## Line Chart: Time Steps to Solution vs. Problem Size and Parameter k ### Overview The image contains two line charts comparing computational performance metrics across different problem sizes and parameter values. The left panel shows time steps to solution as a function of problem size for various algorithms, while the right panel displays time steps as a function of parameter k for percentile-based performance distributions. ### Components/Axes **Left Panel:** - **X-axis (Problem Size):** Logarithmic scale from √100 to √1000 (10 to 31.62 units) - **Y-axis (Time Steps to Solution):** Logarithmic scale from 10³ to 10⁸ - **Legend (Top-left):** - CIM-SFC k=0.2 (magenta circles) - CIM-SFC k=0.15 (purple circles) - CIM-SFC k=0.0 (blue circles) - NMFA (black dashed line with dots) - **Shaded Regions:** Confidence intervals around each line **Right Panel:** - **X-axis (k):** Linear scale from 0.00 to 0.25 - **Y-axis (Time Steps to Solution):** Logarithmic scale from 10⁴ to 10⁸ - **Legend (Top-right):** - 90th percentile (solid red line) - 75th percentile (dashed red line) - Median (dotted red line) - 25th percentile (dash-dot red line) ### Detailed Analysis **Left Panel Trends:** 1. **CIM-SFC k=0.2:** Starts at ~10³.5 steps at √100, rises to ~10⁵.5 at √1000 (magenta) 2. **CIM-SFC k=0.15:** Starts at ~10³.8, rises to ~10⁶ at √1000 (purple) 3. **CIM-SFC k=0.0:** Starts at ~10⁴, rises to ~10⁷ at √1000 (blue) 4. **NMFA:** Starts at ~10⁴, rises to ~10⁷.5 at √1000 (black dashed) - All lines show positive correlation between problem size and time steps - Confidence intervals widen with increasing problem size **Right Panel Trends:** - **90th percentile:** Starts at ~10⁷ at k=0.05, drops to ~10⁵ at k=0.15, then rises to ~10⁶ at k=0.25 - **75th percentile:** Starts at ~10⁶, drops to ~10⁴.5, then rises to ~10⁵.5 - **Median:** Starts at ~10⁵, drops to ~10⁴, then rises to ~10⁴.5 - **25th percentile:** Starts at ~10⁴, drops to ~10³.5, then rises to ~10⁴ ### Key Observations 1. **Algorithm Efficiency:** CIM-SFC with lower k values consistently outperforms NMFA across all problem sizes 2. **Scalability:** Time steps increase exponentially with problem size for all algorithms 3. **Parameter Sensitivity:** Right panel shows non-linear relationship between k and performance, with significant variance in upper percentiles 4. **Confidence Intervals:** Left panel shaded regions indicate increasing uncertainty with larger problem sizes ### Interpretation The data demonstrates that CIM-SFC algorithms with smaller k values (0.0-0.15) achieve better performance than NMFA, particularly at larger problem sizes. The right panel reveals that while median performance improves with increasing k, the 90th percentile shows a U-shaped pattern, suggesting some instances become significantly slower at extreme k values. This indicates that while average performance may improve with parameter tuning, there's a risk of outliers degrading results. The exponential scaling of time steps with problem size highlights the need for algorithmic optimizations to handle larger datasets efficiently.