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The Bit Complexity of Dynamic Algebraic Formulas and their Determinants

Emile Anand, Jan van den Brand, Mehrdad Ghadiri, Daniel Zhang

TL;DR

The paper tackles the stability and bit-complexity of dynamically maintaining algebraic expressions that are expressed as matrix formulas built from inversions, multiplications, additions, and subtractions. It deploys the Sherman–Morrison–Woodbury framework to embed any such formula into a large auxiliary matrix $\mathbf{N}$ whose inverse contains the formula's output, and proves that the constructed matrices stay well-conditioned with condition number $\kappa^{O(s)}$, yielding bit-precision requirements of $O(s \log \kappa)$. It extends this approach to maintain determinants and ranks of matrix formulas, using a ratio of determinants and randomized embeddings, respectively, and provides a suite of dynamic data-structures with polylogarithmic factors that support updates and queries under finite-precision, fixed-point arithmetic. The results have broad applicability to optimization and computational geometry, including dynamic linear programming pivots and dynamic maximum matching via the Tutte matrix, offering a practical framework for implementing iterative algorithms on real hardware with rigorous bit-complexity guarantees. Overall, the paper delivers a unified, stability-aware toolbox for dynamic algebraic computation on the word-RAM, bridging theory and implementable practice in domains where inputs change slightly across iterations.

Abstract

Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next. Although efficient data structures have been proposed for maintaining the solution of such algebraic expressions under low-rank updates, most of these results are only analyzed under exact arithmetic (real-RAM model and finite fields) which may not accurately reflect the complexity guarantees of real computers. In this paper, we analyze the stability and bit complexity of such data structures for expressions that involve the inversion, multiplication, addition, and subtraction of matrices under the word-RAM model. We show that the bit complexity only increases linearly in the number of matrix operations in the expression. In addition, we consider the bit complexity of maintaining the determinant of a matrix expression. We show that the required bit complexity depends on the logarithm of the condition number of matrices instead of the logarithm of their determinant. We also discuss rank maintenance and its connections to determinant maintenance. Our results have wide applications ranging from computational geometry (e.g., computing the volume of a polytope) to optimization (e.g., solving linear programs using the simplex algorithm).

The Bit Complexity of Dynamic Algebraic Formulas and their Determinants

TL;DR

The paper tackles the stability and bit-complexity of dynamically maintaining algebraic expressions that are expressed as matrix formulas built from inversions, multiplications, additions, and subtractions. It deploys the Sherman–Morrison–Woodbury framework to embed any such formula into a large auxiliary matrix whose inverse contains the formula's output, and proves that the constructed matrices stay well-conditioned with condition number , yielding bit-precision requirements of . It extends this approach to maintain determinants and ranks of matrix formulas, using a ratio of determinants and randomized embeddings, respectively, and provides a suite of dynamic data-structures with polylogarithmic factors that support updates and queries under finite-precision, fixed-point arithmetic. The results have broad applicability to optimization and computational geometry, including dynamic linear programming pivots and dynamic maximum matching via the Tutte matrix, offering a practical framework for implementing iterative algorithms on real hardware with rigorous bit-complexity guarantees. Overall, the paper delivers a unified, stability-aware toolbox for dynamic algebraic computation on the word-RAM, bridging theory and implementable practice in domains where inputs change slightly across iterations.

Abstract

Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next. Although efficient data structures have been proposed for maintaining the solution of such algebraic expressions under low-rank updates, most of these results are only analyzed under exact arithmetic (real-RAM model and finite fields) which may not accurately reflect the complexity guarantees of real computers. In this paper, we analyze the stability and bit complexity of such data structures for expressions that involve the inversion, multiplication, addition, and subtraction of matrices under the word-RAM model. We show that the bit complexity only increases linearly in the number of matrix operations in the expression. In addition, we consider the bit complexity of maintaining the determinant of a matrix expression. We show that the required bit complexity depends on the logarithm of the condition number of matrices instead of the logarithm of their determinant. We also discuss rank maintenance and its connections to determinant maintenance. Our results have wide applications ranging from computational geometry (e.g., computing the volume of a polytope) to optimization (e.g., solving linear programs using the simplex algorithm).
Paper Structure (51 sections, 30 theorems, 147 equations, 1 figure, 1 algorithm)

This paper contains 51 sections, 30 theorems, 147 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Notation
  5. Sherman-Morrison-Woodbury Identity woodbury1950inverting.
  6. Schur complement.
  7. Our Computational Model.
  8. Matrix Formula
  9. Technical Overview
  10. Setting the Stage: How to Maintain Dynamic Algebraic Formulas
  11. Bounding the Frobenius Norms
  12. Construction of $\mathbf{N}$
  13. Bounding the Frobenius Norm (\ref{['sec:frobeniusnorm']})
  14. Dynamic Rank and Determinant of Matrix Formulas
  15. Maintaining the Determinant (\ref{['sec:determinant']})
  16. ...and 36 more sections

Key Result

Theorem 1.1

There exists data structures with the following operations. Each data structure initializes in $\widetilde{O}(n^\omega s \log (\kappa/\epsilon))$ time on given accuracy parameters $\epsilon>0,\kappa > n$, matrix formula $f(\mathbf{M}_1,...,\mathbf{M}_s)$, and respective input matrices. Here $n$ is t The outputs are all $\epsilon$-approximate, i.e. each entry is off by at most an additive $\epsilon

Figures (1)

  • Figure 1: Maintaining $\mathbf{N}^{-1}$ allows us to maintain $f(\mathbf{M}_1,...,\mathbf{M}_s)$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof : Proof of \ref{['thm:main']}
  • Definition 1: Basic Solution
  • Theorem 4.1
  • ...and 41 more