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Understanding the shape of a distribution of data is of interest to people in a great variety of fields, as it may affect the types of algorithms used for that data. We study one such problem in the framework of {\em distribution property testing}, characterizing the number of samples required to to distinguish whether a distribution has a certain property or is far from having that property. In particular, given samples from a distribution, we seek to characterize the tail of the distribution, that is, understand how many elements appear infrequently. We develop an algorithm based on a careful bucketing scheme that distinguishes lighttailed distributions from nonlighttailed ones with respect to a definition based on the hazard rate, under natural smoothness and ordering assumptions. We bound the number of samples required for this test to succeed with high probability in terms of the parameters of the problem, showing that it is polynomial in these parameters. Further, we prove a hardness result that implies that this problem cannot be solved without any assumptions.more » « less

For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected dregular graph with eO( 1 1−λ √ n) runtime per query, where λ is the secondlargest eigenvalue of the random walk matrix of the graph in absolute value. Since random dregular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input dregular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on smalldegree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product.more » « less

We consider the problem of sampling and approximately counting an arbitrary given motif H in a graph G, where access to G is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H) = {O_{k₁},...,O_{k_q}, S_{p₁},...,S_{p_𝓁}}. These algorithms were shown to be optimal for the case where H is a clique or an oddlength cycle, but no other lower bounds were known. We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, is always at least as good, and for most graphs G is strictly better. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the pth moment of the degree distribution. Finally, we prove that this algorithm is decompositionoptimal for decompositions that contain at least one odd cycle. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.more » « less

Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing largescale graphs. By now, we have quite fast algorithmsusually sublogarithmictime and often poly(łogłog n)time, or even fasterfor a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widelyadopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an alltoall manner to process largescale data. Contributing to this line of work on MPC graph algorithms, we present poly(łog k) ε poly(łogłog n) round MPC algorithms for computing O(k^1+o(1) )spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmictime MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: For the MPC setting, we get an O(łog^2łog n)round algorithm for O(łog^1+o(1) n) approximation of all pairs shortest paths (APSP) in the nearlinear regime of local memory. To the best of our knowledge, this is the first sublogarithmictime MPC algorithm for distance approximations. Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sublogarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model.more » « less