## Line Graphs: Comparison of AlphaEvolve and Keich Construction for Triangles
### Overview
The image contains two side-by-side line graphs comparing the performance of two algorithms ("AlphaEvolve" and "Keich Construction") across two metrics: "Total Union Area" (left) and "S^T Score" (right). Both graphs plot performance against the "Number of Points" (0–120). The graphs reveal divergent trends in how each algorithm scales with increasing point counts.
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### Components/Axes
#### Left Graph: Total Union Area
- **X-axis**: Number of Points (0, 20, 40, 60, 80, 100, 120)
- **Y-axis**: Total Union Area (0.00–0.35)
- **Legend**:
- Blue line: AlphaEvolve Triangle Areas
- Red line: Keich Construction for Triangles
- **Legend Position**: Top-right corner
#### Right Graph: S^T Score
- **X-axis**: Number of Points (0, 20, 40, 60, 80, 100, 120)
- **Y-axis**: S^T Score (0.50–0.90)
- **Legend**:
- Blue line: AlphaEvolve Triangle Scores
- Red line: Keich Construction for Triangles
- **Legend Position**: Top-right corner
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### Detailed Analysis
#### Left Graph: Total Union Area
- **AlphaEvolve (Blue)**:
- Starts at ~0.35 (0 points), drops sharply to ~0.25 by 20 points, then plateaus.
- Remains stable (~0.24–0.25) from 20 to 120 points.
- **Keich Construction (Red)**:
- Starts at ~0.35 (0 points), drops rapidly to ~0.22 by 20 points.
- Continues a steady decline to ~0.12 by 120 points.
#### Right Graph: S^T Score
- **AlphaEvolve (Blue)**:
- Starts at ~0.9 (0 points), drops to ~0.7 by 20 points.
- Continues declining to ~0.4 by 120 points.
- **Keich Construction (Red)**:
- Starts at ~0.9 (0 points), drops to ~0.85 by 20 points.
- Stabilizes at ~0.82–0.83 from 40 to 120 points.
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### Key Observations
1. **Left Graph**:
- Keich Construction shows a **consistent decline** in union area as points increase, suggesting diminishing efficiency.
- AlphaEvolve stabilizes after an initial drop, indicating better scalability for union area.
2. **Right Graph**:
- AlphaEvolve experiences a **steeper decline** in S^T Score, dropping ~50% from 0 to 120 points.
- Keich Construction maintains a **higher S^T Score** across all point counts, with minimal degradation after 20 points.
3. **Divergence**:
- Keich Construction outperforms AlphaEvolve in S^T Score but underperforms in union area at higher point counts.
- AlphaEvolve sacrifices S^T Score for better union area stability.
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### Interpretation
The data suggests a trade-off between the two algorithms:
- **Keich Construction** prioritizes **S^T Score efficiency**, maintaining high scores even as points increase, but at the cost of reduced union area.
- **AlphaEvolve** sacrifices S^T Score for **union area stability**, which may be critical for applications requiring consistent coverage.
The sharp initial drop in both metrics for both algorithms implies that performance is highly sensitive to early point additions. Keich’s stabilization in S^T Score suggests a robust design for score preservation, while AlphaEvolve’s union area plateau indicates a focus on maintaining spatial coverage. These trends could inform algorithm selection based on whether union area or score efficiency is prioritized in a given use case.
