96 lines
3.1 KiB
Markdown
96 lines
3.1 KiB
Markdown
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# Checkpointed Sorting Experiment
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## Overview
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This experiment demonstrates how external merge sort with limited memory exhibits the space-time tradeoff predicted by Williams' 2025 result.
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## Key Concepts
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### Standard In-Memory Sort
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- **Space**: O(n) - entire array in memory
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- **Time**: O(n log n) - optimal comparison-based sorting
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- **Example**: Python's built-in sort, quicksort
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### Checkpointed External Sort
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- **Space**: O(√n) - only √n elements in memory at once
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- **Time**: O(n√n) - due to disk I/O and recomputation
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- **Technique**: Sort chunks that fit in memory, merge with limited buffers
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### Extreme Space-Limited Sort
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- **Space**: O(log n) - minimal memory usage
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- **Time**: O(n²) - extensive recomputation required
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- **Technique**: Iterative merging with frequent checkpointing
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## Running the Experiments
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### Quick Test
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```bash
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python test_quick.py
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```
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Runs with small input sizes (100-1000) to verify correctness.
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### Full Experiment
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```bash
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python run_final_experiment.py
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```
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Runs complete experiment with:
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- Input sizes: 1000, 2000, 5000, 10000, 20000
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- 10 trials per size for statistical significance
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- RAM disk comparison to isolate I/O overhead
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- Generates publication-quality plots
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### Rigorous Analysis
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```bash
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python rigorous_experiment.py
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```
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Comprehensive experiment with:
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- 20 trials per size
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- Detailed memory profiling
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- Environment logging
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- Statistical analysis with confidence intervals
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## Actual Results (Apple M3 Max, 64GB RAM)
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| Input Size | In-Memory Time | Checkpointed Time | Slowdown | Memory Reduction |
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|------------|----------------|-------------------|----------|------------------|
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| 1,000 | 0.022 ms | 8.2 ms | 375× | 0.1× (overhead) |
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| 5,000 | 0.045 ms | 23.4 ms | 516× | 0.2× |
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| 10,000 | 0.091 ms | 40.5 ms | 444× | 0.2× |
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| 20,000 | 0.191 ms | 71.4 ms | 375× | 0.2× |
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Note: Memory shows algorithmic overhead due to Python's memory management.
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## Key Findings
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1. **Massive Constant Factors**: 375-627× slowdown instead of theoretical √n
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2. **I/O Not Dominant**: Fast NVMe SSDs show only 1.0-1.1× I/O overhead
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3. **Scaling Confirmed**: Power law fits show n^1.0 for in-memory, n^1.4 for checkpointed
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## Real-World Applications
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- **Database Systems**: External sorting for large datasets
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- **MapReduce**: Shuffle phase with limited memory
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- **Video Processing**: Frame-by-frame processing with checkpoints
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- **Scientific Computing**: Out-of-core algorithms
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## Visualization
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The experiment generates:
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1. `paper_sorting_figure.png` - Clean figure for publication
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2. `rigorous_sorting_analysis.png` - Detailed analysis with error bars
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3. `memory_usage_analysis.png` - Memory scaling comparison
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4. `experiment_environment.json` - Hardware/software configuration
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5. `final_experiment_results.json` - Raw experimental data
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## Dependencies
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```bash
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pip install numpy scipy matplotlib psutil
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```
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## Reproducing Results
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To reproduce our results exactly:
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1. Ensure CPU frequency scaling is disabled
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2. Close all other applications
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3. Run on a machine with fast SSD (>3GB/s read)
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4. Use Python 3.10+ with NumPy 2.0+
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