[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control 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  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

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


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

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


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

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

---

[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 line `52` have control dependence over line `70` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes