Hybridization and Optimization Modeling, Analysis, and Comparative Study of Sorting Algorithms: Adaptive Techniques, Parallelization, for Mergesort, Heapsort, Quicksort, Insertion Sort, Selection Sort, and Bubble Sort

This study presents a comprehensive analysis of six classical sorting algorithms—Mergesort, Heapsort, Quicksort (with median-of-three pivot), Insertion Sort, Selection Sort, and Bubble Sort—to evaluate their practical efficiency across diverse input conditions. While theoretical complexity (O(n²) vs. O(n log n)) provides foundational insights, real-world performance depends on implementation-specific factors, input size, and data distribution. The research addresses the critical need to bridge theoretical predictions with empirical benchmarks, particularly as modern computing environments demand optimized algorithm selection for varying workloads. Using Python-based implementations, the methodology systematically tests algorithms on arrays (size 10–100,000) with randomized, sorted, reverse-sorted, and custom patterns, measuring execution times and memory usage. Results reveal quadratic algorithms outperform O(n log n) methods for small datasets (e.g., Selection Sort: 0.000004s at n=10), while Quicksort dominates at scale (0.089s vs. Bubble Sort’s 265.93s at n=100,000). Logarithmic visualizations highlight exponential efficiency divergence, with O(n log n) algorithms achieving 2,900× speedup over O(n²) counterparts for large arrays. Future research should explore adaptive hybrid systems leveraging machine learning for dynamic input adaptation, energy-efficient sorting in embedded systems, and quantum computing architectures. This work provides actionable insights for optimizing algorithm selection in data-intensive applications.