On the Black-Box Complexity of Sperner's Lemma
We present several results on the complexity of various forms of Sperner's Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an O(root n) deterministic query algorithm for the black-box version of the problem 2D-SPERNER, a well studied member of Papadimitriou's complexity class PPAD. This upper bound matches the Omega(root n) deterministic lower bound of Crescenzi and Silvestri. The tightness of this bound was not known before. In another result we prove for the same problem an Omega((4)root n) lower bound for its probabilistic, and an Omega((8)root n) lower bound for its quantum query complexity, showing that all these measures are polynomially related.
sperner's lemma deterministic algorithm query complexity probabilistic and quantum lower bound