Tight cell probe bounds for succinct Boolean matrix-vector multiplication

Chakraborty, Diptarka; Kamma, Lior; Larsen, Kasper Green

doi:10.1145/3188745.3188830

Cited by 16 publications

(12 citation statements)

References 25 publications

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“…Note that the model does not charge the query time for accessing the r precomputed bits, even if t ≪ r. This coincides with the systematic model studied by Chakraborty, Kamma and Larsen [8].…”

Section: Rigid Sets Systematic Linear Model and The Inner Product Par...supporting

confidence: 83%

“…The systematic linear model specializes the systematic model [8,17]. The latter model still stores the input data x ∈ F n 2 verbatim, and it also stores r < n bits that can be precomputed from x, where these need not be linear functions of the input data.…”

Section: Relationship To the Cell Probe Model And Other Modelsmentioning

confidence: 99%

“…The task is to preprocess an n-bit vector v to compute inner products q, v over F 2 for queries q ∈ Q, where Q ⊆ F n 2 is the query set. This problem generalizes the prefix-sum problem [17] and vector-matrix-vector problem [8,23].…”

Section: Introductionmentioning

confidence: 99%

“…The boolean semiring version of this problem has received much recent attention due to connections to the online matrix-vector multiplication conjecture [18]. Moreover, this problem has motivated the study of data structures for a super polynomial number of queries, even when the output is binary [8,9]. Other prior work has either studied binary output problems with poly(n) queries (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Ramamoorthy,

Rashtchian

2019

Preprint

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Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an NP oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the n-dimensional inner product problem with m queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of ω n r log m for r redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time Ω(n 3/2 /r) for redundancy r √ n in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Koucký, Loff, and Mukhopadhyay.

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Section: Rigid Sets Systematic Linear Model and The Inner Product Par...supporting

confidence: 83%

Section: Relationship To the Cell Probe Model And Other Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Ramamoorthy,

Rashtchian

2019

Preprint

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“…We need to preprocess A and then for every i output Av i before seeing v i+1 . It was conjectured in [6] (see also [2,8]) that: Conjecture 2. There exists no algorithm solving the OMv problem in time O(n 3− ), for any constant > 0.…”

Section: Preliminariesmentioning

confidence: 99%

Conditional Lower Bounds for Variants of Dynamic LIS

Gawrychowski,

Janczewski

2021

Preprint

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In this note, we consider the complexity of maintaining the longest increasing subsequence (LIS) of an array under (i) inserting an element, and (ii) deleting an element of an array. We show that no algorithm can support queries and updates in time O(n 1/2− ) and O(n 1/3− ) for the dynamic LIS problem, for any constant > 0, when the elements are weighted or the algorithm supports 1D-queries (on subarrays), respectively, assuming the All-Pairs Shortest Paths (APSP) conjecture or the Online Boolean Matrix-Vector Multiplication (OMv) conjecture. The main idea in our construction comes from the work of Abboud and Dahlgaard [FOCS 2016], who proved conditional lower bounds for dynamic planar graph algorithm. However, this needs to be appropriately adjusted and translated to obtain an instance of the dynamic LIS problem.

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