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Improved Bounds for Rectangular Monotone Min-Plus Product and Applications

Anita Dürr

TL;DR

A generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values is presented and the running time for Unweighted Tree Edit Distance is improved.

Abstract

In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $\tilde{O}(n^{\frac{3 + ω}{2}})$ time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions $n \times n$ and entries bounded by $O(n)$. This greatly improves upon the previous $\tilde O(n^{\frac{12 + ω}{5}})$ time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: $M$-bounded Single Source Replacement Path, Batch Range Mode, $k$-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al.

Improved Bounds for Rectangular Monotone Min-Plus Product and Applications

TL;DR

A generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values is presented and the running time for Unweighted Tree Edit Distance is improved.

Abstract

In a recent breakthrough paper, Chi et al. (STOC'22) introduce an time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions and entries bounded by . This greatly improves upon the previous time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: -bounded Single Source Replacement Path, Batch Range Mode, -Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al.
Paper Structure (31 sections, 7 theorems, 15 equations, 1 table)

This paper contains 31 sections, 7 theorems, 15 equations, 1 table.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Definitions
  4. Results
  5. Monotone and Bounded-Difference Min-Plus Product
  6. Related work
  7. Rectangular Monotone Min Plus Product
  8. Overview
  9. Algorithm
  10. Compute the quotient matrix $\lfloor C /p \rfloor$
  11. Compute the remainder matrix $(C \bmod p)$
  12. First iteration
  13. Multiply matrices of polynomials
  14. Filter candidates and subtract erroneous terms
  15. Computing $T_{b}^{(l)}$ from $T_{b}^{(l + 1)}$.
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\beta, \mu$ be non-negative real numbers. Let $A$ be an $n \times n^\beta$ integer matrix and $B$ an $n^\beta \times n$ integer monotone matrix with non-negative entries bounded by $O(n^\mu)$. Then the Min-Plus Product $A \star B$ can be computed in $\Tilde{O}(n^{\frac{1 + \beta + \mu + \omega(

Theorems & Definitions (13)

  • Theorem 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 3 more