Problem Setting
---------------

Consider a CSV file with $N$ rows and $M$ columns, where $M \leq 10$. We feed each row to an LLM inference engine (with a prefix KV cache) by concatenating all column values in that row. For the $i$-th row with entries $A[i,1], A[i,2], \ldots, A[i,M]$, we construct the input string:

```math
S_i = \text{Concat}(\text{string}(A[i,1]), \text{string}(A[i,2]), \ldots, \text{string}(A[i,M]))
````

When requesting $S_i$ for $i > 1$, the prefix KV-cache hit rate depends on the longest common prefix with any previously seen request:

```math
\text{hit\_rate}_i = 
\frac{\max_{1 \le j < i} \text{LCP}(S_i, S_j)}{|S_i|}
```

where $LCP(S, T)$ is the length of the longest common prefix between strings $S$ and $T$.

You are allowed to reorder the CSV columns. Let $p$ be a permutation of $\{1, 2, ..., M\}$. The reordered string for row $i$ becomes:

```math
S'_i = \text{Concat}(\text{string}(A[i,p_1]), \text{string}(A[i,p_2]), \ldots, \text{string}(A[i,p_M]))
```

The goal is to choose a permutation $p$ that maximizes the overall KV-cache hit rate:

```math
\max_p\;
\frac{\sum_{i=2}^N \max_{1 \le j < i} \text{LCP}(S'_i, S'_j)}
     {\sum_{i=1}^N |S'_i|}
```




Target
---
Maximize prefix hit rate shown above (higher is better)

- **Hard Constraint**: Average runtime per dataset must be $\leq 10$ seconds (score = 0 if exceeded) and correctly handle column merge constraint.

**Column Merges**:
- Column merge specs are provided per dataset
- Columns in each merge group are concatenated into a single column
- The merged column replaces the original columns
- Merge operations are applied before column reordering

API Specification
---
Implement a `Solution` class:

```python
import pandas as pd

class Solution:
    def solve(
        self,
        df: pd.DataFrame,
        early_stop: int = 100000,
        row_stop: int = 4,
        col_stop: int = 2,
        col_merge: list = None,
        one_way_dep: list = None,
        distinct_value_threshold: float = 0.7,
        parallel: bool = True,
    ) -> pd.DataFrame:
        """
        Reorder columns in the DataFrame to maximize prefix hit rate.
        
        Args:
            df: Input DataFrame to optimize
            early_stop: Early stopping parameter (default: 100000)
            row_stop: Row stopping parameter (default: 4)
            col_stop: Column stopping parameter (default: 2)
            col_merge: List of column groups to merge (columns in each group are merged into one)
            one_way_dep: List of one-way dependencies (not used in this variant)
            distinct_value_threshold: Threshold for distinct values (default: 0.7)
            parallel: Whether to use parallel processing (default: True)
        
        Returns:
            DataFrame with reordered columns (same rows, different column order)
        """
        # Your implementation
        pass
```

**Evaluation Process**:
1. Column merges are applied if specified
2. Your `solve()` method reorders the remaining columns
3. Rows are concatenated (no spaces) and prefix hit rate is calculated

Scoring (0-100)
---

baseline_hit_rate = Average prefix hit rate using original column order (0-point anchor)
avg_hit_rate = Your solution's average prefix hit rate across all datasets

For each dataset:
    dataset_score = ((hit_rate - baseline_hit_rate) / (1.0 - baseline_hit_rate)) × 100

final_score = Average of individual dataset scores

Score is clamped to [0, 100] range


**Runtime Constraint**:
- Average runtime per dataset must be ≤ 10 seconds
- If average runtime exceeds 10 seconds, score = 0.0

**Scoring Examples**:
- baseline_hit_rate = 0.0 (worst), avg_hit_rate = 1.0 (perfect) → Score = 100
- baseline_hit_rate = 0.5, avg_hit_rate = 0.5 → Score = 0
- baseline_hit_rate = 0.5, avg_hit_rate = 0.75 → Score = 50
- baseline_hit_rate = 0.5, avg_hit_rate = 1.0 → Score = 100

Implementation Notes
---
- Row values are concatenated without spaces: `"".join(row.values)`
- Column reordering should optimize for maximum prefix overlap in the concatenated string representation
- Consider column dependencies, distinct value distributions, and merge requirements when reordering
- Large datasets with $M > 10$ columns require efficient algorithms due to larger search space
- In our larger dataset, $50k \leq N \leq 100k$ and $4 \leq M \leq 9$ 

**Example input**
please ignore the $> 10$ column number here
---
```csv
ID,LIMIT_BAL,SEX,EDUCATION,MARRIAGE,AGE,PAY_0,PAY_2,PAY_3,PAY_4,PAY_5,PAY_6,BILL_AMT1,BILL_AMT2,BILL_AMT3,BILL_AMT4,BILL_AMT5,BILL_AMT6,PAY_AMT1,PAY_AMT2,PAY_AMT3,PAY_AMT4,PAY_AMT5,PAY_AMT6,default payment next month
1,20000,2,2,1,24,2,2,-1,-1,-2,-2,3913,3102,689,0,0,0,0,689,0,0,0,0,1
2,120000,2,2,2,26,-1,2,0,0,0,2,2682,1725,2682,3272,3455,3261,0,1000,1000,1000,0,2000,1
3,90000,2,2,2,34,0,0,0,0,0,0,29239,14027,13559,14331,14948,15549,1518,1500,1000,1000,1000,5000,0
4,50000,2,2,1,37,0,0,0,0,0,0,46990,48233,49291,28314,28959,29547,2000,2019,1200,1100,1069,1000,0
5,50000,1,2,1,57,-1,0,-1,0,0,0,8617,5670,35835,20940,19146,19131,2000,36681,10000,9000,689,679,0
...
```

**Example output**
---
```
ID,LIMIT_BAL,SEX,EDUCATION,MARRIAGE,AGE,PAY_0,PAY_2,PAY_3,PAY_4,PAY_5,PAY_6,BILL_AMT1,BILL_AMT2,BILL_AMT3,BILL_AMT4,BILL_AMT5,BILL_AMT6,PAY_AMT1,PAY_AMT2,PAY_AMT3,PAY_AMT4,PAY_AMT5,PAY_AMT6,default payment next month
1,20000,2,2,1,24,2,2,-1,-1,-2,-2,3913,3102,689,0,0,0,0,689,0,0,0,0,1
2,120000,2,2,2,26,-1,2,0,0,0,2,2682,1725,2682,3272,3455,3261,0,1000,1000,1000,0,2000,1
3,90000,2,2,2,34,0,0,0,0,0,0,29239,14027,13559,14331,14948,15549,1518,1500,1000,1000,1000,5000,0
4,50000,2,2,1,37,0,0,0,0,0,0,46990,48233,49291,28314,28959,29547,2000,2019,1200,1100,1069,1000,0
5,50000,1,2,1,57,-1,0,-1,0,0,0,8617,5670,35835,20940,19146,19131,2000,36681,10000,9000,689,679,0
```
($p$ = $1, 2, \ldots M$)
