Estimating Graph Parameters from Random Order Streams

Pan, Peng; Sohler, Christian

doi:10.1137/1.9781611975031.157

Cited by 18 publications

(25 citation statements)

References 22 publications

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“…The latter consider worst case edge arrivals, but operate under a bounded arboricity assumption on the input graph. Very recently constant space algorithms for approximating some graph parameters (such as number of connected components and weight of the minimum spanning tree) from random order streams were obtained in [PS18] (see also [MMPS17]).…”

Section: Related Workmentioning

confidence: 99%

“…It has recently been shown that randomly ordered streams allow for surprisingly space efficient estimation of graph parameters by nontrivial memory vs sample complexity tradeoffs (see, e.g. [KKS14,MMPS17,PS18] for approximating matching size and other graph properties such as number of connected components, weight of MST and independent set size). Memory vs sample complexity tradeoffs for learning problems have also recently received a lot of attention in literature [Raz16,Raz17,KRT17,GRT18,BGY18].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Space Efficient Approximation to Maximum Matching Size from Uniform Edge Samples

Kapralov¹,

Mitrović²,

Norouzi-Fard³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a source of iid samples of edges of an input graph G with n vertices and m edges, how many samples does one need to compute a constant factor approximation to the maximum matching size in G? Moreover, is it possible to obtain such an estimate in a small amount of space? We show that, on the one hand, this problem cannot be solved using a nontrivially sublinear (in m) number of samples: m 1−o(1) samples are needed. On the other hand, a surprisingly space efficient algorithm for processing the samples exists: O(log 2 n) bits of space suffice to compute an estimate.Our main technical tool is a new peeling type algorithm for matching that we simulate using a recursive sampling process that crucially ensures that local neighborhood information from 'dense' regions of the graph is provided at appropriately higher sampling rates. We show that a delicate balance between exploration depth and sampling rate allows our simulation to not lose precision over a logarithmic number of levels of recursion and achieve a constant factor approximation. The previous best result on matching size estimation from random samples was a log O(1) n approximation [Kapralov et al'14], which completely avoided such delicate trade-offs due to the approximation factor being much larger than exploration depth.Our algorithm also yields a constant factor approximate local computation algorithm (LCA) for matching with O(d log n) exploration starting from any vertex. Previous approaches were based on local simulations of randomized greedy, which take O(d) time in expectation over the starting vertex or edge (Yoshida et al'09, Onak et al'12), and could not achieve a better than d 2 runtime. Interestingly, we also show that unlike our algorithm, the local simulation of randomized greedy that is the basis of the most efficient prior results does take Ω(d 2 )O(d log n) time for a worst case edge even for d = exp(Θ( √ log n)).Remark 4. Note that here L(w) is a random variable independently sampled from the distribution of L(w), similarly to L a in Eq. (11).Remark 5. Here U (E) denotes the uniform distribution over E. Also we denote by v ∈ e the fact that e is adjacent to v, where v is a vertex and e is an edge. In this case we denote the other endpoint of e by e\v. We use this notation heavily throughout the analysis. ones where q is not in the middle, will be very simple.Case 2: r ≤ p ≤ q. Then,On the other hand, if r = ,by Fact 3 since ≤ 1/2.Case 5: p ≤ q ≤ r. We consider two subcases.(a.) p ≤ 4 . Then q cannot be greater than 8 . Indeed this would mean by Fact 4 thatwhich is a contradiction. Similarly, r cannot be greater than 16 . Indeed this would mean by Fact 4 that D KL (q r) > D KL (8 16 ) ≥ 2 , which is also a contradiction. Ultaminately, r ≤ 16 , so D KL (p r) ≤ D KL (0 16 ) ≤ 32by Fact 3 since 16 ≤ 1/2.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Space Efficient Approximation to Maximum Matching Size from Uniform Edge Samples

Kapralov¹,

Mitrović²,

Norouzi-Fard³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Traditionally, these problems have been studied extensively in the query complexity model and for sublinear-time algorithms. However, starting from the pioneering work of [55], these problems have been receiving increasing attention in the streaming literature as well [37,55,77,81]. In particular, [55] gave single-pass streaming algorithms for these problems with O(n 1−poly(ε) ) space (for bipartiteness testing, the input graph is assumed to be planar).…”

Section: Property Testing Of Connectivity Bipartiteness and Cycle-fre...mentioning

confidence: 99%

“…• Streaming Property Testing: Solving property testing problems in the streaming setting (as opposed to the more familiar query algorithms) has been receiving increasing attention lately [37,55,77,81] starting from the pioneering work of [55]. On this front, BHH has been used to prove n 1−O(ε) -space lower bounds for ε-property testing of connectivity, cycle-freeness, and bipartiteness [55].…”

Section: B3 Streaming Lower Boundsmentioning

confidence: 99%

Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

Assadi

Kol

Saxena

et al. 2020

Preprint

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Consider the following gap cycle counting problem in the streaming model: The edges of a 2-regular n-vertex graph G are arriving one-by-one in a stream and we are promised that G is a disjoint union of either k-cycles or 2k-cycles for some small k; the goal is to distinguish between these two cases using a limited memory. Verbin and Yu [SODA 2011] introduced this problem and showed that any single-pass streaming algorithm solving it requires n 1−Ω( 1 /k) space. This result and the proof technique behind it-the Boolean Hidden Hypermatching communication problem-has since been used extensively for proving streaming lower bounds for various problems, including approximating MAX-CUT, matching size, property testing, matrix rank and Schatten norms, streaming unique games and CSPs, and many others.Despite its significance and broad range of applications, the lower bound technique of Verbin and Yu comes with a key weakness that is also inherited by all subsequent results: the Boolean Hidden Hypermatching problem is hard only if there is exactly one round of communication and, in fact, can be solved with logarithmic communication in two rounds. Therefore, all streaming lower bounds derived from this problem only hold for single-pass algorithms. Our goal in this paper is to remedy this state-of-affairs.We prove the first multi-pass lower bound for the gap cycle counting problem: Any p-pass streaming algorithm that can distinguish between disjoint union of k-cycles vs 2k-cycles-or even k-cycles vs one Hamiltonian cycle-requires n 1− 1 /k Ω(1/p) space. This makes progress on multiple open questions in this line of research dating back to the work of Verbin and Yu.

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“…A closely related model of property testing is the one where the graph arrives as a random order stream and the property testing algorithm is required to use sublinear space. Although this appears to be a less powerful model because the algorithm no longer has the ability to execute whatever queries it wants, interestingly, Peng and Sohler [PS18] show that sublinear property testing algorithms give rise to sublinear space algorithms for random order streams.…”

Section: Related Workmentioning

confidence: 99%

Testing Graph Clusterability: Algorithms and Lower Bounds

Chiplunkar

Kapralov

Khanna

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We consider the problem of testing graph cluster structure: given access to a graph G = (V, E), can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph? This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish between the graph having good expansion (i.e. being a good single cluster) and the graph having a sparse cut (i.e. being a union of at least two clusters). A recent work of Czumaj, Peng, and Sohler (STOC'15) gave an ingenious sublinear time algorithm for testing k-clusterability in timeÕ(n 1/2 poly(k)): their algorithm implicitly embeds a random sample of vertices of the graph into Euclidean space, and then clusters the samples based on estimates of Euclidean distances between the points. This yields a very efficient testing algorithm, but only works if the cluster structure is very strong: it is necessary to assume that the gap between conductances of accepted and rejected graphs is at least logarithmic in the size of the graph G. In this paper we show how one can leverage more refined geometric information, namely angles as opposed to distances, to obtain a sublinear time tester that works even when the gap is a sufficiently large constant. Our tester is based on the singular value decomposition of a natural matrix derived from random walk transition probabilities from a small sample of seed nodes.We complement our algorithm with a matching lower bound on the query complexity of testing clusterability. Our lower bound is based on a novel property testing problem, which we analyze using Fourier analytic tools. As a byproduct of our techniques, we also achieve new lower bounds for the problem of approximating MAX-CUT value in sublinear time.

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Estimating Graph Parameters from Random Order Streams

Cited by 18 publications

References 22 publications

Space Efficient Approximation to Maximum Matching Size from Uniform Edge Samples

Space Efficient Approximation to Maximum Matching Size from Uniform Edge Samples

Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

Testing Graph Clusterability: Algorithms and Lower Bounds

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