A worst-case optimal algorithm to compute the Minkowski sum of convex polytopes

Authors

Sandip Das, Indian Statistical Institute, Kolkata
Subhadeep Ranjan Dev, Indian Statistical Institute, Kolkata
Sarvottamananda, Ramakrishna Mission Vivekananda Educational and Research Institute

Article Type

Research Article

Publication Title

Discrete Applied Mathematics

Abstract

We propose algorithms to compute the Minkowski sum of two or more convex polytopes represented by their face lattices in R^d. The time and space complexities of the pair-wise algorithm are O(d^ωnm) and O(n+m+M), respectively, where n, m and M are the face lattice sizes of the two summands and the sum, respectively, and ω≈2.37, currently, is the matrix multiplication exponent. We also show that this algorithm is worst-case optimal for fixed d. Next, we generalize the pair-wise algorithm to r>2 convex polytopes in R^d. The time and space complexities of the r-summands generalized algorithm are O(d^ωmin{MN,r+∏i=1^rNi}) and O(M+N), respectively, where Ni, 1≤i≤r, is the size of ith summand, N=∑i=1^rNi is the total input size and M is the output size of the Minkowski sum. We show that the r-summands generalized algorithm is worst-case optimal for fixed d≥3 and r

First Page

44

Last Page

61

DOI

10.1016/j.dam.2024.02.004

Publication Date

6-15-2024

Recommended Citation

Das, Sandip; Dev, Subhadeep Ranjan; and Sarvottamananda, "A worst-case optimal algorithm to compute the Minkowski sum of convex polytopes" (2024). Journal Articles. 4575.
https://digitalcommons.isical.ac.in/journal-articles/4575