### Treaps versus red-black trees

In a previous blog entry, I discussed the difficulties associated with implementing left-leaning red-black trees. A couple of readers commented that treaps might be superior to red-black trees, and as part of some recent jemalloc optimization work, I had occasion to implement treaps in order to measure tree operation overhead.

Most of this article discusses performance, but let me first mention implementation difficulty. It took me about 90 hours to design/implement/test/benchmark left-leaning red-black trees, and less than 10 hours for treaps. Search/insert/delete for red-black trees is O(lg n), versus O(n) for treaps. However, the average case for treaps is (lg n), and the chances of worst case behavior are vanishingly small, thanks to (pseudo-)randomness. Thus, real-world performance differences are only incremental. To be fair, I made red-black trees harder by avoiding recursion. Regardless however, treaps are way easier to implement than red-black trees.

As for benchmarking, I wrote functionally identical benchmark programs for three red-black tree implementations and two treap implementations. The tree implementations are:

**rb_new**: Left-leaning red-black trees.**rb_old**: Standard red-black trees.**RB**: Standard red-black trees, as implemented by the *BSD sys/tree.h.**trp_hash**: Treaps, with priorities computed via pointer hashing.**trp_prng**: Treaps, with priorities computed via pseudo-random number generation (PRNG).

The following table summarizes benchmark results as measured on a 2.2 GHz amd64 Ubuntu 8.10 system. The benchmarks were all compiled with "gcc -O3", and the times are user+system time (fastest of three runs):

NNODES, NSETS, NSEARCH, NITER (focus) | rb_new | rb_old | RB | trp_hash | trp_prng |
---|---|---|---|---|---|

1500,25,0,0 (ins/del) | 7.60 | 3.99 | 4.25 | 17.57 | 7.58 |

125,25,10,0 (search) | 17.74 | 18.61 | 16.60 | 17.84 | 17.77 |

250,25,0,100 (iteration) | 18.45 | 21.06 | 19.19 | 20.45 | 20.40 |

Insertion/deletion is fastest for the red-black tree implementations that do lazy fixup (rb_old and RB). rb_new uses a single-pass algorithm, which requires more work. trp_prng is about the same speed as rb_new, but trp_hash is way slower, due to the repeated hash computations that are required to avoid explicitly storing node priorities.

Search performance is similar for all implementations, which indicates that there are no major disparities in tree balance.

Iteration performance is similar for all implementations, even though they use substantially different algorithms. If tree size were much larger, rb_old and RB would suffer, since they use an O(n lg n) algorithm, whereas rb_new and trp_* use O(n) algorithms. rb_new uses a rather complicated iterative algorithm, but trp_* use recursion and callback functions due to the weak upper bound on tree depth.

Sadly, there is no decisive winner, though any of the five tree implementations is perfectly adequate for the vast majority of applications. The winners according to various criteria are:

**Space**: rb_new and trp_hash.**Speed**(insertion/deletion): rb_old and RB.**Ease of implementation**: trp_prng.