Assembly Addition Chains

TL;DR

This paper generalizes the concept of Addition Chains from positive integers to arbitrary sets using algebraic structures called Assembly Multi-Magma, providing bounds and examples for various combinatorial objects.

Contribution

It introduces Assembly Addition Chains over arbitrary sets with a new algebraic framework and establishes bounds for their lengths, extending classical integer addition chain results.

Findings

Defined Assembly Addition Chains over sets with algebraic structure

Proved bounds for the lengths of optimal Assembly Addition Chains

Applied the framework to j-Strings, colored graphs, and polyominoes

Abstract

In this paper we extend the notion of Addition Chains over Z+ to a general set S. We explain how the algebraic structure of Assembly Multi-Magma over the pairs (S,BB proper subset of S) allows to define the concept of Addition Chain over S, called Assembly Addition Chains of S with Building Blocks BB. Analogously to the Z+ case, we introduce the concept of Optimal Assembly Addition Chains over S and prove lower and upper bounds for their lengths, similar to the bounds found by Schonhage for the Z+ case. In the general case the unit 1 is in set Z+ is replaced by the subset BB and the mentioned bounds for the length of an Optimal Assembly Addition Chain of O is in set S are defined in terms of the size of O (i.e. the number of Building Blocks required to construct O). The main examples of S that we consider through this papers are (i) j-Strings (Strings with an alphabeth of j letters), (ii) Colored Connected Graphs and (iii) Colored Polyominoes.