Incremental Algorithms for Lattice Problems

TL;DR

This paper introduces incremental algorithms for key lattice problems, providing bounds on update steps and efficient implementation of orthogonal decomposition, advancing computational methods in lattice theory.

Contribution

It presents novel incremental algorithms for lattice basis, successive minima, and orthogonal decomposition, with proven update bounds and efficient implementation techniques.

Findings

Bounded the number of update steps for lattice basis insertions

Developed an efficient implementation of Kneser's orthogonal decomposition argument

Enhanced computational methods for lattice problems

Abstract

In this short note we give incremental algorithms for the following lattice problems: finding a basis of a lattice, computing the successive minima, and determining the orthogonal decomposition. We prove an upper bound for the number of update steps for every insertion order. For the determination of the orthogonal decomposition we efficiently implement an argument due to Kneser.