Fast Set Operations Using Treaps

We present parallel algorithms for union, intersection and difference on ordered sets using random balanced binary trees (treaps [26]). For two sets of size n and m (m ≤ n) the algorithms run in expected O(mlg(n=m)) work and O(lg n) depth (parallel time) on an EREW PRAM with scan operations (implying O(lg2 n) depth on a plain EREW PRAM). As with the sequential algorithms on treaps for insertion and deletion, the main advantage of our algorithms are their simplicity. In fact, our algorithms for set operations seem simpler than previous sequential algorithms with the same work bounds, and might therefore also be useful in a sequential context. To analyze the effectiveness of the algorithms we implemented both sequential and parallel versions of the algorithms and ran several experiments on them. Our parallel implementation uses the Cilk [5] shared memory runtime system on a 16 processor SGI Power Challenge and a 6 processor Sun Ultra Enterprise 3000. It shows reasonable speedup: 6.3 to 6.8 speedup on 8 processors of the SGI, and 4.1 to 4.4 speedup on 5 processors of the Sun.