[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**. 

```C
1   #include <stdio.h>
2   #include <stdlib.h>
3   int min(int a, int b){
4   	if(a>b){
5   		return b;
6   	}
7   	return a;
8   }
9   int max(int a, int b){
10  	if(a>b){
11  		return a;
12  	}
13  	return b;
14  }
15  int main(){
16  	int t;
17  	scanf("%d",&t);
18  	for (int m = 1; m <= t; ++m)
19  	{
20  		int n,k;
21  		scanf("%d%d",&n,&k);
22  		int a[n],b[n],l_a[n],l_b[n],r_a[n],r_b[n];
23  		for (int i = 0; i < n; ++i)
24  		{
25  			l_a[i]=l_b[i]=r_a[i]=r_b[i] = i;
26  		}
27  		for (int i = 0; i < n; ++i)
28  		{
29  			scanf("%d",a+i);
30  		}
31  		for (int i = 0; i < n; ++i)
32  		{
33  			scanf("%d",b+i);
34  		}
35  		for (int i = 0; i < n; ++i)
36  		{
37  			int l_temp = i;
38  			while(l_temp>-1){
39  				if(a[l_temp]<=a[i]){
40  					l_temp--;
41  				}
42  				else{
43  					break;
44  				}
45  			}
46  			l_a[i] = l_temp;
47  			l_temp = i;
48  			while(l_temp>-1){
49  				if(b[l_temp]<=b[i]){
50  					l_temp--;
51  				}
52  				else{
53  					break;
54  				}
55  			}
56  			l_b[i] = l_temp;
57  			l_temp = i;
58  			while(l_temp<n){
59  				if(a[l_temp]<=a[i]){
60  					l_temp++;
61  				}
62  				else{
63  					break;
64  				}
65  			}
66  			r_a[i] = l_temp;
67  			l_temp = i;
68  			while(l_temp<n){
69  				if(b[l_temp]<=b[i]){
70  					l_temp++;
71  				}
72  				else{
73  					break;
74  				}
75  			}
76  			r_b[i] = l_temp;
77  		}
78  		long long int tot = 0;
79  		for (int i = 0; i < n; ++i)
80  		{
81  			for (int j = i; j < r_a[i]; ++j)
82  			{
83  				if(l_b[j] < i){
84  					if(a[i]-b[j] <= k && a[i]-b[j] >= -k){
85  						tot += (i-max(l_a[i],l_b[j]))*(min(r_a[i],r_b[j])-j);
86  					}
87  				}
88  			}
89  		}
90  		for (int i = 0; i < n; ++i)
91  		{
92  			for (int j = i+1; j < r_b[i]; ++j)
93  			{
94  				if(l_a[j] < i){
95  					if(a[j]-b[i] <= k && a[j]-b[i] >= -k){
96  						tot += (i-max(l_b[i],l_a[j]))*(min(r_b[i],r_a[j])-j);
97  					}
98  				}
99  			}
100  		}
101  		printf("Case #%d: %lld\n", m,tot);
102  	}
103  }
```


**Question**: Does line `31` have control dependence over line `66` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no