[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.ArrayList;
2   import java.util.Arrays;
3   import java.util.HashMap;
4   import java.util.List;
5   import java.util.Map;
6   import java.util.Scanner;
7   import java.util.Set;
8   public class Solution {
9   	public static void main(String[] args) {
10  		Scanner sc = new Scanner(System.in);
11  		int t = sc.nextInt();
12  		for (int i = 0; i < t; i++) {
13  			solve(sc, i + 1);
14  		}
15  	}
16  	static Map<Integer, Set<Integer>> all;
17  	static int l;
18  	private static void solve(Scanner sc, int num) {
19  		boolean result = false;
20  		int c = sc.nextInt();
21  		int[] b = new int[c];
22  		int[] r = new int[c];
23  		for (int i = 0; i < c; i++) {
24  			b[i] = sc.nextInt();
25  			r[i] = 1;
26  		}
27  		int[] d = new int[c];
28  		List<String> l = new ArrayList<>();
29  		boolean good = false;
30  		String s = "";
31  		while (true) {
32  			result = true;
33  			for (int i = 0; i < c; i++) {
34  				if (b[i] != r[i]) {
35  					result = false;
36  				}
37  			}
38  			if (result) {
39  				break;
40  			}
41  			int cl = 0;
42  			int last = 0;
43  			int[] z = new int[c];
44  			 good = false;
45  			  s = "";
46  			  int[] rr = new int[c];
47  			for (int i = 0; i < c; i++) {
48  				if (b[i] > 0) {
49  					int want = b[i];
50  					int nexttmp = 0;
51  					for (int j = last; j < c ; j++) {
52  						nexttmp = 0;
53  						if (r[j] == 0) {
54  							s += ".";
55  							continue;
56  						}
57  						want-=r[j];
58  						if (j > i && j > 0&& j  < c-1) {
59  							z[j] = -1;
60  							rr[j-1] += r[j];
61  							good = true;
62  							s += "/";
63  						} else 
64  						if (j < i && j > 0&& j  < c-1) {
65  							z[j] = 1;
66  							rr[j+1]+= r[j];
67  							good = true;
68  							s += "\\";
69  						} else {
70  							s += ".";
71  							rr[j]+=r[j];
72  						}
73  						if (want <= 0) {
74  							last = j+1;
75  							break;
76  						}
77  					}
78  				}
79  			}
80  			r = rr;
81  			l.add(s);
82  			if (!good)
83  			break;
84  		}
85  		if (!result) {
86  			System.out.println("Case #" + num + ": IMPOSSIBLE");
87  		} else {
88  			s = "";
89  			for (int i = 0; i < c ; i++) {
90  				s+=".";
91  			}
92  			l.add(s);
93  			System.out.println("Case #" + num + ": " + l.size());
94  			for (String string : l) {
95  				System.out.println(string);
96  			}
97  		}
98  	}
99  	private static String find(Node root, int h) {
100  		if (h == l) {
101  			return null;
102  		}
103  		if (root.map.size() < all.get(h).size()) {
104  			for (Integer ii : all.get(h)) {
105  				if (root.map.get(ii) == null) {
106  					char rr = (char) ('A' + ii);
107  					return Character.toString(rr);
108  				}
109  			}
110  		} else {
111  			for (Map.Entry<Integer, Node> node : root.map.entrySet()) {
112  				String res = find(node.getValue(), h + 1);
113  				if (res != null) {
114  					char rr = (char) ('A' + node.getKey());
115  					return Character.toString(rr) + res;
116  				}
117  			}
118  		}
119  		return null;
120  	}
121  }
122  class Node {
123  	public Map<Integer, Node> map = new HashMap<>();
124  }
```


**Question**: Does `(c,20)` have data dependence over `(r,80)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes