  Many disciplines of science and engineering deal with problems related to
compositions, ranging from chemical compositions in materials science to
portfolio compositions in economics. They exist in non-Euclidean simplex
spaces, causing many standard tools to be incorrect or inefficient, which is
significant in combinatorically or structurally challenging spaces exemplified
by Compositionally Complex Materials (CCMs) and Functionally Graded Materials
(FGMs). Here, we explore them conceptually in terms of problem spaces and
quantitatively in terms of computational feasibility.
  This work implements several essential methods specific to the compositional
(simplex) spaces through a high-performance open-source library nimplex. Most
significantly, we derive and implement an algorithm for constructing a novel
n-dimensional simplex graph data structure, which contains all discretized
compositions and all possible neighbor-to-neighbor transitions as pointer
arrays. Critically, no distance or neighborhood calculations are performed,
instead leveraging pure combinatorics and the ordering in procedurally
generated simplex grids, keeping the algorithm $\mathcal{O}(N)$, so that graphs
with billions of transitions take seconds to construct on a laptop.
Furthermore, we demonstrate how such graph representations can be combined to
express path-planning problem spaces and to incorporate prior knowledge while
keeping the problem space homogeneous. This allows for efficient deployment of
existing high-performance gradient descent, graph traversal search, and other
path optimization algorithms.
