## Line Chart: Percentage of Problems Solved vs. Number of Solutions per Problem ### Overview The chart illustrates the performance of five algorithms in solving problems as the number of solutions per problem (N) increases. The y-axis represents the percentage of problems solved (89%–94%), while the x-axis shows N in powers of 2 (2² to 2⁶). Each algorithm is represented by a distinct line with markers and shaded confidence intervals. ### Components/Axes - **X-axis (N)**: Number of solutions per problem, labeled as $ N = \text{number of solutions per problems} $, with values $ 2^2, 2^3, 2^4, 2^5, 2^6 $. - **Y-axis**: Percentage of problems solved, ranging from 89% to 94%. - **Legend**: Located on the right, mapping colors and markers to algorithms: - **Green +**: Majority Vote - **Red ▲**: +OmegaPRM - **Purple ○**: +PRM800K - **Blue ○**: +Shepherd - **Dark Purple ○**: +Shepherd (ours) ### Detailed Analysis 1. **Majority Vote (Green +)**: - Starts at ~89.2% at $ 2^2 $, rising to ~92.7% at $ 2^6 $. - Confidence interval widens at lower N values, narrowing as N increases. 2. **+OmegaPRM (Red ▲)**: - Highest performance, starting at ~92.5% at $ 2^2 $, peaking at ~93.6% at $ 2^6 $. - Confidence interval remains relatively narrow across all N values. 3. **+PRM800K (Purple ○)**: - Begins at ~91.8% at $ 2^2 $, increasing to ~92.9% at $ 2^6 $. - Confidence interval widens slightly at lower N but stabilizes. 4. **+Shepherd (Blue ○)**: - Starts at ~91.1% at $ 2^2 $, rising to ~92.7% at $ 2^6 $. - Confidence interval is moderate, with minimal variation. 5. **+Shepherd (ours) (Dark Purple ○)**: - Lowest performance, starting at ~89.1% at $ 2^2 $, reaching ~91.8% at $ 2^6 $. - Confidence interval is the widest, indicating higher uncertainty. ### Key Observations - **+OmegaPRM** consistently outperforms all other algorithms, maintaining the highest percentage of problems solved. - **+Shepherd (ours)** shows the lowest performance, with a significant gap compared to other methods. - All algorithms improve as N increases, but the rate of improvement varies. +OmegaPRM and +PRM800K show steeper growth. - Confidence intervals (shaded regions) are widest at lower N values, suggesting greater uncertainty in early results. ### Interpretation The data demonstrates that **+OmegaPRM** is the most effective algorithm for solving problems, particularly as the number of solutions per problem increases. The shaded confidence intervals indicate that performance estimates are less reliable at lower N values, with uncertainty decreasing as N grows. The **+Shepherd (ours)** algorithm lags behind others, suggesting potential limitations in its design or implementation. The trend highlights the importance of solution quantity in problem-solving efficiency, with +OmegaPRM leveraging this advantage most effectively.