[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data Dependence
   All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.

## 4. Example Code Snippet

### Example 1

```python
1    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.util.*;
2   import java.io.*;
3   public class Solution
4   {
5     public static ArrayList<ArrayList<Integer>> table;
6     public static void main(String [] args)
7     {
8       Scanner input = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
9       int t = Integer.parseInt(input.nextLine());
10      for(int iii = 1; iii <= t; iii++)
11      {
12        int n = input.nextInt();
13        int[] aw = new int[n];
14        table = new ArrayList<ArrayList<Integer>>();
15        table.add( new ArrayList<Integer>());
16        for(int i = 0; i < n; i++)
17        {
18          aw[i]=input.nextInt();
19          table.add( new ArrayList<Integer>());
20        }
21        for(int i = 1; i <= n; i++)
22        {
23          table.set(i,recurse(aw,i));
24        }
25        int max = table.get(n).size();
26        for(int i = 0; i < n; i++ )
27        {
28          table = new ArrayList<ArrayList<Integer>>();
29          for(int ii = 0; ii < n; ii++)
30          {
31            table.add( new ArrayList<Integer>());
32          }
33          int[] aww = new int[n-1];
34          for(int ii = 0; ii < i; ii++ )
35          {
36            aww[ii]=aw[ii];
37          }
38          for(int ii = i+1; ii < n; ii++)
39          {
40            aww[ii-1]=aw[ii];
41          }
42          for(int ii = 1; ii <= n-1; ii++)
43          {
44            table.set(ii,recurse(aww,ii));
45          }
46          int tempmax = table.get(n-1).size();
47          if(tempmax>max)
48            max=tempmax;
49        }
50        for(int i = 0; i < n; i++ )
51        {
52          for(int j = i+1; j < n; j++)
53          {
54            table = new ArrayList<ArrayList<Integer>>();
55            for(int ii = 0; ii < n-1; ii++)
56            {
57              table.add( new ArrayList<Integer>());
58            }
59            int[] aww = new int[n-2];
60            for(int ii = 0; ii < i; ii++ )
61            {
62              aww[ii]=aw[ii];
63            }
64            for(int ii = i+1; ii < j; ii++)
65            {
66              aww[ii-1]=aw[ii];
67            }
68            for(int ii = j+1; ii < n; ii++)
69            {
70              aww[ii-2]=aw[ii];
71            }
72            for(int ii = 1; ii <= n-2; ii++)
73            {
74              table.set(ii,recurse(aww,ii));
75            }
76            int tempmax = table.get(n-2).size();
77            if(tempmax>max)
78              max=tempmax;
79          }
80        }
81        System.out.println("Case #"+iii+": "+max);
82      }
83    }
84    public static ArrayList<Integer> recurse(int[] aw, int n)
85    {
86      ArrayList<Integer> resul = new ArrayList<Integer>();
87      if(n==1){resul.add(0);}
88      else
89      {
90        int maxlength = -1;
91        int minweight = 100000000;
92        int goodindex = -1;
93        boolean addon = false;
94        for(int i = 0; i < n; i++)
95        {
96          boolean tempaddon = false;
97          ArrayList<Integer> result = table.get(i);
98          int leng = result.size();
99          if (leng < maxlength) continue;
100          int weight = 0;
101          for(int ii : result)
102          {
103            weight += aw[ii];
104          }
105          if(weight<=6*aw[n-1]){leng ++;weight += aw[n-1];tempaddon=true;}
106          if(maxlength==leng){if(weight<minweight){minweight=weight;goodindex=i;addon=tempaddon;}}else{maxlength=leng; minweight=weight; goodindex=i;addon=tempaddon;}
107        }
108        for(int i : table.get(goodindex))
109        {
110          resul.add(i);
111        }
112        if(addon)resul.add(n-1);
113      }
114      return resul;
115    }
116  }
```


**Question**: Does `(aw,13)` have data dependence over `(aww,70)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes