[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   #define inf 2e30+1
3   typedef long long ll;
4   int compare(const void *a, const void *b) {
5     return *(int*)a - *(int*)b;
6   }
7   int main(void) {
8     int i, j, k, n, m;
9     scanf("%d%d", &n, &m);
10    int x, y, uni[n], a[n];
11    for(i = 0; i < n; ++i) 
12      scanf("%d", &a[i]);
13    for(i = 0; i < n; ++i) 
14      uni[i] = i;
15    for(i = 0; i < m; ++i) {
16      scanf("%d%d", &x, &y);
17      int t;
18      if(uni[x] != x) {
19        t = uni[y];
20        uni[y] = x;
21        for(j = y; uni[j] != j; ) {
22          j = uni[j];
23        }
24        uni[y] = j;
25        for(j = 0; j < n; ++j) {
26          if(j == y) 
27            continue;
28          if(uni[j] == t) 
29            uni[j] = uni[y];
30        }
31      } else {
32        t = uni[x];
33        uni[x] = y;
34        for(j = x; uni[j] != j; ) {
35          j = uni[j];
36        }
37        uni[x] = j;
38        for(j = 0; j < n; ++j) {
39          if(j == x) 
40            continue;
41          if(uni[j] == t) 
42            uni[j] = uni[x];
43        }
44      }
45    }
46    int num = 0;
47    for(i = 0; i < n; ++i) {
48      if(uni[i] == i) 
49        ++num;
50    }
51    if(num == 1) {
52      printf("0");
53      return 0;
54    }
55    if(num * 2 - 2 > n) {
56      printf("Impossible");
57      return 0;
58    }
59    if(num * 2 - 2 == n) {
60      int t = 0;
61      for(i = 0; i < n; ++i) 
62        t += a[i];
63      printf("%d", t);
64      return 0;
65    }
66    int min[n], minn[n];
67    for(i = 0; i < n; ++i) 
68      min[i] = inf;
69    for(i = 0; i < n; ++i) 
70      if(min[uni[i]] > a[i]) {
71        minn[uni[i]] = i;
72        min[uni[i]] = a[i];
73      }
74    ll ssum = 0;
75    int tnum = num - 2;
76    for(i = 0; i < n; ++i) {
77      int y = inf;
78      if(min[i] != y) {
79        ssum += (ll)min[i];
80        a[minn[i]] = inf;
81      }
82    }
83    qsort(a, n, sizeof(int), compare);
84    for(i = 0; i < tnum; ++i) {
85      ssum += a[i];
86    }
87    printf("%lld", ssum);
88    return 0;
89  }
```


**Question**: Does line `51` have control dependence over line `67` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes