An optimal randomized parallel algorithm for the single function coarsest partition problem

We describe a randomized parallel algorithm to solve the single function coarsest partition problem. The algorithm runs in O(log n) time using O(n) operations with high probability on the Priority CRCW PRAM. The previous best known algorithms run in O(log2 n) time using O(n log2 n) operations on the CREW PRAM and O(log n) time using O(n log log n) operations on the Arbitrary CRCW PRAM. The technique presented can be used to generate the Euler tour of a rooted tree optimally from the parent representation.