Very fast optimal parallel algorithms for heap construction

We give two algorithms for permuting n items in an array into heap order on a CRCW PRAM. The first is deterministic and runs in O(log log n) time and performs O(n) operations and is time- and work-optimal. The second is randomized and runs in O(log log log n) time with high probability, performing O(n) operations. No PRAM algorithm with o(log n) run-time was previously known for this problem. We also study the parallel complexity of selecting the kth smallest of n elements on the CRCW PRAM, a problem of independent interest. We show that this problem can be solved deterministically in O(log log n+log k/log log n) time and O(n) operations for all 1⩽k⩽n/2, improving on existing algorithms when k is small compared to n. This run-time is also shown to be optimal