Polynomial pass lower bounds for graph streaming algorithms

Assadi, Sepehr; Chen, Yu; Khanna, Sanjeev

doi:10.1145/3313276.3316361

“…Our reductions can be modified to give reductions from HPC 4 as well. Thus Theorem 3.8 and Theorem 3.11 also make progress on Assadi, Chen, and Khanna [ACK19a]'s conjecture that hidden pointer chasing would lead to multi-pass lower bounds for other graph problems in the streaming setting.…”

Section: Our Contributionsmentioning

confidence: 80%

“…This in turn, retains the pass lower bounds in Theorem 1.2 (Section 3.1.1). Surprisingly, our communication model also allows us to use a simple direct-sum argument in our proofs, as opposed to a more involved one in [ACK19a]. This is our main technical contribution in designing a two player version of HPC 4 .…”

Section: Our Contributionsmentioning

confidence: 99%

“…Our formulation allows us to lower bound its communication complexity with a simple direct-sum argument. Using this lower bound on the communication complexity of HPC 2 , we retain the streaming and query complexity lower bounds by [ACK19a].Further, by giving reductions from HPC 2 , we prove new multi-pass space lower bounds for graph problems in turnstile streams. In particular, we show that any algorithm which computes the exact weight of the maximum weighted matching in an n-vertex graph requires O(n 2 ) space unless it makes ω(log n) passes over the turnstile stream, and that any algorithm which computes the minimum s-t distance in an n-vertex graph requires n 2−o(1) space unless it makes n Ω(1) passes over the turnstile stream.…”

mentioning

confidence: 91%

“…Our formulation allows us to lower bound its communication complexity with a simple direct-sum argument. Using this lower bound on the communication complexity of HPC 2 , we retain the streaming and query complexity lower bounds by [ACK19a].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Two Player Hidden Pointer Chasing and Multi-Pass Lower Bounds in Turnstile Streams

Mehrotra¹,

Porwal²,

Tewari³

2020

Preprint

0

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Assadi, Chen, and Khanna [ACK19a] define a 4-player hidden-pointer-chasing (HPC 4 ), and using it, give strong multi-pass lower bounds for graph problems in the streaming model of computation and a lower bound on the query complexity of sub-modular minimization. We present a two-player version (HPC 2 ) of HPC 4 that has matching communication complexity to HPC 4 . Our formulation allows us to lower bound its communication complexity with a simple direct-sum argument. Using this lower bound on the communication complexity of HPC 2 , we retain the streaming and query complexity lower bounds by [ACK19a].Further, by giving reductions from HPC 2 , we prove new multi-pass space lower bounds for graph problems in turnstile streams. In particular, we show that any algorithm which computes the exact weight of the maximum weighted matching in an n-vertex graph requires O(n 2 ) space unless it makes ω(log n) passes over the turnstile stream, and that any algorithm which computes the minimum s-t distance in an n-vertex graph requires n 2−o(1) space unless it makes n Ω(1) passes over the turnstile stream. Our reductions can be modified to use HPC 4 as well.

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“…We note a few results which are related to submodular function minimization. Assadi, Chen, and Khanna [ACK19] considered the problem of finding the minimum s-t-cut in an undirected graph in the streaming setting. They showed that any p-pass algorithm must take Ω(n 2 /p 5 )-space, where n is the number of vertices.…”

Section: Related Workmentioning

confidence: 99%

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Chakrabarty¹,

Chen²,

Khanna³

2021

Preprint

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Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making poly(N ) queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where N denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least N rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with poly(N ) queries using only poly-logarithmic rounds of adaptivity.An important step towards understanding the adaptivity needed for efficient parallel SFM was taken recently in the work of Balkanski and Singer who showed that any SFM algorithm making poly(N ) queries necessarily requires Ω(log N/ log log N ) rounds. This left open the possibility of efficient SFM algorithms in poly-logarithmic rounds. For matroid intersection, even the possibility of a constant round, poly(N ) query algorithm was not hitherto ruled out.In this work, we prove that any, possibly randomized, algorithm for submodular function minimization or matroid intersection making poly(N ) queries requires 1 Ω N 1/3 rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make at most 2 N 1−δ queries, for any constant δ > 0. Therefore, even though SFM and matroid intersection are efficiently solvable, they are not highly parallelizable in the oracle model.

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Parameterized Complexity of Streaming Diameter and Connectivity Problems

Oostveen,

van Leeuwen

2024

Algorithmica

0

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We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is $$\mathcal {O}(\log n)$$ O ( log n ) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in $$\mathcal {O}(k)$$ O ( k ) passes and $$\mathcal {O}(k \log n)$$ O ( k log n ) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where $$\Omega (n/p)$$ Ω ( n / p ) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, $$\Omega (n \log n)$$ Ω ( n log n ) bits of memory lower bounds. We also prove a much stronger $$\Omega (n^2/p)$$ Ω ( n 2 / p ) lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with $$\mathcal {O}(k^2)$$ O ( k 2 ) edges) produced as a stream in $$\text {poly}(k)$$ poly ( k ) passes and only $$\mathcal {O}(k \log n)$$ O ( k log n ) bits of memory.

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Polynomial pass lower bounds for graph streaming algorithms

Cited by 27 publications

References 142 publications

Two Player Hidden Pointer Chasing and Multi-Pass Lower Bounds in Turnstile Streams

Two Player Hidden Pointer Chasing and Multi-Pass Lower Bounds in Turnstile Streams

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Parameterized Complexity of Streaming Diameter and Connectivity Problems

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