[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**. 

```Java
1   import java.util.*;
2   public class Main {
3   	public static boolean bfs (int sx,int sy, int[] spx, int[] spy) {
4   		int size = 10, spCnt = 0;
5   		int[] dx = {-1,0,1,-1,0,1,-2,-2,-2,2,2,2};
6   		int[] dy = {-2,-2,-2,2,2,2,-1,0,1,-1,0,1};
7   		int[] dx2 = {0,0,1,1,1,0,-1,-1,-1};
8   		int[] dy2 = {0,-1,-1,0,1,1,1,0,-1,-1};
9   		boolean[][] park = new boolean[size][size];
10  		Queue<Integer> queueX = new LinkedList<Integer>();
11  		Queue<Integer> queueY = new LinkedList<Integer>();
12  		queueX.add(sx);
13  		queueY.add(sy);
14  		while (!queueX.isEmpty()) {
15  			if (spCnt == spx.length) {
16  				return true;
17  			}
18  			for (int i = 0; i < dx2.length; i++) {
19  				int y = spy[spCnt] + dy2[i];
20  				int x = spx[spCnt] + dx2[i];
21  				if (0 <= y && y < size &&
22  					0 <= x && x < size) {
23  					park[y][x] = true;
24  				}
25  			}
26  			Queue<Integer> nx = new LinkedList<Integer>();
27  			Queue<Integer> ny = new LinkedList<Integer>();
28  			while (!queueX.isEmpty()) {
29  				int x = queueX.poll();
30  				int y = queueY.poll();
31  				for (int i = 0; i < dx.length; i++) {
32  					int nextX = x + dx[i];
33  					int nextY = y + dy[i];
34  					if (0 <= nextX && nextX < 10 &&
35  						0 <= nextY && nextY < 10 &&
36  						park[nextY][nextX]) {
37  						nx.add(nextX);
38  						ny.add(nextY);
39  					}
40  				}
41  			}
42  			spCnt++;
43  			park = new boolean[size][size];
44  			queueX = nx;
45  			queueY = ny;
46  		}
47  		return false;
48  	}
49  	public static void main(String[] args) {
50  		Scanner sc = new Scanner(System.in);
51  		while (true) {
52  			int sx = sc.nextInt();
53  			int sy = sc.nextInt();
54  			if (sx == 0 && sy == 0) break;
55  			sc.nextLine();
56  			int n = sc.nextInt();
57  			int[] spx = new int[n];
58  			int[] spy = new int[n];
59  			sc.nextLine();
60  			for (int i = 0; i < n; i++) {
61  				spx[i] = sc.nextInt();
62  				spy[i] = sc.nextInt();
63  			}
64  			sc.nextLine();
65  			System.out.println(bfs(sx,sy,spx,spy) ? "OK" : "NA");
66  		}
67  	}
68  }
```


**Question**: Does line `31` have control dependence over line `42` in function `bfs`? If so, provide a trace.

**Output**:


Ground truth answer: no