Left-leaning Red-Black Trees

The red-black tree model for implementing balanced search trees, introduced by Guibas and Sedgewick thirty years ago, is now found throughout our computational infrastructure. Red-black trees are described in standard textbooks and are the underlying data structure for symbol-table implementations within C++, Java, Python, BSD Unix, and many other modern systems. However, many of these implementations have sacrificed some of the original design goals (primarily in order to develop an effective implementation of the delete operation, which was incompletely specified in the original paper), so a new look is worthwhile. In this paper, we describe a new variant of redblack trees that meets many of the original design goals and leads to substantially simpler code for insert/delete, less than one-fourth as much code as in implementations in common use. All red-black trees are based on implementing 2-3 or 2-3-4 trees within a binary tree, using red links to bind together internal nodes into 3-nodes or 4-nodes. The new code is based on combining three ideas: Use a recursive implementation. • Require that all 3-nodes lean left. • Perform rotations on the way up the tree (after the recursive calls). • Not only do these ideas lead to simple code, but they also unify the algorithms: for example, the leftleaning versions of 2-3 trees and top-down 2-3-4 trees differ in the position of one line of code. All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of lg N in a tree of N keys, and the behavior observed in practice is typically that same multiple faster than the worst-case bound, close the to optimal lg N nodes examined that would be observed in a perfectly balanced tree. This performance is also conjectured (but not yet proved) for trees built from random keys, for all the major variants of red-black trees. Can we analyze average-case performance with random keys for this new, simpler version? This paper describes experimental results that shed light on the fascinating dynamic behavior of the growth of these trees. Specifically, in a left-leaning red-black 2-3 tree built from N random keys: A random successful search examines lg • N – 0.5 nodes. The average tree height is about 2 ln • N (!) The average size of left subtree exhibits log-oscillating behavior. • The development of a mathematical model explaining this behavior for random keys remains one of the outstanding problems in the analysis of algorithms. From a practical standpoint, left-leaning red-black trees (LLRB trees) have a number of attractive characteristics: Experimental studies have not been able to distinguish these algorithms from optimal. • They can be implemented by adding just a few lines of code to standard BST algorithms. • Unlike hashing, they support ordered operations such as • select, rank, and range search. Thus, LLRB trees are useful for a broad variety of symbol-table applications and are prime candidates to serve as the basis for symbol tables in software libraries in the future.