[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   import java.io.*;
3   public class Solution {
4       public static void main(String[] args) {
5           int[] dx = { 0, 1, 0, -1 };
6           int[] dy = { 1, 0, -1, 0 };
7           Scanner sc = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
8           StringBuilder ans = new StringBuilder();
9           int T = sc.nextInt();
10          for (int t = 1; t <= T; t++) {
11              int X = sc.nextInt();
12              int Y = sc.nextInt();
13              String M = sc.next();
14              int[][] g = new int[2001][2001];
15              int px = 1000;
16              int py = 1000;
17              List<List<Integer>> tour = new ArrayList<>();
18              List<Integer> te = new ArrayList<>();
19              te.add(px);
20              te.add(py);
21              tour.add(te);
22              g[py][px] = 1;
23              for (int i = 0; i < M.length(); i++) {
24                  char d = M.charAt(i);
25                  if (d == 'N') 
26                      py--;
27                  if (d == 'S') 
28                      py++;
29                  if (d == 'E') 
30                      px++;
31                  if (d == 'W') 
32                      px--;
33                  g[py][px] = 1;
34                  te = new ArrayList<>();
35                  te.add(px);
36                  te.add(py);
37                  tour.add(te);
38              }
39              boolean[][] visited = new boolean[2001][2001];
40              int myx = 1000-X;
41              int myy = 1000+Y;
42              LinkedList<Pair> q = new LinkedList<>();
43              q.offer(new Pair(myx,myy,0));
44              int tmp = -1;
45              int nearX = -1;
46              int nearY = -1;
47              while (!q.isEmpty()) {
48                  Pair now = q.poll();
49                  if (g[now.y][now.x] == 1) {
50                      tmp = now.dist;
51                      nearX = now.x;
52                      nearY = now.y;
53                      break;
54                  }
55                  visited[now.y][now.x] = true;;
56                  for (int v = 0; v < 4; v++) {
57                      int nx = now.x+dx[v];
58                      int ny = now.y+dy[v];
59                      if (nx < 0 || visited[0].length <= nx || ny < 0 || visited.length <= ny) 
60                              continue;
61                      if (visited[ny][nx]) 
62                          continue;
63                      q.add(new Pair(nx,ny,now.dist+1));
64                  }
65              }
66              ans.append("Case #"+t+": "); 
67              if (tmp == -1) {
68                  ans.append("IMPOSSIBLE\n");
69                  continue;
70              }
71              int ti = 0;
72              for (int i = 0; i < tour.size(); i++) {
73                  List<Integer> ll = tour.get(i);
74                  if (ll.get(0) == nearX && ll.get(1) == nearY) {
75                      ti = i;
76                      break;
77                  }
78              }
79              boolean[][] visited2 = new boolean[2001][2001];
80              if (tmp <= ti) {
81                  ans.append(tmp+(ti-tmp+1)/2).append('\n');
82              } else {
83                  boolean success = false;
84                  for (int i = ti; i < tour.size() && tmp-ti+i < tour.size(); i++) {
85                      int mfx = tour.get(tmp+i-ti).get(0);
86                      int mfy = tour.get(tmp+i-ti).get(1);
87                      visited2[mfy][mfx] = true;
88                      int ffx = tour.get(i).get(0);
89                      int ffy = tour.get(i).get(1);
90                      if (visited2[ffy][ffx]) {
91                          ans.append(i).append('\n');
92                          success = true;
93                          break;
94                      }
95                  }
96                  if (!success) ans.append("IMPOSSIBLE\n");
97              }
98          }
99          System.out.print(ans);
100      }
101      private static class Pair {
102          private final int x;
103          private final int y;
104          private final int dist;
105          Pair(int x, int y, int dist) {
106              this.x = x;
107              this.y = y;
108              this.dist = dist;
109          }
110      }
111  }
```


**Question**: Does `(v,56)` have data dependence over `(q,63)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes