## Line Charts: AlphaEvolve Performance vs. Keich Construction ### Overview The image contains four line charts comparing the performance of "AlphaEvolve" and "Keich Construction" algorithms, measured by "Total Union Area" against the "Number of Points". The charts are arranged in a 2x2 grid. The top-left and bottom-left charts show "AlphaEvolve Performance" over a range of 0-500 points. The top-right and bottom-right charts compare "AlphaEvolve" and "Keich Construction" over a range of 0-2000 points. ### Components/Axes * **X-axis (Horizontal):** "Number of Points". The top-left and bottom-left charts range from 0 to 500. The top-right and bottom-right charts range from 0 to 2000. * **Y-axis (Vertical):** "Total Union Area". All charts range from 0.10 to 0.35. * **Legends (Top-Right and Bottom-Right Charts):** * Light Blue: "AlphaEvolve" * Red: "Keich Construction" * **Legends (Top-Left and Bottom-Left Charts):** * Red: "AlphaEvolve Performance" ### Detailed Analysis **Top-Left Chart: AlphaEvolve Performance (0-500 Points)** * **AlphaEvolve Performance (Red):** The line starts at approximately 0.33 at 0 points, rapidly decreases to around 0.15 by 50 points, then continues to decrease more gradually with some upward spikes, reaching approximately 0.10 at 500 points. The line has a jagged appearance between 50 and 300 points, with several small upward jumps. **Bottom-Left Chart: AlphaEvolve Performance (0-500 Points)** * **AlphaEvolve Performance (Red):** The line starts at approximately 0.33 at 0 points, rapidly decreases to around 0.15 by 50 points, then continues to decrease more gradually with some upward spikes, reaching approximately 0.10 at 500 points. The line has a jagged appearance between 50 and 300 points, with several small upward jumps. This chart is nearly identical to the top-left chart. **Top-Right Chart: AlphaEvolve vs. Keich Construction (0-2000 Points)** * **AlphaEvolve (Light Blue):** The line starts at approximately 0.36 at 0 points, rapidly decreases to around 0.11 by 250 points, then continues to decrease more gradually, reaching approximately 0.08 at 2000 points. * **Keich Construction (Red):** The line starts at approximately 0.36 at 0 points, rapidly decreases to around 0.13 by 250 points, then continues to decrease more gradually, reaching approximately 0.08 at 2000 points. The Keich Construction line is consistently slightly above the AlphaEvolve line. **Bottom-Right Chart: AlphaEvolve vs. Keich Construction (0-2000 Points)** * **AlphaEvolve (Light Blue):** The line starts at approximately 0.36 at 0 points, rapidly decreases to around 0.11 by 250 points, then continues to decrease more gradually, reaching approximately 0.08 at 2000 points. * **Keich Construction (Red):** The line starts at approximately 0.36 at 0 points, rapidly decreases to around 0.13 by 250 points, then continues to decrease more gradually, reaching approximately 0.08 at 2000 points. The Keich Construction line is consistently slightly above the AlphaEvolve line. This chart is nearly identical to the top-right chart. ### Key Observations * The "Total Union Area" decreases as the "Number of Points" increases for all algorithms. * The decrease in "Total Union Area" is most rapid in the initial phase (0-250 points). * "Keich Construction" consistently has a slightly higher "Total Union Area" than "AlphaEvolve" in the top-right and bottom-right charts. * The top-left and bottom-left charts show the same data. * The top-right and bottom-right charts show the same data. ### Interpretation The charts suggest that both "AlphaEvolve" and "Keich Construction" algorithms improve their performance (lower "Total Union Area") as the number of points increases. The initial increase in points leads to a significant performance boost, with diminishing returns as the number of points continues to rise. "Keich Construction" appears to have a slightly higher "Total Union Area" than "AlphaEvolve", indicating potentially slightly worse performance. The duplication of charts suggests a possible error or an intention to highlight specific aspects of the data. The jaggedness of the AlphaEvolve Performance line in the top-left and bottom-left charts between 50 and 300 points indicates instability or variability in the algorithm's performance within that range.

## Charts: AlphaEvolve Performance vs. Keich Construction ### Overview The image contains four line charts comparing the performance of "AlphaEvolve" and "Keich Construction" algorithms. Each chart plots "Total Union Area" against "Number of Points". The charts appear to be examining how the algorithms scale with increasing data points. ### Components/Axes * **X-axis:** "Number of Points" - Ranges from 0 to approximately 2000. * **Y-axis:** "Total Union Area" - Ranges from approximately 0.08 to 0.35. * **Data Series:** * "AlphaEvolve" (represented by a blue dashed line with triangle markers) * "Keich Construction" (represented by a red solid line with circle markers) * **Legends:** Located in the top-right corner of the top two charts and bottom-right corner of the bottom two charts. ### Detailed Analysis or Content Details **Chart 1 (Top-Left):** * **AlphaEvolve Performance (Red):** The line starts at approximately 0.28 at 0 points, rapidly decreases to around 0.12 at 50 points, and then plateaus around 0.10-0.11 from approximately 200 points onwards. * **Trend:** The line initially shows a steep downward slope, indicating a rapid decrease in Total Union Area with increasing points, then flattens out. **Chart 2 (Top-Right):** * **AlphaEvolve (Blue):** The line starts at approximately 0.32 at 0 points, decreases to around 0.15 at 200 points, and then continues to decrease slowly, reaching approximately 0.10 at 2000 points. * **Keich Construction (Red):** The line starts at approximately 0.30 at 0 points, decreases rapidly to around 0.12 at 200 points, and then plateaus around 0.10-0.11 from approximately 500 points onwards. * **Trends:** Both lines show a decreasing trend, but AlphaEvolve decreases more gradually over the entire range. Keich Construction shows a steeper initial decrease, then levels off. **Chart 3 (Bottom-Left):** * **AlphaEvolve Performance (Red):** Similar to Chart 1, the line starts at approximately 0.28 at 0 points, rapidly decreases to around 0.12 at 50 points, and then plateaus around 0.10-0.11 from approximately 200 points onwards. * **Trend:** Identical to Chart 1. **Chart 4 (Bottom-Right):** * **AlphaEvolve (Blue):** The line starts at approximately 0.32 at 0 points, decreases to around 0.15 at 200 points, and then continues to decrease slowly, reaching approximately 0.10 at 2000 points. * **Keich Construction (Red):** The line starts at approximately 0.30 at 0 points, decreases rapidly to around 0.12 at 200 points, and then plateaus around 0.10-0.11 from approximately 500 points onwards. * **Trends:** Identical to Chart 2. ### Key Observations * Both algorithms show a decreasing Total Union Area as the Number of Points increases. * Keich Construction appears to converge to a lower Total Union Area faster than AlphaEvolve. * The charts 1 and 3 are identical, as are charts 2 and 4. * AlphaEvolve exhibits a more gradual decrease in Total Union Area, especially at higher point counts. ### Interpretation The data suggests that both AlphaEvolve and Keich Construction algorithms reduce the Total Union Area as more data points are considered. However, Keich Construction achieves this reduction more quickly, potentially indicating a faster convergence rate. AlphaEvolve maintains a more gradual reduction, which might be desirable in scenarios where a smoother, less abrupt change in Total Union Area is preferred. The identical nature of the top two charts and the bottom two charts suggests that these are either independent runs of the same experiment or different views of the same underlying data. The initial high Total Union Area values could represent the area before any optimization or construction process is applied, and the decreasing trend represents the improvement achieved by the algorithms. The plateauing of the lines indicates that the algorithms reach a point of diminishing returns, where adding more points does not significantly reduce the Total Union Area.

## Line Charts: AlphaEvolve vs. Keich Construction Performance ### Overview The image contains four line charts arranged in a 2x2 grid. Each chart plots "Total Union Area" (y-axis) against "Number of Points" (x-axis). The charts compare the performance of a method called "AlphaEvolve" against a baseline or alternative method called "Keich Construction." The two charts on the left display only the "AlphaEvolve Performance" line, while the two charts on the right compare "AlphaEvolve" and "Keich Construction" directly. The charts on the right use a larger x-axis scale (up to 2000 points) compared to the charts on the left (up to 500 points). ### Components/Axes * **Chart Type:** Line Charts (2x2 grid). * **Y-Axis Label (All Charts):** "Total Union Area". The scale ranges from approximately 0.08 to 0.38, with major tick marks at 0.10, 0.15, 0.20, 0.25, 0.30, and 0.35. * **X-Axis Label (All Charts):** "Number of Points". * **Left Column Charts (Top & Bottom):** Scale from 0 to 500, with major tick marks at 0, 100, 200, 300, 400, and 500. * **Right Column Charts (Top & Bottom):** Scale from 0 to 2000, with major tick marks at 0, 500, 1000, 1500, and 2000. * **Legends:** * **Top-Left Chart:** Legend in the top-right corner. Contains one entry: a solid red line labeled "AlphaEvolve Performance". * **Bottom-Left Chart:** Legend in the top-right corner. Contains one entry: a solid red line labeled "AlphaEvolve Performance". * **Top-Right Chart:** Legend in the top-right corner. Contains two entries: 1. A light blue line with circle markers labeled "AlphaEvolve". 