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Limits RegistryA foundational result in computer science proves that no comparison-based sorting algorithm can sort n items in fewer than Omega(n log n) comparisons in the worst case, since a decision tree distinguishing all n! possible input orderings must have at least log2(n!) levels. Algorithms including merge sort and heapsort achieve this bound asymptotically, making them worst-case optimal among comparison-based methods; non-comparison-based methods like radix sort can beat this bound only by exploiting extra structure (such as bounded key size) unavailable to a general comparison-based algorithm.
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A foundational result in computer science proves that no comparison-based sorting algorithm can sort n items in fewer than Omega(n log n) comparisons in the worst case, since a decision tree distinguishing all n! possible input orderings must have at least log2(n!) levels. Algorithms including merge sort and heapsort achieve this bound asymptotically, making them worst-case optimal among comparison-based methods; non-comparison-based methods like radix sort can beat this bound only by exploiting extra structure (such as bounded key size) unavailable to a general comparison-based algorithm.
Any comparison-based sorting algorithm must perform at least ceiling(log2(n!)) = Omega(n log n) pairwise comparisons in the worst case to sort n distinct items, a bound proven via a decision-tree argument counting the n! possible orderings.
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Permanent ID limitsregistry.com/limits/LR-COMPARISON-SORT-LOWER-BOUND
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Limits Registry. LR-COMPARISON-SORT-LOWER-BOUND. Information-theoretic lower bound on comparison-based sorting. 2026.