The core of the Nondeterminator is an asymptotically efficient serial algorithm (inspired by Tarjan's nearly linear-time least-common-ancestors algorithm) for detecting determinacy races in series-parallel directed acyclic graphs. For a Cilk program that runs in T time on one processor and uses v shared-memory locations, the Nondeterminator runs in O(T α(v,v)) time, where α is Tarjan's functional inverse of Ackermann's function, a very slowly growing function which, for all practical purposes, is bounded above by 4 . The Nondeterminator uses at most a constant factor more space than does the original program. On a variety of Cilk program benchmarks, the Nondeterminator exhibits a slowdown of less than 12 compared with the serial execution time of the original optimized code, which we contend is an acceptable slowdown for debugging purposes.