[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   #ifndef _GNU_SOURCE
2   #define _GNU_SOURCE
3   #endif
4   #include <stdio.h>
5   #include <stdlib.h>
6   #include <string.h>
7   #include <limits.h>
8   #include <ctype.h>
9   #include <stdarg.h>
10  #include <math.h>
11  static int dpr (const char *fmt, ...)
12  {
13    int res = 0;
14    va_list ap;
15    va_start (ap, fmt);
16  #if 0
17    res = vprintf (fmt, ap);
18  #endif
19    va_end (ap);
20    return res;
21  }
22  int main ()
23  {
24    int T;
25    scanf ("%d\n", &T);
26    for (int i = 0; i < T; i++)
27      {
28        int N, P;
29        scanf ("%d %d\n", &N, &P);
30        int P_orig = P;
31        int *w = (int *)malloc (sizeof (int) * N);
32        int *h = (int *)malloc (sizeof (int) * N);
33        int *min_add = (int *)malloc (sizeof (int) * N);
34        double *max_add = (double *)malloc (sizeof (double) * N);
35        double *best_min = (double *)malloc (sizeof (double) * N);
36        double *best_max = (double *)malloc (sizeof (double) * N);
37        double total = 0;
38        for (int j = 0; j < N; j++)
39          {
40            scanf ("%d %d\n", w + j, h + j);
41            int circ = (w[j] + h[j]) * 2;
42            P -= circ;
43            total += circ;
44            min_add[j] = 2 * (w[j] < h[j] ? w[j] : h[j]);
45            max_add[j] = 2 * sqrt (w[j] * w[j] + h[j] * h[j]);
46          }
47        dpr ("looking for %d\n", P);
48        double best_valid = 0;
49        for (int j = 0; j < N; j++)
50          {
51            int min1 = 0;
52            int max1 = 0;
53            int k1 = -1;
54            for (int k = 0; k < j; k++)
55              if (best_min[k] + min_add[j] < P && best_max[k] > max1) {
56                k1 = k;
57                min1 = best_min[k];
58                max1 = best_max[k];
59              }
60            best_min[j] = min1 + min_add[j];
61            best_max[j] = max1 + max_add[j];
62            dpr ("cookie %d: using prev %d [%f..%f]\n", j, k1, best_min[j], best_max[j]);
63            if (best_min[j] <= P && best_max[j] > best_valid)
64              best_valid = best_max[j];
65          }
66        if (best_valid > P)
67  	      best_valid = P;
68        printf ("Case #%d: %.6f\n", i + 1, total + best_valid);
69        free (max_add);
70        free (min_add);
71        free (h);
72        free (w);
73      }
74    exit (0);
75  }
```


**Question**: Does line `55` have control dependence over line `51` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no