AI 中文总结
研究组合可离散化距离几何问题，当前驱集不连续时对称计数方法不足。通过定位骨架无种子连通分量识别部分反射，用二进制掩码编码，经图操作和秩计算得出结果计数，建立精确计数定理，适用于各欧几里得维度。
AI 中文摘要
组合可离散化距离几何问题将有限二进制定位过程与附加距离约束相结合。当前驱集不连续时，这些剪枝约束的相互作用使得适用于分子实例的基于对称的计数方法不足。我们在严格离散化和一般可行框架假设下，为种子固定的可行实现建立了一个精确的、维度统一的计数定理。我们的方法通过定位骨架的无种子连通分量识别部分反射。这些反射由二进制掩码编码，而标记的约束矩阵检测保留所有剪枝距离的组合。可行分支选择分为由兼容部分反射生成的受限选择和剪枝端点前驱闭包之外的无约束选择。结果计数通过二进制域上的图操作和秩计算获得，无需枚举定位树。完备性源于通过部分反射对相关连接图的一般实现的刻画。该定理适用于每个欧几里得维度，包括一维。
英文摘要
The Combinatorial Discretizable Distance Geometry Problem combines a finite binary lateration process with additional distance constraints. When predecessor sets are not consecutive, these pruning constraints interact in ways that make the symmetry-based counting methods available for molecular instances insufficient. We establish an exact, dimension-uniform counting theorem for seed-fixed feasible realizations under strict discretization and a generic feasible framework assumption. Our approach identifies partial reflections through seed-free connected components of the lateration skeleton. These reflections are encoded by binary masks, while a labeled constraint matrix detects combinations that preserve all pruning distances. The feasible branch choices split into constrained choices generated by compatible partial reflections and unconstrained choices outside the predecessor closure of the pruning endpoints. The resulting count is obtained from graph operations and rank computations over the binary field, without enumerating the lateration tree. Completeness follows from the characterization of generic realizations of the relevant joined graphs by partial reflections. The theorem applies in every Euclidean dimension, including dimension one.
Comments12 pages and 1 figure
