191 research outputs found

    Equivalence Classes and Conditional Hardness in Massively Parallel Computations

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    The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., P ? NP), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by MPC(o(log N)), and some standard classes concerning space complexity, namely L and NL, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model

    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 εn\varepsilon n and (1+ε)(1+\varepsilon)-approximating the weight of minimum spanning tree, for any small constant ε>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 ε\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−ε)\tilde{O}(n^{1-\varepsilon}) space, which is o(n)o(n) for any constant ε\varepsilon. To complement our algorithms, we present Ω(n1−O(ε))\Omega(n^{1-O(\varepsilon)}) space lower bounds for these problems, which show that such a dependence on ε\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−ϵ)(|P^{-1}(0)|/2^k-\epsilon)-far from satisfiability requires Ω(n1/2+δ)\Omega(n^{1/2+\delta}) queries where nn is the number of variables and δ>0\delta>0 is a constant that depends on PP and ϵ\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+δ)\Omega(n^{1/2+\delta}) lower bound for distinguishing instances between ϵ\epsilon-close to and (1/2−ϵ)(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−ϵ)(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 ϵn\epsilon n requires Ω(n)\Omega(n) queries. Previously, only super-constant lower bounds were known
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