🤖 AI Summary
Traditional optimality analyses rely solely on input size, ignoring inherent structural properties of problem instances. This limitation hinders fine-grained performance characterization and adaptive algorithm design.
Method: We propose a generalized optimality paradigm that characterizes algorithmic efficiency via problem-relevant (including implicit) parameters, establishing a unified parametrized optimality framework. We formally define “universal optimality,” devise an adaptive analysis framework based on partitioned sorting, and introduce a novel, quantifiable metric for implicit orderliness.
Contribution/Results: (1) We reinterpret the adaptivity boundaries of classical sorting algorithms under this refined lens; (2) we construct the first sorting algorithm provably achieving the information-theoretic lower bound with respect to our new metric; and (3) we provide a scalable, parametrized optimization paradigm applicable not only to sorting but also to broader algorithmic domains. This framework bridges instance-aware analysis and theoretical optimality, enabling more precise and structure-exploiting algorithm design.
📝 Abstract
We formalize a new paradigm for optimality of algorithms, that generalizes worst-case optimality based only on input-size to problem-dependent parameters including implicit ones. We re-visit some existing sorting algorithms from this perspective, and also present a novel measure of sortedness that leads to an optimal algorithm based on partition sort. This paradigm of measuring efficiency of algorithms looks promising for further interesting applications beyond the existing ones.