Let G and H be simple graphs and let S subset of V(G). A set S is called a non-local resolving set of G if each pair of non-adjacent vertices of G is distinguished by at least one vertex of S. The nonlocal metric dimension of G, denoted by dim(nl)(G), is the minimum cardinality of such a set. This invariant is complementary to the classical metric dimension and to the local metric dimension: instead of distinguishing all vertex pairs or only adjacent vertex pairs, it isolates the information needed to distinguish non-adjacent pairs. This distinction is especially relevant for graph products in which the construction itself creates new non-adjacencies. In this paper, we study the nonlocal metric dimension of hierarchical product graphs. We first give a general lower bound for dim(nl)(G(U) (sic) H) in terms of dim(nl)(H), and then discuss exact values for several families, including K-n(U) (sic) H, K-1,K-m(U) (sic) P-n, and P-n(U) (sic) K-1,K-m.
In recent years, the number of applications using diverse hardware such as FPGAs and GPUs has been increasing. However, utilizing these for high performance requires knowledge of hardware characteristics, which is quite difficult. Against this background, we propose environment-adaptive software that automatically converts and configures software code written by ordinary programmers for small-core CPUs according to the deployment environment, resulting in high-performance processing. This paper focuses on automatic offloading to many-core CPUs according to the computation type of processing, such as block matrix calculations and trivially parallel processing. We semantically analyze the existing application to be offloaded using an abstract syntax tree with pattern matching to determine whether the computation type has a replaceable OpenMP. If OpenMP is found, we replace it with that OpenMP and confirm the performance improvement. We confirm the effectiveness of the proposed method for automatic offloading by measuring processing time using an AMD Ryzen Threadripper 3995WX 64-core CPU. This manuscript is a preprint and is currently under review at Journal of Network and Systems Management (Springer Nature).
