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Lazy Kronecker Product

Zhao Song

Abstract

In this paper, we show how to generalize the lazy update regime from dynamic matrix product [Cohen, Lee, Song STOC 2019, JACM 2021] to dynamic kronecker product. We provide an algorithm that uses $n^{ω( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a}$ amortized update time and $ n^{ω( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )}$ worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both $n^{ω( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a-Ω(1)}$ amortized update time, and $ n^{ω( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )-Ω(1)}$ worst case query time.

Lazy Kronecker Product

Abstract

In this paper, we show how to generalize the lazy update regime from dynamic matrix product [Cohen, Lee, Song STOC 2019, JACM 2021] to dynamic kronecker product. We provide an algorithm that uses amortized update time and worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both amortized update time, and worst case query time.
Paper Structure (1 section, 2 theorems)

This paper contains 1 section, 2 theorems.

Table of Contents

  1. Introduction

Key Result

Theorem 1.2

Let $k \geq s \geq 1$ be positive integers. For any $a>0$, there is an algorithm that takes $O( n^{ \omega( \lceil k/2 \rceil, \lfloor k/2 \rfloor,a ) - a} )$ amortized update time and $O( n^{\omega( \lceil ( k - s ) / 2 \rceil, \lfloor ( k - s ) / 2 \rfloor, a )} )$ worst case query time.

Theorems & Definitions (7)

  • Definition 1.1: Dynamic kronecker product
  • Theorem 1.2: Our result, algorithm
  • proof
  • Definition 1.3: A Tensor Version of Hinted Mv, s26
  • Conjecture 1.4: s26
  • Theorem 1.5: Our result, hardness
  • proof