A Geometric Approach to Dipath Classification on Process Graphs

We show that on a process graph defined by 1-semaphores, the space of directed paths (or dipaths) deformation retracts to a discrete set correlated by homotopy class relative to endpoints. This result extends to the set of ‘fastest executables’ (time-optimal dipaths). These results are obtained via techniques from cat(0) geometry and spaces of nonpositive curvature. keywords: CAT(0), nonpositive curvature, dipath, process graph MSRC 2000: Primary: 68Q85,51K10; Secondary: 68M20 1. Process graphs and directed spaces 1.1. Background. Process graphs were introduced in [3, 19] (and there attributed to Dijkstra) as a model for processors with shared resources. As geometric spaces, these are rectangular prisms with certain obstruction sets — where independent processes conflict — removed. A large collection of work on the algebraic topology of process graphs and related spaces has arisen [4, 10, 11, 16, 17, 18, 20], led by the work of Fajstrup, Goubault, Raussen, and others. This work focuses on dihomotopy and the peculiar constraints it imposes on the topology of paths. More specifically: processes are oriented and paths in the process graph must respect all of the orientations. Therefore, when classifying paths on a process graph, it is necessary and proper to work in the class of dipaths (directed paths) up to dihomotopy (homotopy through directed paths). A wide variety of directed topological invariants has been created, both homotopy-theoretic and homological in nature, applicable to process graphs and/or generalizations thereof. See, e.g., [5, 6, 10, 11, 16, 17, 18]. In particular, [18] gives a general introduction to ditopology in the context of process graphs and contains a brief classification in the 2-dimensional case. The paper [11] begins with process graphs and generalizes to cubical complexes with a Research supported by DARPA # HR0011-05-1-0008 and by NSF PECASE Grant # DMS 0337713.