'$-matchoid constraint imposed on the edges, with the same goal of finding a $b$-matching that maximizes a submodular function. When the extra constraint…
Recent breakthroughs in graph streaming have led to the design of single-pass semi-streaming algorithms for various graph coloring problems such as $(Δ+1)$-coloring, degeneracy-coloring, coloring triangle-free graphs, and others. These algorithms are all randomized in crucial ways and whether or not there is any deterministic analogue of them has remained an important open question in this line of work. We settle this fundamental question by proving that there is no deterministic single-pass semi-streaming algorithm that given a graph $G$ with maximum degree $Δ$, can output a proper coloring o…
While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of $2$ for the \emph{unweighted} matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor $2$ achieved by the…
Spectral sparsification is a technique that is used to reduce the number of non-zero entries in a positive semidefinite matrix with little changes to its spectrum. In particular, the main application of spectral sparsification is to construct sparse graphs whose spectra are close to a given dense graph. We study spectral sparsification under the assumption that the edges of a graph are allocated among sites which can communicate among each other. In this work we show that if a graph is allocated among several sites, the union of the spectral sparsifiers of each induced subgraph give us an spec…
Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical DFS algorithm requires $O(m+n)$ time for a graph having $n$ vertices and $m$ edges. In the streaming model, an algorithm is allowed several passes (preferably single) over the input graph having a restriction on the size of local space used. Trivially, a DFS tree can be computed using a single pass using $O(m)$ space. In the semi-streaming model allowing $O(n)$ space, it can be computed in $O(n)$ passes, where each pass adds one vertex to the DFS tree. However, it remains an open pro…
We reduce the best known approximation ratio for finding a weighted matching of a graph using a one-pass semi-streaming algorithm from 5.828 to 5.585. The semi-streaming model forbids random access to the input and restricts the memory to O(n*polylog(n)) bits. It was introduced by Muthukrishnan in 2003 and is appropriate when dealing with massive graphs.
A motif is a frequently occurring subgraph of a given directed or undirected graph $G$. Motifs capture higher order organizational structure of $G$ beyond edge relationships, and, therefore, have found wide applications such as in graph clustering, community detection, and analysis of biological and physical networks to name a few. In these applications, the cut structure of motifs plays a crucial role as vertices are partitioned into clusters by cuts whose conductance is based on the number of instances of a particular motif, as opposed to just the number of edges, crossing the cuts. In this …
This paper studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph $G = (V, E)$ whose edges arrive one-by-one and the goal is to construct an edge cover $F \subseteq E$ with the objective of minimizing the cardinality (or cost in the weighted case) of $F$. We consider a parameterized relaxation of this problem, where given some $0 \leq ε< 1$, the goal is to construct an edge $(1 - ε)$-cover, namely, a subset of edges incident to all but an $ε$-fraction of the vertices (or their benefit in the weighted case). The key limitat…
We provide $\widetilde{O}(ε^{-1})$-pass semi-streaming algorithms for computing $(1-ε)$-approximate maximum cardinality matchings in bipartite graphs. Our most efficient methods are deterministic and use optimal, $O(n)$, space, improving upon the space complexity of the previous state-of-the-art $\widetilde{O}(ε^{-1})$-pass algorithm of Ahn and Guha. To obtain our results we provide semi-streaming adaptations of more general continuous optimization tools. Further, we leverage these techniques to obtain improvements for streaming variants of approximate linear programming, optimal transport, ex…
In this paper we consider the problem of maximizing a non-negative submodular function subject to a cardinality constraint in the data stream model. Previously, the best known algorithm for this problem was a $5.828$-approximation semi-streaming algorithm based on a local search technique (Feldman et al., 2018). For the special case of this problem in which the objective function is also monotone, the state-of-the-art semi-streaming algorithm is an algorithm known as Sieve-Streaming, which is based on a different technique (Badanidiyuru, 2014). Adapting the technique of Sieve-Streaming to non-…
We present the first semi-streaming algorithms to determine k-connectivity of an undirected graph with k being any constant. The semi-streaming model for graph algorithms was introduced by Muthukrishnan in 2003 and turns out to be useful when dealing with massive graphs streamed in from an external storage device. Our two semi-streaming algorithms each compute a sparse subgraph of an input graph G and can use this subgraph in a postprocessing step to decide k-connectivity of G. To this end the first algorithm reads the input stream only once and uses time O(k^2*n) to process each input edge. T…
We prove a lower bound on the space complexity of two-pass semi-streaming algorithms that approximate the maximum matching problem. The lower bound is parameterized by the density of Ruzsa-Szemeredi graphs: * Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at least $(1- Ω(\frac{\log{RS(n)}}{\log{n}}))$, where $RS(n)$ denotes the maximum number of induced matchings of size $Θ(n)$ in any $n$-vertex graph, i.e., the largest density of a Ruzsa-Szemeredi graph. Currently, it is known that $n^{Ω(1/\!\log\log{n})} \leq RS(n) \leq \frac{n}{2^{O(\log^*{\!(n)})}}$ and …
We present semi-streaming algorithms for basic graph problems that have optimal per-edge processing times and therefore surpass all previous semi-streaming algorithms for these tasks. The semi-streaming model, which is appropriate when dealing with massive graphs, forbids random access to the input and restricts the memory to O(n*polylog n) bits. Particularly, the formerly best per-edge processing times for finding the connected components and a bipartition are O(alpha(n)), for determining k-vertex and k-edge connectivity O(k^2n) and O(n*log n) respectively for any constant k and for computing…
