Lazy Kronecker Product

TL;DR

This paper extends lazy update techniques from dynamic matrix products to dynamic Kronecker products, providing algorithms with specific amortized and worst-case times, and establishing complexity lower bounds based on the tensor MV conjecture.

Contribution

It generalizes lazy update regimes to dynamic Kronecker products and offers algorithms with defined time complexities, along with conditional lower bounds.

Findings

Provides an algorithm with $n^{ ext{omega}(...)}$ amortized update time.

Offers a worst-case query time algorithm with specific complexity.

Establishes lower bounds based on the tensor MV conjecture.

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 $n^{\omega( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a}$ amortized update time and $ n^{\omega( \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 $n^{\omega( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a-\Omega(1)}$ amortized update time, and $ n^{\omega( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )-\Omega(1)}$ worst case query time.