2. A red line with square markers labeled "Keich Construction". * **Bottom-Right Chart:** Legend in the top-right corner. Contains two entries: 1. A light blue line with circle markers labeled "AlphaEvolve". 2. A red line with square markers labeled "Keich Construction". ### Detailed Analysis **Top-Left Chart (AlphaEvolve Performance, 0-500 Points):** * **Trend:** The red "AlphaEvolve Performance" line shows a steep, near-exponential decay from a high initial value. The decline is not perfectly smooth; it exhibits several small, sharp upward jumps or discontinuities (e.g., near x=50, x=120, x=250) before resuming its downward trend. * **Data Points (Approximate):** * At x ≈ 0, y ≈ 0.33. * At x ≈ 50, y ≈ 0.16 (just before a jump). * At x ≈ 100, y ≈ 0.13. * At x ≈ 200, y ≈ 0.11. * At x ≈ 300, y ≈ 0.10. * At x ≈ 500, y ≈ 0.09. **Bottom-Left Chart (AlphaEvolve Performance, 0-500 Points):** * **Trend:** Similar overall decaying trend as the top-left chart, but the line appears noisier or more jagged, especially in the range of x=0 to x=200. The discontinuities/jumps are less pronounced or absent. * **Data Points (Approximate):** * At x ≈ 0, y ≈ 0.33. * At x ≈ 100, y ≈ 0.13. * At x ≈ 200, y ≈ 0.11. * At x ≈ 300, y ≈ 0.10. * At x ≈ 500, y ≈ 0.10 (appears to plateau or slightly increase at the very end). **Top-Right Chart (AlphaEvolve vs. Keich Construction, 0-2000 Points):** * **Trend Verification:** * **AlphaEvolve (Light Blue, Circles):** Shows a very steep initial drop, then flattens out considerably. The line is consistently below the Keich Construction line. * **Keich Construction (Red, Squares):** Also shows a steep initial drop, but from a higher starting point. It remains above the AlphaEvolve line across the entire plotted range. * **Data Points & Cross-Reference (Approximate):** * At x ≈ 0: AlphaEvolve y ≈ 0.15, Keich Construction y ≈ 0.37. * At x ≈ 100: AlphaEvolve y ≈ 0.11, Keich Construction y ≈ 0.13. * At x ≈ 500: AlphaEvolve y ≈ 0.09, Keich Construction y ≈ 0.10. * At x ≈ 1000: AlphaEvolve y ≈ 0.085, Keich Construction y ≈ 0.09. * At x ≈ 2000: AlphaEvolve y ≈ 0.08, Keich Construction y ≈ 0.085. **Bottom-Right Chart (AlphaEvolve vs. Keich Construction, 0-2000 Points):** * **Trend Verification:** The trends are nearly identical to the top-right chart. AlphaEvolve (light blue) maintains a consistent, slight performance advantage (lower Total Union Area) over Keich Construction (red) across all point counts. * **Data Points & Cross-Reference (Approximate):** The values appear visually identical to the top-right chart at the sampled points (x=0, 100, 500, 1000, 2000). ### Key Observations 1. **Consistent Superiority:** In the direct comparison charts (right column), the "AlphaEvolve" method consistently achieves a lower "Total Union Area" than the "Keich Construction" method for the same number of points. 2. **Diminishing Returns:** All charts demonstrate a law of diminishing returns. The most significant reduction in Total Union Area occurs with the first 100-200 points. Adding more points beyond 500 yields progressively smaller improvements. 3. **Left vs. Right Column Discrepancy:** The "AlphaEvolve Performance" lines in the left-column charts (red lines) start at a much higher y-value (~0.33 at x≈0) compared to the "AlphaEvolve" lines in the right-column charts (light blue lines, starting at ~0.15 at x≈0). This suggests the left and right columns may represent different experimental conditions, datasets, or parameter settings for the AlphaEvolve method. 4. **Chart Noise:** The bottom-left chart's AlphaEvolve line is noticeably more jagged than the top-left chart's line, indicating potential variability in the process or measurement for that specific trial. ### Interpretation The data demonstrates the efficacy of the "AlphaEvolve" method for minimizing "Total Union Area" as a function of the number of points used. The primary finding is that AlphaEvolve outperforms the "Keich Construction" baseline, achieving a better (lower) score with the same computational budget (number of points). The steep initial descent in all curves indicates that the most critical optimizations are achieved early in the process. The performance gap between the two methods is established early and maintained, suggesting AlphaEvolve has a more efficient optimization trajectory. The plateauing of the curves implies a fundamental limit or convergence point for the problem being solved, where adding more points provides negligible benefit. The discrepancy between the left-column and right-column AlphaEvolve results is a critical anomaly. It strongly implies that the "AlphaEvolve Performance" shown in isolation (left charts) is from a different, more challenging scenario (resulting in higher initial union area) than the AlphaEvolve variant compared directly to Keich Construction (right charts). This could be due to different problem instances, hyperparameter settings, or versions of the algorithm. Without additional context, the exact reason is unclear, but it highlights the importance of the comparative charts (right column) for a fair evaluation against the baseline.

