[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.io.*;
2   import java.lang.*;
3   import java.util.*;
4   public class Solution{
5       static class Pair implements Comparable<Pair>{
6           long value;
7           int count;
8           public Pair(long value,int count){
9               this.value = value;
10              this.count = count;
11          }
12          @Override
13          public int compareTo(Pair o){
14              return o.count - this.count;
15          }
16      }
17      public static void main(String[] args)throws IOException{
18          BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
19          int t = new Integer(br.readLine());
20          StringBuffer sb = new StringBuffer();
21          for(int z=1;z<=t;z++){
22              sb.append("Case #"+z+": ");
23              StringTokenizer st = new StringTokenizer(br.readLine());
24              StringTokenizer st1 = new StringTokenizer(br.readLine());
25              int n = new Integer(st.nextToken());
26              int d = new Integer(st.nextToken());
27              if(n==1&&d==3){
28                  sb.append("2\n");
29                  continue;
30              }
31              else if(n==1&&d==2){
32                  sb.append("1\n");
33                  continue;
34              }
35              long[] array = new long[n];
36              for(int i=0;i<n;i++){
37                  array[i]=new Long(st1.nextToken());
38              }
39              Arrays.sort(array);
40              Pair[] ps = new Pair[n];
41              int w=0;
42              int count = 1;
43              for(int i=1;i<n;i++){
44                  if(array[i]==array[i-1]){
45                      count++;
46                  }
47                  else{
48                      ps[w++]=new Pair(array[i-1],count);
49                      count=1;
50                  }
51              }
52              ps[w++]=new Pair(array[n-1],count);
53              Arrays.sort(ps,0,w);
54              if(ps[0].count>=d){
55                  sb.append("0\n");
56                  continue;
57              }
58              else{
59                  if(d==2&&ps[0].count==1){
60                      sb.append("1\n");
61                      continue;
62                  }
63                  else if(d==3&&ps[0].count==2&&ps[0].value!=array[n-1]){
64                      sb.append("1\n");
65                      continue;
66                  }
67                  else if(d==3&&ps[0].count==2&&ps[0].value==array[n-1]){
68                      if(ps[1].count==2){
69                          sb.append("1\n");
70                          continue;
71                      }
72                      boolean flag = false;
73                      outer:for(int i=0;i<n;i++){
74                          for(int j=0;j<n;j++){
75                              if(array[j]==2*array[i]||array[i]==2*array[j]){
76                                  sb.append("1\n");
77                                  flag = true;
78                                  break outer;
79                              }
80                          }
81                      }
82                      if(!flag){
83                          sb.append("2\n");
84                          continue;
85                      }
86                  }
87                  else if(d==3&&ps[0].count==1){
88                      boolean flag = false;
89                      outer:for(int i=0;i<n;i++){
90                          for(int j=0;j<n;j++){
91                              if(array[j]==2*array[i]||array[i]==2*array[j]){
92                                  sb.append("1\n");
93                                  flag = true;
94                                  break outer;
95                              }
96                          }
97                      }
98                      if(!flag){
99                          sb.append("2\n");
100                          continue;
101                      }
102                  }
103              }
104          }
105          br.close();
106          System.out.println(sb);
107      }
108  }
```


**Question**: Does line `31` have control dependence over line `45` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes