Dynamic Order Statistics

Figure 14.2: Updating subtree sizes during rotations. The link around which the rotation is performed is incident on the two nodes whose size fields need to be updated. The updates are local, requiring only the size information stored in x, y, and the roots of the subtrees shown as triangles.

Since at most two rotations are performed during insertion into a red-black tree, only O(1) additional time is spent updating size fields in the second phase. Thus, the total time for insertion into an n-node order-statistic tree is O(lg n), which is asymptotically the same as for an ordinary red-black tree.

Deletion from a red-black tree also consists of two phases: the first operates on the underlying search tree, and the second causes at most three rotations and otherwise performs no structural changes. The first phase splices out one node y. To update the subtree sizes, we simply traverse a path from node y up to the root, decrementing the size field of each node on the path. Since this path has length O(lg n) in an n-node red-black tree, the additional time spent maintaining size fields in the first phase is O(lg n). The O(1) rotations in the second phase of deletion can be handled in the same manner as for insertion. Thus, both insertion and deletion, including the maintenance of the size fields, take O(lg n) time for an n-node order-statistic tree.