These algorithms prioritize transactions based on fee rates while maintaining topological order, occasionally using post-processing techniques to refine the outcomes. The complexity of identifying high-fee-rate subsets within these clusters is acknowledged, highlighting that while straightforward methods exist for managing larger clusters, advanced techniques yield significant benefits for smaller, more common ones. Advanced linearization strategies are detailed, emphasizing the optimization of connected components within a cluster by treating separable groups individually. This includes bottleneck splitting, which targets central transactions for isolated processing, and an iterative search approach aimed at finding the highest-feerate subsets. This approach involves evaluating potential subsets, refining them through branching paths, and employing techniques like 'jump ahead' to expedite the process based on transaction ancestry.

Efficiency improvements are sought through various means, including bounding evaluations with conservative quality estimates and utilizing Last-In-First-Out (LIFO) stack approaches for processing work items. Caching strategies are implemented to reduce recalculations, and early comparisons between potential set feerates and current best subsets help avoid unnecessary computations. The algorithm's initialization with the best ancestor set ensures performance at least on par with simpler linearization methods, guiding further optimizations. The current implementation diverges slightly from its theoretical model by not universally applying connected-component splitting but incorporates many proposed ideas, such as managing multiple LIFO stacks and strategically selecting transactions to minimize the search space.

Discussion also extends to alternative methodologies for linearization, suggesting a shift from traditional approaches to one where the transaction graph is initially transformed into a graph of valid chunks. This method implies a fundamental change in transaction processing, potentially offering improved processing times or enhanced scalability. Assessing the feasibility of this transformation involves evaluating technical challenges, comparing it against existing algorithms, and considering factors like computational complexity and system performance impact. The practical implications for existing systems and the necessary modifications to accommodate this new methodology are also critical considerations for its viability and effectiveness in real-world scenarios.