Best of two worlds: Cartesian sampling and volume computation for distance-constrained configuration spaces using Cayley coordinates

Volume calculation of configurational spaces acts as a vital part in configurational entropy calculation, which contributes towards calculating free energy landscape for molecular systems. In this article, we present our sampling-based volume computation method using distance-based Cayley coordinate, mitigating drawbacks: our method guarantees that the sampling procedure stays in lower-dimensional coordinate space (instead of higher-dimensional Cartesian space) throughout the whole process; and our mapping function, utilizing Cayley parameterization, can be applied in both directions with low computational cost. Our method uniformly samples and computes a discrete volume measure of a Cartesian configuration space of point sets satisfying systems of distance inequality constraints. The systems belong to a large natural class whose feasible configuration spaces are effectively lower dimensional subsets of high dimensional ambient space. Their topological complexity makes discrete volume computation challenging, yet necessary in several application scenarios including free energy calculation in soft matter assembly modeling. The algorithm runs in linear time and empirically sub-linear space in the number of grid hypercubes (used to define the discrete volume measure) \textit{that intersect} the configuration space. In other words, the number of wasted grid cube visits is insignificant compared to prevailing methods typically based on gradient descent. Specifically, the traversal stays within the feasible configuration space by viewing it as a branched covering, using a recent theory of Cayley or distance coordinates to convexify the base space, and by employing a space-efficient, frontier hypercube traversal data structure. A software implementation and comparison with existing methods is provided.