  Higher-dimensional rewriting is founded on a duality of rewrite systems and
cell complexes, connecting computational mathematics to higher categories and
homotopy theory: the two sides of a rewrite rule are two halves of the boundary
of an (n+1)-cell, which are diagrams of n-cells. We study higher-dimensional
diagram rewriting as a mechanism of computation, focussing on the matching
problem for rewritable subdiagrams within the combinatorial framework of
diagrammatic sets. We provide an algorithm for subdiagram matching in arbitrary
dimensions, based on new results on layerings of diagrams, and derive upper
bounds on its time complexity. We show that these superpolynomial bounds can be
improved to polynomial bounds under certain acyclicity conditions, and that
these conditions hold in general for diagrams up to dimension 3. We discuss the
challenges that arise in dimension 4.
