Abstract
Red–black (RB) trees are one of the most efficient variants of balanced binary search trees. However, they have often been criticized for being too complicated, hard to explain, and unsuitable for pedagogical purposes, particularly their delete operation. Sedgewick (in: Dagstuhl Workshop on Data Structures, 2008. https://sedgewick.io/wp-content/themes/sedgewick/papers/2008LLRB.pdf) identified the length of code as the root of the problems and introduced left-leaning red–black (LLRB) trees. The delete operation of LLRB trees has a compact recursive code. Unfortunately, it may perform many unnecessary operations. The crux of the deletion algorithm is dealing with a “deficient” subtree, that is one whose black-height has become one less than that of its sibling subtree. In this paper, we revisit 2–3 red–black trees and propose a parity-seeking delete algorithm with the basic idea of making a deficient subtree on a par with its sibling: either by fixing the deficient subtree or by turning the sibling deficient as well, ascending deficiency to the parent node. Interestingly, the proposed parity-seeking delete algorithm also works for 2–3–4 RB trees. Our experiments show that the proposed parity-seeking delete algorithm is as efficient as the best preceding RB trees. The proposed parity-seeking delete algorithm is easily understandable and suitable for pedagogical and practical purposes.
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Notes
Before, with the same goal of having simple code, Andersson [1] had considered “right-leaning” binary versions of 2–3 trees, in which only right links can connect nodes of the same level.
We have visualized the steps taken by each variant of RB trees on randomly inserting and deleting 30 numbers. This visualization is available at https://k-ghiasi.github.io/RedBlackTrees/doc/index.html.
We can also define the root deficient when its black-height has become one less than its value before deletion started. However, the two definitions are equivalent since the root would be red at that time.
We borrowed this view from Sedgewick [17].
Alternatively, one can choose the smallest value in the right subtree.
In place of a deficient node, [9] introduced the notion of an “extra” black for the node x. Here, we use the notion of a deficient subtree to state the same concept.
We remind the reader that the root node, or the whole tree, is defined to be deficient if the tree is balanced and the root is red.
Note: Case (b) might look complicated, but conceptually it is simple. The path “g-p-x” has four subtrees of the same black-height, all with black roots (see Fig. 18b.1, b.2). Applying one or two rotations changes this path into a configuration with one root, two children, and the four subtrees. The root is colored black, and its two children are colored red, and we are done.
Note: Again, case (b) might look complicated, but conceptually it is simple. The path “g-p-x” has four subtrees of the same black-height, all with black roots (Fig. 20b.1, b.2). Applying one or two rotations changes this path into a configuration with one root, two children, and the four subtrees. The root and its two children are colored black, and the root is the new overfull node with which the iteration is continued.
The equality of the number of top–down visits is clear.
To see how this happens in an example, compare page 66 of https://k-ghiasi.github.io/RedBlackTrees/doc/2-3-psrbt-insert-images.pdf and page 62 of https://k-ghiasi.github.io/RedBlackTrees/doc/clrs-insert-images.pdf.
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The authors are deeply grateful to the anonymous reviewers for generously devoting their precious time to improving this paper and giving constructive valuable comments that highly improved the quality of the manuscript.
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Ghiasi-Shirazi, K., Ghandi, T., Taghizadeh, A. et al. Revisiting 2–3 red–black trees with a pedagogically sound yet efficient deletion algorithm: parity-seeking. Acta Informatica (2024). https://doi.org/10.1007/s00236-023-00452-6
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DOI: https://doi.org/10.1007/s00236-023-00452-6