[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   #include <string.h>
4   #include <math.h>
5   int descending_compare(const void *a, const void *b){
6       if (*(int*)a > *(int*)b){
7           return -1;
8       }else if (*(int*)a == *(int*)b){
9           return 0;
10      }else{
11          return 1;
12      }
13  }
14  int ascending_compare(const void *a, const void *b){
15      if (*(int*)a < *(int*)b){
16          return -1;
17      }else if (*(int*)a == *(int*)b){
18          return 0;
19      }else{
20          return 1;
21      }
22  }
23  int lower_bound(int *a, int n, int key){
24      int left, mid, right;
25      left = 0, right = n;
26      mid = (left + right)/2;
27      while ((left+1 != mid || mid+1 != right) && mid != left){
28          if (key > a[mid]){
29              left = mid;
30          }else{
31              right = mid+1;
32          }
33          mid = (left + right)/2;
34      }
35      if (a[left] >= key)return left;
36      if (a[mid] >= key)return mid;
37      if (a[right] >= key)return right;
38      return n;
39  }
40  unsigned long  gcd(unsigned long x, unsigned long y){
41      if (y == 0){ 
42          return x;
43      }else if (x > y){
44          return gcd(y, x % y);
45      }else{
46          return gcd(x, y % x);
47      }
48  }
49  long long factorial(int x){
50      long long rtn = 1;
51      int i;
52      for (i = x; i > 1; i--){
53          rtn = (rtn*i);
54      }
55      return rtn;
56  }
57  int main(void){
58      int h, w;
59      long long square[3];
60      long long min, max;
61      long long ans = 10000000000000;
62      scanf("%d %d", &h, &w);
63      for (int i = 1; i < h; i++){
64          square[0] = (long long)i * w;
65          square[1] = (long long)(h-i)*(w/2);
66          square[2] = (long long)(h-i)*(w-w/2);
67          min = 10000000000000, max = 0;
68          for (int j = 0; j < 3; j++){
69              if (min > square[j]){
70                  min = square[j];
71              }
72              if (max < square[j]){
73                  max = square[j];
74              }
75          }
76          if (ans > llabs(max-min)){
77              ans = llabs(max-min);
78          }
79          square[1] = (long long)(h-i)/2*w;
80          square[2] = (long long)((h-i)-(h-i)/2)*w;
81          min = 10000000000000, max = 0;
82          for (int j = 0; j < 3; j++){
83              if (min > square[j]){
84                  min = (long long)square[j];
85              }
86              if (max < square[j]){
87                  max = (long long)square[j];
88              }
89          }
90          if (ans > llabs(max-min)){
91              ans = llabs(max-min);
92          }
93      }
94      for (int i = 0; i < w; i++){
95          square[0] = (long long)i*h;
96          square[1] = (long long)(w-i)*(h/2);
97          square[2] = (long long)(w-i)*(h-h/2);
98          min = 10000000000000, max = 0;
99          for (int j = 0; j < 3; j++){
100              if (min > square[j]){
101                  min = (long long)square[j];
102              }
103              if (max < square[j]){
104                  max = (long long)square[j];
105              }
106          }
107          if (ans > llabs(max-min)){
108              ans = llabs(max-min);
109          }
110          square[1] = (long long)(w-i)/2*h;
111          square[2] = (long long)((w-i)-(w-i)/2)*h;
112          min = 10000000000000, max = 0;
113          for (int j = 0; j < 3; j++){
114              if (min > square[j]){
115                  min = (long long)square[j];
116              }
117              if (max < square[j]){
118                  max = (long long)square[j];
119              }
120          }
121          if (ans > llabs(max-min)){
122              ans = llabs(max-min);
123          }
124      }
125      printf("%lld\n", ans);
126      return 0;
127  }
```


**Question**: Does line `69` have control dependence over line `84` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no