Assembly Addition Chains
Leroy Cronin, Juan Carlos Morales Parra, Keith Y. Patarroyo
TL;DR
This work generalizes Addition Chains to Assembly Addition Chains over arbitrary sets by modeling gluing operations with Assembly Multi-Magma and Assembly Spaces. It defines AACs, Building Blocks, Building-Block–driven size $s(O)$, Assembly Index $a(O)$, and Optimal AACs $OAAC(O)$, linking them to Assembly Pathways and DAG representations. It proves coarse bounds $\log_2(s(O)) \le a(O) \le s(O)-1$ and formulates structure-based upper bounds (e.g., $Ma_{BD}$ and $Ma_{2PD}$) for Binary Decomposable and 2-Piece Decomposable objects, with concrete bounds for $j$-Strings, Colored Connected Graphs, and Colored Polyominoes. The framework enables intrinsic complexity estimates for combinatorial objects and clarifies the computational hardness (NP-completeness) of exactly computing the Assembly Index, while connecting to broader applications in strings, 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.