In this paper we introduce and study a new family of combinatorial simplicial complexes, which we call immediate snapshot complexes. Our construction and terminology is strongly motivated by theoretical distributed computing, as these complexes are combinatorial models of the standard protocol complexes associated to immediate snapshot read/write shared memory communication model. In order to define the immediate snapshot complexes we need a new combinatorial object, which we call a witness structure. These objects are indexing the simplices in the immediate snapshot complexes, while a special operation on them, called ghosting, describes the combinatorics of taking simplicial boundary. In general, we develop the theory of witness structures and use it to prove several combinatorial as well as topological properties of the immediate snapshot complexes. Comment: full paper version of the 1st part of the preprint arXiv:1402.4707; to appear in DMTCS
Discrete Mathematics & Theoretical Computer Science