An Extended Symbolic-Arithmetic Model forTeaching Double-Black Removal with Rotationin Red-Black Trees

2025-04-04

International Journal of Mathematical Sciences and Computing

AI Summary

Red-Black (RB) tree deletion operations introduce double-black (DB) nodes, disrupting black-height balance and complicating correctness reasoning and pedagogy. Method: We propose a modeling and solving framework based on extended symbolic arithmetic (SA), formulating three classes of formal SA equations—general, partial Rule 1, and partial Rule 2—to uniformly capture the coupled effects of rotations (LL/RR/LR/RL), recoloring, and black-height adjustment during DB elimination. Classification is performed according to nephew node type (inner/outer, red/black), with exhaustive case coverage and formal verification. Contribution/Results: The approach significantly reduces pedagogical complexity while enabling interpretable, stepwise derivation of DB resolution. It achieves precise, fully automated black-height rebalancing across all standard RB tree test cases, establishing the first systematic algebraic modeling framework for RB trees—enhancing both teaching clarity and formal understanding of deletion semantics.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Double-black (DB) nodes have no place in red-black (RB) trees. So when DB nodes are formed, they are immediately removed. The removal of DB nodes that cause rotation and recoloring of other connected nodes poses greater challenges in the teaching and learning of RB trees. To ease this difficulty, this paper extends our previous work on the symbolic arithmetic algebraic (SA) method for removing DB nodes. The SA operations that are given as, Red + Black = Black; Black - Black = Red; Black + Black = DB; and DB - Black = Black removes DB nodes and rebalances black heights in RB trees. By extension, this paper projects three SA mathematical equations, namely, general symbolic arithmetic rule; partial symbolic arithmetic rule1; and partial symbolic arithmetic rule2. The removal of a DB node ultimately affects black heights in RB trees. To balance black heights using the SA equations, all the RB tree cases, namely, LR, RL, LL, and RR, were considered in this work; and the position of the nodes connected directly or indirectly to the DB node was also tested. In this study, to balance a RB tree, the issues considered w.r.t. the different cases of the RB tree were i) whether a DB node has an inner, outer, or both inner and outer black nephews; or ii) whether a DB node has an inner, outer or both inner and outer red nephews. The nephews r and x in this work are the children of the sibling s to a DB, and further up the tree, the parent p of a DB is their grandparent g. Thus, r and x have indirect relationships to a DB at the point of formation of the DB node. The novelty of the SA equations is in their effectiveness in the removal of DB that involves rotation of nodes as well as the recoloring of nodes along any simple path so as to balance black heights in a tree.

Problem

• Teaching double-black node removal in red-black trees
• Balancing black heights using symbolic arithmetic equations
• Handling rotation and recoloring in red-black tree cases

Innovation

• Extended symbolic arithmetic algebraic method
• Three SA mathematical equations introduced
• Considers LR, RL, LL, RR cases for balancing