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PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Simple in-place yet comparison-optimal Mergesort
Christian Siebert · 2025-09-29 · via cs.DS updates on arXiv.org

Mergesort is one of the few efficient sorting algorithms and, despite being the oldest one, often still the method of choice today. In contrast to some alternative algorithms, it always runs efficiently using O(n log n) element comparisons and usually works in a stable manner. Its only practical disadvantage is the need for a second array, and thus twice the amount of memory. This can be an impeding factor, especially when handling large amounts of data, where it is often either impractical or even impossible to fall back to slower or unstable sorting alternatives. Therefore, many attempts have been made to fix this problem by adapting Mergesort to work in place. While it is known that such algorithms exist, the previously published solutions are mostly not efficient, become unstable, and/or are very complex. This renders them practically useless for real-world applications. In this paper, we propose a novel in-place Mergesort algorithm that is stable by design. Albeit its running time of O(n log^2 n) is not quite optimal, it still works efficiently, both in theory using the optimal number of O(n log n) comparisons and in practice with low constants, while being easily comprehensible. The baseline for this new algorithm includes just two prerequisites: 1) an optimal array rotation; and 2) the co-ranking idea, published in 2014 and originally intended to parallelize Mergesort. Although it would certainly be possible to parallelize the presented algorithm, this paper focuses on the sequential aspect of this efficient, stable and in-place Mergesort algorithm. Additionally, we implemented our algorithm and present performance results measured on one of the largest shared memory systems currently available.