[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 Solution {
3   	public static void main(String[] args) {
4   		Scanner sc = new Scanner(System.in);
5   		int t = sc.nextInt();
6   		for (int i = 1; i <= t; i++) {
7   			int p = sc.nextInt();
8   			int q = sc.nextInt();
9   			ArrayList<Person> ppl = new ArrayList<Person>(p);
10  			for (int j = 0; j < p; j++) {
11  				int b = sc.nextInt(); 
12  				int a = q - sc.nextInt() - 1;
13  				String c = sc.next();
14  				int d = 0;
15  				int e = 0;
16  				if (c.equals("N")) {
17  					e = -1;
18  				}
19  				else if (c.equals("S")) {
20  					e = 1;
21  				}
22  				else if (c.equals("E")) {
23  					d = 1;
24  				}
25  				else {
26  					d = -1;
27  				}
28  				ppl.add(new Person(a, b, d, e));
29  			}
30  			int[][] b = new int[q][q];
31  			for (Person a : ppl) {
32  				int x = a.x;
33  				int y = a.y;
34  				if (a.xDir == 0) {
35  					for (int j = x+a.yDir; j < q && j >= 0; j += a.yDir) {
36  						for (int k = 0; k < q; k++) {
37  							b[j][k]++;
38  						}
39  					}
40  				} else {
41  					for (int j = 0; j < q; j++) {
42  						for (int k = y+a.xDir; k < q && k >= 0; k += a.xDir) {
43  							b[j][k]++;
44  						}
45  					}
46  				}
47  			}
48  			int max = 0;
49  			for (int j = 0; j < q; j++) {
50  				for (int k = 0; k < q; k++) {
51  					if (b[j][k] > max)
52  						max = b[j][k];
53  				}
54  			}
55  			int row[] = new int[q*q];
56  			int col[] = new int[q*q];
57  			int num = 0;
58  			for (int c = 0; c < q; c++) {
59  				for (int r = 0; r < q; r++) {
60  					if (b[r][c] == max) {
61  						row[num] = q - r - 1;
62  						col[num] = c;
63  						num++;
64  					}
65  				}
66  			}
67  			int min = Integer.MAX_VALUE;
68  			ArrayList<Person> f = new ArrayList<Person>(q*q);
69  			for(int j = 0; j < num; j++) {
70  				if(row[j] + col[j] < min) {
71  					min = row[j] + col[j];
72  					f.clear();
73  					f.add(new Person(row[j],col[j],0,0));
74  				}
75  				else if(row[j] + col[j] == min) {
76  					f.add(new Person(row[j],col[j],0,0));
77  				}
78  			}
79  			ArrayList<Person> g = new ArrayList<Person>(f.size());
80  			int m = Integer.MAX_VALUE;
81  			for(int j = 0; j < f.size(); j++) {
82  				if(m > f.get(j).x) {
83  					m = f.get(j).x;
84  					g.clear();
85  					g.add(f.get(j));
86  				}
87  				else if(m == f.get(j).x) {
88  					g.add(f.get(j));
89  				}
90  			}
91  			Person z = g.get(0);
92  			for(int j = 1; j < g.size(); j++) {
93  				if(g.get(j).y < z.y) {
94  					z = g.get(j);
95  				}
96  			}
97  			System.out.println("Case #" + i + ": " + z.y + " " + z.x);
98  		}
99  	}
100  	public static class Person {
101  		int x;
102  		int y;
103  		int xDir;
104  		int yDir;
105  		public Person(int a, int b, int c, int d) {
106  			x = a;
107  			y = b;
108  			xDir = c;
109  			yDir = d;
110  		}
111  	}
112  }
```


**Question**: Does line `10` have control dependence over line `20` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes