We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constant-query testable in the adjacency list model can be tested with constant space in a single-pass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space. We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the adjacency list model: As an example, we obtain a constant-space single-pass random order streaming algorithms for approximating the size of a maximum matching with additive error @math ( @math is the number of nodes). Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while @math space is needed for many graph streaming problems for adversarial orders.
@cite_27 initiated the study of property testing in streaming model, and they gave efficient testers for some properties of a sequence of data items (rather than graphs as we consider here). Bury and Schwiegelshohn @cite_16 gave a lower bound of @math on the space complexity of any algorithm that @math -approximates the size of maximum matching in adversarial streams. @cite_10 showed that in random streams, @math space is necessary to distinguish if a graph is bipartite or @math -far from being bipartite. Previous work has extensively studied streaming graph algorithms in both the insertion-only and dynamic models, see the recent survey @cite_18 .
@cite_27 @cite_16 @cite_10 @cite_18
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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 @math passes or @math time to compute a @math approximation. In this paper we provide the first FPTAS (polynomial in @math ) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the size of the matching in @math passes using slightly superlinear space. To achieve both results, we use the structural properties of the matching polytope such as the laminarity of the tight sets and total dual integrality. The algorithms are iterative, and are based on the fractional packing and covering framework. However the formulations herein require exponentially many variables or constraints. We use laminarity, metric embeddings and graph sparsification to reduce the space required by the algorithms in between and across the iterations. This is the first use of these ideas in the semi-streaming model to solve a combinatorial optimization problem. We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass @math -approximation streaming algorithm can be easily obtained with space @math , where @math denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space @math is granted, for every @math . Specifically, for every @math , we show that in the one-pass turnstile streaming model, in order to compute a @math -approximation, space @math is required for constant error randomized algorithms, and, up to logarithmic factors, space @math is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest. We investigate the importance of space when solving problems based on graph distance in the streaming model. In this model, the input graph is presented as a stream of edges in an arbitrary order. The main computational restriction of the model is that we have limited space and therefore cannot store all the streamed data; we are forced to make space-efficient summaries of the data as we go along. For a graph of n vertices and m edges, we show that testing many graph properties, including connectivity (ergo any reasonable decision problem about distances) and bipartiteness, requires Ω(n) bits of space. Given this, we then investigate how the power of the model increases as we relax our space restriction. Our main result is an efficient randomized algorithm that constructs a (2t + 1)-spanner in one pass. With high probability, it uses O(t .n1+1 t log2n) bits of space and processes each edge in the stream in O(t2·n1 t log n) time. We find approximations to diameter and girth via the constructed spanner. For t = Ω(log n log log n), the space requirement of the algorithm is O(n .polylog n), and the per-edge processing time is O(polylog n). We also show a corresponding lower bound of t for the approximation ratio achievable when the space restriction is O(t.n1+1 t log2n).We then consider the scenario in which we are allowed multiple passes over the input stream. Here, we investigate whether allowing these extra passes will compensate for a given space restriction. We show that finding vertices at distance d from a particular vertex will always take d passes, for all d ∈ 1,...,t 2 , when the space restriction is o(n1+1 t). For girth, we show the existence of a direct trade-off between space and passes in the form of a lower bound on the product of the space requirement and number of passes. Finally, we conclude with two general techniques for speeding up the per-edge computation time of streaming algorithms while increasing the space by at most a log factor. In most algorithmic applications which compare two distributions, information theoretic distances are more natural than standard l p norms. In this paper we design streaming and sublinear time property testing algorithms for entropy and various information theoretic distances. posed the problem of property testing with respect to the Jensen-Shannon distance. We present optimal algorithms for estimating bounded, symmetric f-divergences (including the Jensen-Shannon divergence and the Hellinger distance) between distributions in various property testing frameworks. Along the way, we close a (log n) H gap between the upper and lower bounds for estimating entropy H, yielding an optimal algorithm over all values of the entropy. In a data stream setting (sublinear space), we give the first algorithm for estimating the entropy of a distribution. Our algorithm runs in polylogarithmic space and yields an asymptotic constant factor approximation scheme. An integral part of the algorithm is an interesting use of an F 0 (the number of distinct elements in a set) estimation algorithm; we also provide other results along the space time approximation tradeoff curve.Our results have interesting structural implications that connect sublinear time and space constrained algorithms. The mediating model is the random order streaming model, which assumes the input is a random permutation of a multiset and was first considered by Munro and Paterson in 1980. We show that any property testing algorithm in the combined oracle model for calculating a permutation invariant functions can be simulated in the random order model in a single pass. This addresses a question raised by regarding the relationship between property testing and stream algorithms. Further, we give a polylog-space PTAS for estimating the entropy of a one pass random order stream. This bound cannot be achieved in the combined oracle (generalized property testing) model.