137 research outputs found

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

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    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 Ω(n)\Omega(n) space, where nn is the number of vertices, existing works mainly focused on designing O~(n)\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. nn 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)o(n) space algorithms for estimating the number of connected components with additive error Δn\varepsilon n and (1+Δ)(1+\varepsilon)-approximating the weight of minimum spanning tree, for any small constant Δ>0\varepsilon>0. The latter improves previous O~(n)\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 Δ\varepsilon-far from having the property. We consider the problem of testing kk-edge connectivity, kk-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly O~(n1−Δ)\tilde{O}(n^{1-\varepsilon}) space, which is o(n)o(n) for any constant Δ\varepsilon. To complement our algorithms, we present Ω(n1−O(Δ))\Omega(n^{1-O(\varepsilon)}) space lower bounds for these problems, which show that such a dependence on Δ\varepsilon is necessary.Comment: ICALP 201

    Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

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    In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate P:0,1k→0,1P:{0,1}^{k} \to {0,1} except \equ where k≄3k\geq 3, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances (∣P−1(0)∣/2k−ϔ)(|P^{-1}(0)|/2^k-\epsilon)-far from satisfiability requires Ω(n1/2+ÎŽ)\Omega(n^{1/2+\delta}) queries where nn is the number of variables and ÎŽ>0\delta>0 is a constant that depends on PP and Ï”\epsilon. This breaks a natural lower bound Ω(n1/2)\Omega(n^{1/2}), which is obtained by the birthday paradox. We also show that every one-sided error tester requires Ω(n)\Omega(n) queries for such PP. These results are hereditary in the sense that the same results hold for any predicate QQ such that P−1(1)⊆Q−1(1)P^{-1}(1) \subseteq Q^{-1}(1). For EQU, we give a one-sided error tester whose query complexity is O~(n1/2)\tilde{O}(n^{1/2}). Also, for 2-XOR (or, equivalently E2LIN2), we show an Ω(n1/2+ÎŽ)\Omega(n^{1/2+\delta}) lower bound for distinguishing instances between Ï”\epsilon-close to and (1/2−ϔ)(1/2-\epsilon)-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances (1−2k/2k−ϔ)(1-2k/2^k-\epsilon)-far from satisfiability requires Ω(n)\Omega(n) queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the dd-to-11 Conjecture. As a corollary, for Maximum Independent Set on graphs with nn vertices and a degree bound dd, we show that every approximation algorithm within a factor d/\poly\log d and an additive error of Ï”n\epsilon n requires Ω(n)\Omega(n) queries. Previously, only super-constant lower bounds were known

    Semi-Streaming Algorithms for Annotated Graph Streams

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    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(n2)\Omega(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⋅polylog(n))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 fraction of the input, and hence cannot perform the desired computation locally. The client therefore accesses a powerful but untrusted service provider, who not only performs the requested computation, but also proves that the answer is correct. We put forth the notion of semi-streaming algorithms for annotated graph streams (semi-streaming annotation schemes for short). These are protocols in which both the client's space usage and the length of the proof are O(n⋅polylog(n))O(n \cdot \text{polylog}(n)). We give evidence that semi-streaming annotation schemes represent a substantially more robust solution concept than does the standard semi-streaming model. On the positive side, we give semi-streaming annotation schemes for two dynamic graph problems that are intractable in the standard model: (exactly) counting triangles, and (exactly) computing maximum matchings. The former scheme answers a question of Cormode. On the negative side, we identify for the first time two natural graph problems (connectivity and bipartiteness in a certain edge update model) that can be solved in the standard semi-streaming model, but cannot be solved by annotation schemes of "sub-semi-streaming" cost. That is, these problems are just as hard in the annotations model as they are in the standard model.Comment: This update includes some additional discussion of the results proven. The result on counting triangles was previously included in an ECCC technical report by Chakrabarti et al. available at http://eccc.hpi-web.de/report/2013/180/. That report has been superseded by this manuscript, and the CCC 2015 paper "Verifiable Stream Computation and Arthur-Merlin Communication" by Chakrabarti et a

    Hamilton cycles in dense vertex-transitive graphs

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    A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series
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