We present a systematic comparison and combination of two orthogonal techniques for efficient parsing of Combinatory Categorial Grammar (CCG). First we consider adaptive supertagging, a widely used approximate search technique that prunes most lexical categories from the parser's search space using a separate sequence model. Next we consider several variants on A*, a classic exact search technique which to our knowledge has not been applied to more expressive grammar formalisms like CCG. In addition to standard hardware-independent measures of parser effort we also present what we believe is the first evaluation of A* parsing on the more realistic but more stringent metric of CPU time. By itself, A* substantially reduces parser effort as measured by the number of edges considered during parsing, but we show that for CCG this does not always correspond to improvements in CPU time over a CKY baseline. Combining A* with adaptive supertagging decreases CPU time by 15% for our best model.