Encoding and Decoding Algorithms for Arbitrary Dimensional Hilbert Order

TL;DR

This paper introduces eight algorithms for encoding and decoding Hilbert order in arbitrary dimensions, expanding beyond low-dimensional cases with efficient methods based on arithmetic and bit operations.

Contribution

It presents new algorithms for arbitrary dimensional Hilbert order encoding and decoding, including both arithmetic and bit-based methods with varying complexities.

Findings

Four algorithms are linear in complexity.

Four algorithms are constant in complexity.

Algorithms are applicable to arbitrary dimensions.

Abstract

Hilbert order is widely applied in many areas. However, most of the algorithms are confined to low dimensional cases. In this paper, algorithms for encoding and decoding arbitrary dimensional Hilbert order are presented. Eight algorithms are proposed. Four algorithms are based on arithmetic operations and the other four algorithms are based on bit operations. For the algorithms complexities, four of them are linear and the other four are constant for given inputs. In the end of the paper, algorithms for two dimensional Hilbert order are presented to demonstrate the usage of the algorithms introduced.