[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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 `(e,20)` have data dependence over `(a,31)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes