Graph streaming lower bounds for parameter estimation and property testing via a streaming XOR lemma

Assadi, Sepehr; N, Vishvajeet

doi:10.1145/3406325.3451110

Cited by 14 publications

(18 citation statements)

References 33 publications

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“…• [AN21] proves that in the streaming setting, any p-pass s-space algorithm for ⊕ k i=1 P (x i ) on the stream x 1 • x 2 . .…”

Section: Rs Gadget Rs Gadgetmentioning

confidence: 90%

“…. • x k can only gain an advantage of 1/2 Ω(k) over random guessing (as shown in [AN21], this is the strongest form of XOR lemma possible in that setting).…”

Section: Rs Gadget Rs Gadgetmentioning

confidence: 99%

“…The author is grateful to Soheil Behnezhad, Michael Kapralov, Raghuvansh Saxena, and Huacheng Yu for illuminating conversations, and to Christian Konrad for helpful discussions on his recent work in [KN21]. The author is also indebted to his collaborators Ran Raz in [AR20], Gillat Kol, Raghuvansh Saxena, and Huacheng Yu in [AKSY20], Vishvajeet N. in [AN21], and Soheil Behnezhad in [AB21] for their previous collaborations that formed various building blocks and inspirations for this work.…”

Section: Acknowledgementmentioning

confidence: 99%

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A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

Assadi

2021

Preprint

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We prove a lower bound on the space complexity of two-pass semi-streaming algorithms that approximate the maximum matching problem. The lower bound is parameterized by the density of Ruzsa-Szemerédi graphs:• Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at least 1 − Ω( log RS(n) log n ) , where RS(n) denotes the maximum number of induced matchings of size Θ(n) in any n-vertex graph, i.e., the largest density of a Ruzsa-Szemerédi graph.and closing this (large) gap between upper and lower bounds has remained a notoriously difficult problem in combinatorics.Under the plausible hypothesis that RS(n) = n Ω(1) , our lower bound is the first to rule out small-constant approximation two-pass semi-streaming algorithms for the maximum matching problem, making progress on a longstanding open question in the graph streaming literature.

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“…• [AN21] proves that in the streaming setting, any p-pass s-space algorithm for ⊕ k i=1 P (x i ) on the stream x 1 • x 2 . .…”

Section: Rs Gadget Rs Gadgetmentioning

confidence: 90%

“…. • x k can only gain an advantage of 1/2 Ω(k) over random guessing (as shown in [AN21], this is the strongest form of XOR lemma possible in that setting).…”

Section: Rs Gadget Rs Gadgetmentioning

confidence: 99%

Section: Acknowledgementmentioning

confidence: 99%

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A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

Assadi

2021

Preprint

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“…Base case of algorithm (2). We perform a breadth first search starting from ∇(G) (note that these vertices are already colored) and whenever we see a vertex v for the first time, we give it color opposite to the color of the vertex from which we discovered v. We then verify that the resulting 2-coloring is proper.…”

Section: Base Case Of Algorithm (1) Assume That |G| ≤ O(s)mentioning

confidence: 99%

Massively Parallel Computation and Sublinear-Time Algorithms for Embedded Planar Graphs

Holm¹,

Tětek²

2022

Preprint

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While algorithms for planar graphs have received a lot of attention, few papers have focused on the additional power that one gets from assuming an embedding of the graph is available. While in the classic sequential setting, this assumption gives no additional power (as a planar graph can be embedded in linear time), we show that this is far from being the case in other settings. We assume that the embedding is straight-line, but our methods also generalize to nonstraight-line embeddings. Specifically, we focus on sublinear-time computation and massively parallel computation (MPC).Our main technical contribution is a sublinear-time algorithm for computing a relaxed version of an r-division. We then show how this can be used to estimate Lipschitz additive graph parameters. This includes, for example, the maximum matching, maximum independent set, or the minimum dominating set. We also show how this can be used to solve some property testing problems with respect to the vertex edit distance.In the second part of our paper, we show an MPC algorithm that computes an r-division of the input graph. We show how this can be used to solve various classical graph problems with space per machine of O(n 2/3+ ) for some > 0, and while performing O(1) rounds. One needs Ω(log n) rounds with n 1−Ω(1) space per machine without the embedding to solve these problems, assuming the 1-vs-2-cycles conjecture. Among the problems solved by our approach are: counting connected components, bipartition, minimum spanning tree problem, O(1)-approximate shortest paths, O(1)-approximate diameter/radius. Our techniques however also apply to other problems. In conjunction with existing algorithms for computing the Delaunay triangulation, our results imply an MPC algorithm for Euclidean minimum spanning three that uses O(n 2/3+ ) space per machine and performs O(1) rounds.Finally, we show two generalizations of our approach. First, we show that that our approach also works for non-planar embeddings. Specifically, we get non-trivial results whenever the number of crossings is n 2 . Second, we show that our algorithms can be modified to work with embeddings consisting of curves that are not "too squiggly" (as formalized by the total absolute curvature). This results in an algorithm parameterized by the total absolute curvature of all edges. We do this via a new lemma which we believe is of independent interest.

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“…On the other hand, due to a result of Kapralov [Kap21], any single-pass streaming algorithm for matching with an approximation ratio better than 0.59 requires n 1+Ω(1/ log log n) space. In another work [AN21], space-pass tradeoffs in graph streaming are analyzed, proving that an (1 + ε)-approximation needs either n Ω(1) space or Ω(1/ε) passes, even for restricted graph families.…”

mentioning

confidence: 99%

Deterministic $(1+\varepsilon)$-Approximate Maximum Matching with $\mathsf{poly}(1/\varepsilon)$ Passes in the Semi-Streaming Model and Beyond

Fischer¹,

Mitrović²,

Uitto³

2021

Preprint

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We present a deterministic (1+ε)-approximate maximum matching algorithm in poly( 1/ε) passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on 1/ε. Our algorithm exponentially improves on the well-known randomized (1/ε) O(1/ε) -pass algorithm from the seminal work by McGregor [APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity [FSTTCS18], as well as the deterministic log n • poly(1/ε)-pass algorithm by Ahn and Guha [ICALP11].

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Graph streaming lower bounds for parameter estimation and property testing via a streaming XOR lemma

Cited by 14 publications

References 33 publications

A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

Massively Parallel Computation and Sublinear-Time Algorithms for Embedded Planar Graphs

Deterministic $(1+\varepsilon)$-Approximate Maximum Matching with $\mathsf{poly}(1/\varepsilon)$ Passes in the Semi-Streaming Model and Beyond

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