A proof of a theorem in algebraic-topology by a distributed algorithm

The celebrated Simplicial Approximation Theorem of Algebraic Topology essentially says that any subdivided simplex A can be approximated by a subdivided simplex B which is a simplex that has been subdivided enough times by a Barycentric Subdivision. The chromatic counterpart of the Simplicial Approximation was proved in [1]. It says that any chromatic subdivided simplex Ac can chromatically be approximated by a chromatic simplex that has been SUbdivided enough times by the subdivision corresponding to one-shot immediate snapshot [3], B.. In this note the formal notion of Bc approximating A is that there exists a color preserving simplicial map from B. to A. which preserves boundaries. The proof in [1], as well as parts of the proof of the convergence algorithm in [2], use nontrivial e, 6 arguments. Such arguments are the domain of continuous mathematics. It would have been nice if we could use a text-book theorem extending it with purely distributed algorithms arguments to derive these results. Here we prove precisely that. We substitute the c argument in [1] and [2], with the Simplicial Approximation Theorem. We then use the simplex convergence algorithm in [2] to show the result. In fact we show a stronger result. A subdivided simplex is chromatically multi-colored if all its corners are colored by distinct colors, and a vertex is colored by a subset of the colors of its carrier. Let Am. be a multi colored subdivided simplex such that the union of the color of each simplex is the set of the colors of the carrier. Then there exist a simplicial map from B= to Am= preserving boundary and each vertex of color z’ is mapped to a vertex that contains color Z. It is easy to see that the theorem in [I] is a special case of this result. Herlihy and Rajsbaum informed us that they independently understood that the convergence algorithm of [2] implies this. We first substitute the c argumentation in [2] for the case when we have a single processor waking up in any corner of the subdivided simplex. We notice that there is a simplicial map from the subdivided simplex of one-shot immediate snapshot to the first Barycentric Subdivision preserving boundaries. Thus a k iterated one-shot immediate snapshot subdivision is simplicially mapped to the kth Barycentric Subdivision. Given a chromatic subdivided simplex, take its first barycent ric subdivision. By the Simplicial Approximation theorem we can approximate it by the barycentric