## Line Graphs: Comparison of AlphaEvolve and Keich Construction Performance ### Overview The image contains four line graphs arranged in a 2x2 grid, comparing the performance of two algorithms ("AlphaEvolve" and "Keich Construction") across varying numbers of data points. Each graph plots "Total Union Area" against "Number of Points," with distinct trends observed for each algorithm. --- ### Components/Axes - **X-Axis**: "Number of Points" (ranges from 0 to 500 in top-left/bottom-left graphs, 0 to 2000 in top-right/bottom-right graphs). - **Y-Axis**: "Total Union Area" (ranges from 0.10 to 0.35 across all graphs). - **Legends**: Positioned in the top-right corner of each graph, with: - **Red line**: "AlphaEvolve Performance" - **Blue line**: "Keich Construction" - **Gridlines**: Present in all graphs for reference. --- ### Detailed Analysis #### Top-Left Graph (0–500 Points) - **AlphaEvolve (Red)**: - Starts at ~0.35 (0 points), drops sharply to ~0.15 at 50 points. - Fluctuates between ~0.10 and ~0.15 for 100–500 points. - **Keich Construction (Blue)**: - Starts at ~0.25 (0 points), drops to ~0.10 at 50 points. - Stabilizes near ~0.08 for 100–500 points. #### Top-Right Graph (0–2000 Points) - **AlphaEvolve (Red)**: - Sharp initial drop from ~0.35 to ~0.15 at 50 points. - Gradual decline to ~0.09 by 2000 points. - **Keich Construction (Blue)**: - Steeper initial drop from ~0.35 to ~0.12 at 50 points. - Stabilizes near ~0.08 by 2000 points. #### Bottom-Left Graph (0–500 Points) - **AlphaEvolve (Red)**: - Sharp drop from ~0.35 to ~0.15 at 50 points. - Fluctuates between ~0.10 and ~0.15, with a slight rise to ~0.12 at 500 points. - **Keich Construction (Blue)**: - Sharp drop from ~0.25 to ~0.10 at 50 points. - Stabilizes near ~0.08 for 100–500 points. #### Bottom-Right Graph (0–2000 Points) - **AlphaEvolve (Red)**: - Sharp drop from ~0.35 to ~0.15 at 50 points. - Minor fluctuations, ending near ~0.09 at 2000 points. - **Keich Construction (Blue)**: - Steeper initial drop from ~0.35 to ~0.12 at 50 points. - Stabilizes near ~0.07 at 2000 points. --- ### Key Observations 1. **Initial Sharp Decline**: Both algorithms exhibit a steep reduction in "Total Union Area" for the first 50–100 points. 2. **Stabilization**: Keich Construction consistently stabilizes at lower values than AlphaEvolve across all graphs. 3. **Fluctuations**: AlphaEvolve shows irregular fluctuations (e.g., ~0.10–0.15 in top-left graph), while Keich Construction trends are smoother. 4. **Scalability**: Keich Construction maintains lower values at higher point counts (e.g., ~0.07 vs. ~0.09 in bottom-right graph). --- ### Interpretation - **Performance Implications**: Keich Construction appears more efficient at reducing "Total Union Area" as the number of points increases, suggesting better scalability or optimization for large datasets. - **AlphaEvolve Variability**: The fluctuations in AlphaEvolve’s performance may indicate sensitivity to input variability or algorithmic instability. - **Threshold Behavior**: Both methods show a critical threshold (50–100 points) where performance stabilizes, implying diminishing returns beyond this range. - **Practical Use**: Keich Construction may be preferable for applications requiring minimal union area at scale, while AlphaEvolve’s fluctuations warrant further investigation for reliability. --- ### Spatial Grounding & Trend Verification - **Legend Alignment**: Red (AlphaEvolve) and blue (Keich Construction) lines match legend labels across all graphs. - **Trend Consistency**: Initial sharp declines align with the described trends (e.g., top-left graph’s ~0.35 → ~0.15 drop for AlphaEvolve). - **Outliers**: No significant outliers; fluctuations in AlphaEvolve are within expected variability. --- ### Content Details - **Numerical Approximations**: - Top-left: AlphaEvolve ~0.35 → ~0.15 (50 points); Keich Construction ~0.25 → ~0.10 (50 points). - Bottom-right: AlphaEvolve ~0.09 (2000 points); Keich Construction ~0.07 (2000 points). - **Axis Scales**: Y-axis increments of ~0.05; X-axis increments vary (50–2000 points). --- ### Summary The graphs demonstrate that Keich Construction consistently outperforms AlphaEvolve in minimizing "Total Union Area" as the number of points increases, with smoother and more stable performance. AlphaEvolve’s fluctuations suggest potential trade-offs between initial efficiency and long-term reliability.