[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   int max(int a, int b){
3   	if(a>b){
4   		return a;
5   	}
6   	return b;
7   }
8   int min(int a, int b){
9   	if(a>b){
10  		return b;
11  	}
12  	return a;
13  }
14  int main(){
15  	int H;
16  	scanf("%d",&H);
17  	int W;
18  	scanf("%d",&W);
19  	char c;
20  	char d;
21  	char l[1001][1001];
22  	int i;
23  	int j;
24  	for(i=1;i<=H;i++){
25  		scanf("%c",&d);
26  		for(j=1;j<=W;j++){
27  			scanf("%c",&c);
28  			l[i][j]=c;
29  		}
30  	}
31  	int m;
32  	m=H+W;
33  	int p[1001][1001];
34  	int q[1002][1001];
35  	int r[1001][1002];
36  	int s[1002][1002];
37  	for(i=1;i<=H;i++){
38  		p[i][0]=m;
39  		q[i][W+1]=m;
40  		r[i][0]=m;
41  		s[i][W+1]=m;
42  	}
43  	for(j=1;j<=W;j++){
44  		p[0][j]=m;
45  		q[0][j]=m;
46  		r[H+1][j]=m;
47  		s[H+1][j]=m;
48  	}
49  	for(i=1;i<=H;i++){
50  		for(j=1;j<=W;j++){
51  			if(l[i][j]=='#'){
52  				p[i][j]=0;
53  			}else{
54  				p[i][j]=min(p[i-1][j],p[i][j-1])+1;
55  			}
56  		}
57  	}
58  	for(i=1;i<=H;i++){
59  		for(j=W;j>=1;j--){
60  			if(l[i][j]=='#'){
61  				q[i][j]=0;
62  			}else{
63  				q[i][j]=min(q[i-1][j],q[i][j+1])+1;
64  			}
65  		}
66  	}
67  	for(i=H;i>=1;i--){
68  		for(j=1;j<=W;j++){
69  			if(l[i][j]=='#'){
70  				r[i][j]=0;
71  			}else{
72  				r[i][j]=min(r[i+1][j],r[i][j-1])+1;
73  			}
74  		}
75  	}
76  	for(i=H;i>=1;i--){
77  		for(j=W;j>=1;j--){
78  			if(l[i][j]=='#'){
79  				s[i][j]=0;
80  			}else{
81  				s[i][j]=min(s[i+1][j],s[i][j+1])+1;
82  			}
83  		}
84  	}
85  	int n[1001][1001];
86  	int ans;
87  	ans=0;
88  	for(i=1;i<=H;i++){
89  		for(j=1;j<=W;j++){
90  			n[i][j]=min(p[i][j],q[i][j]);
91  			n[i][j]=min(n[i][j],r[i][j]);
92  			n[i][j]=min(n[i][j],s[i][j]);
93  			ans=max(ans,n[i][j]);
94  		}
95  	}
96  	printf("%d",ans);
97  }
```


**Question**: Does line `51` have control dependence over line `63` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no