The maximum discrete surface-to-volume ratio of space-filling curve partitions

Space-filling curves (SFCs) are used in high performance computing to distribute a computational domain or its mesh, respectively, amongst different compute units, i.e.~cores or nodes or accelerators. The part of the domain allocated to each compute unit is called a partition. Besides the balancing of the work, the communication cost to exchange data between units determines the quality of a chosen partition. This cost can be approximated by the surface-to-volume ratio of partitions: the volume represents the amount of local work, while the surface represents the amount of data to be transmitted. Empirical evidence suggests that space-filling curves yield advantageous surface-to-volume ratios. Formal proofs are available only for regular grids. We investigate the surface-to-volume ratio of space-filling curve partitions for adaptive grids and derive the maximum surface-to-volume ratio as a function of the number of cells in the partition. In order to prove our main theorem, we construct a new framework for the study of adaptive grids, notably introducing the concepts of a shape and of classified partitions. The new methodological framework yields insight about the SFC-induced partition character even if the grids refine rather aggressively in localised areas: it quantifies the obtained surface-to-volume ratio. This framework thus has the potential to guide the design of better load balancing algorithms on the long term.