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README.md

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+# Quick-Median
+
+## Why this project?
+
+Many existing median-finding packages on npm are not optimized for performance. This project implements the Floyd-Rivest algorithm, a highly efficient selection algorithm that outperforms traditional quickselect in practice.
+
+## Algorithm
+
+Quick-Median uses the Floyd-Rivest algorithm, an optimized selection algorithm with the following characteristics:
+
+- Expected running time: $O(n)$
+- Expected number of comparisons: $n + \min(k, n - k) + O(n^{1/2} \log^{1/2} n)$
+
+
+The algorithm works by:
+1. Selecting a small random sample from the input array
+2. Recursively selecting two pivot elements
+3. Partitioning the array into three sets based on these pivots
+4. Recursively applying the algorithm to the appropriate partition
+
+This approach significantly reduces the number of comparisons needed, especially for large datasets.
+
+### Comparison with Other Algorithms
+
+| Algorithm | Average Case | Worst Case | Space Complexity |
+|-----------|--------------|------------|------------------|
+| Floyd-Rivest | $O(n)$ | $O(n^2)$ | $O(1)$ |
+| Quickselect | $O(n)$ | $O(n^2)$ | $O(1)$ |
+| Median of Medians | $O(n)$ | $O(n)$ | $O(1)$ |
+| Sorting-based | $O(n \log n)$ | $O(n \log n)$ | $O(1)$ to $O(n)$ |
+
+While Floyd-Rivest and Quickselect have the same big-O complexity, Floyd-Rivest performs fewer comparisons on average, leading to better real-world performance.
+
+The worst-case scenario for Floyd-Rivest (and Quickselect) occurs when the pivot choices consistently result in unbalanced partitions. However, this is extremely rare in practice due to the algorithm's use of random sampling.
+
+
+## Benchmarks
+
+This implementation consistently outperforms other popular median-finding packages on npm:
+
+
+| Algorithm | 10 | 100 | 1000 | 10000 | 100000 | 1000000 | 10000000 |
+|-----------|----|----|------|-------|--------|---------|----------|
+| median | 0.00 | 0.00 | 0.02 | 0.27 | 0.99 | 9.52 | 97.26 |
+| faster-median | 0.00 | 0.01 | 0.05 | 0.24 | 2.76 | 68.41 | 693.28 |
+| compute-median | 0.00 | 0.01 | 0.03 | 0.19 | 1.85 | 17.04 | 205.33 |
+| **quick-median** | **0.00** | **0.00** | **0.01** | **0.04** | **0.28** | **1.70** | **17.04** |
+| ml-array-median | 0.00 | 0.00 | 0.01 | 0.08 | 0.67 | 2.24 | 25.22 |
+| median-quickselect | 0.00 | 0.00 | 0.01 | 0.03 | 0.26 | 1.59 | 17.91 |
+
+(Times in milliseconds)
+
+As shown, Quick-Median is consistently faster, especially for larger datasets. For an input size of 10,000,000 elements, Quick-Median is:
+- 5.7x faster than the 'median' package
+- 40.7x faster than 'faster-median'
+- 12x faster than 'compute-median'
+- 1.48x faster than 'ml-array-median'
+
+## TypeScript Support
+
+This package is written in TypeScript and includes type definitions, ensuring type safety in TypeScript projects.
+
+## Acknowledgements
+
+This implementation is based on the Floyd-Rivest algorithm, originally described in:
+
+Floyd, Robert W.; Rivest, Ronald L. (March 1975). "Expected time bounds for selection". Communications of the ACM. 18 (3): 165–172. 
+
+For more information about the algorithm, see the [Wikipedia article on the Floyd-Rivest algorithm](https://en.wikipedia.org/wiki/Floyd%E2%80%93Rivest_algorithm).
+
+## License
+
+This project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.