Diamonds are Forever in the Blockchain: Geometric Polyhedral Point-Set Pattern Matching

TL;DR

This paper introduces approximation algorithms for geometric pattern matching of polyhedral point sets, motivated by blockchain-based supply chain tracing of ethically sourced diamonds, providing efficient solutions under translations and rotations.

Contribution

It presents novel $(1 + \varepsilon)$-approximation schemes for polyhedral pattern matching under transformations, with complexity bounds in multiple dimensions, including handling rotations.

Findings

Efficient $(1 + \varepsilon)$-approximation algorithms for pattern matching.

Algorithms with complexity depending on dimension, $\varepsilon$, and point set properties.

Applicable to supply chain tracing of ethically sourced diamonds.

Abstract

Motivated by blockchain technology for supply-chain tracing of ethically sourced diamonds, we study geometric polyhedral point-set pattern matching as minimum-width polyhedral annulus problems under translations and rotations. We provide two $(1 + \varepsilon)$-approximation schemes under translations with $O(\varepsilon^{-d} n)$-time for $d$ dimensions and $O(n\log \varepsilon^{-1} + \varepsilon^{-2})$-time for two dimensions, and we give an $O(f^{d-1}\varepsilon^{1-2d}n)$-time algorithm when also allowing for rotations, parameterized on $f$, which we define as the slimness of the point set.