[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**. 

```C
1   #include <stdio.h>
2   #include <stdlib.h>
3   #include <string.h>
4   #include <stdarg.h>
5   #include <math.h>
6   char* readline() {
7   	size_t bufLen = 1024, dataLen = 0;
8   	char* data = malloc(bufLen);
9   	while (1) {
10  		char* cursor = data + dataLen;
11  		char* line = fgets(cursor, bufLen - dataLen, stdin);
12  		if (!line) { break; }
13  		dataLen += strlen(cursor);
14  		if (dataLen < bufLen - 1 || data[dataLen - 1] == '\n') { break; }
15  		size_t new_length = bufLen << 1;
16  		data = realloc(data, new_length);
17  		if (!data) { break; }
18  		bufLen = new_length;
19  	}
20  	if (data[dataLen - 1] == '\n') {
21  		data[dataLen - 1] = '\0';
22  	}
23  	else
24  	{
25  		dataLen++;
26  		data[dataLen - 1] = '\0';
27  	}
28  	data = realloc(data, dataLen);
29  	return data;
30  }
31  int G[1000][1000];
32  int targetX, targetY;
33  void main(int argc, char **argv)
34  {
35  	unsigned int u_w;
36  	int T;
37  	char *str;
38  	char *retStr;
39  	str = readline();
40  	sscanf(str, "%d", &T);
41  	free(str);
42  	u_w = 1;
43  	while (T-- != 0)
44  	{
45  		int R, C, H, V;
46  		int i, j, k, l, m;
47  		char *Waffle[10];
48  		int ChocChipCount[10];
49  		for (i = 0; i < 10; i++)
50  			ChocChipCount[i] = 0;
51  		str = readline();
52  		sscanf(str, "%d %d %d %d", &R, &C, &H, &V);
53  		free(str);
54  		for (i = 0; i < R; i++)
55  		{
56  			Waffle[i] = readline();
57  			for (j = 0; j < C; j++)
58  			{
59  				if (Waffle[i][j] == '@')
60  				{
61  					ChocChipCount[i]++;
62  				}
63  			}
64  		}
65  		retStr = "IMPOSSIBLE";
66  		for (i = 1; (i < R && (strcmp(retStr, "IMPOSSIBLE") == 0)); i++)
67  		{
68  			int topCount, bottomCount;
69  			int topleftCount, toprightCount;
70  			int bottomleftCount, bottomrightCount;
71  			topCount = 0;
72  			bottomCount = 0;
73  			for (j = 0; j < i; j++)
74  				topCount += ChocChipCount[j];
75  			for (j = i; j < R; j++)
76  				bottomCount += ChocChipCount[j];
77  			if (topCount != bottomCount)
78  				continue;
79  			else
80  			{
81  				for (k = 1; k < C; k++)
82  				{
83  					topleftCount = 0;
84  					toprightCount = 0;
85  					for (l = 0; l < i; l++)
86  					{
87  						for (m = 0; m < k; m++)
88  							if (Waffle[l][m] == '@')
89  								topleftCount++;
90  						toprightCount += ChocChipCount[l];
91  					}
92  					toprightCount -= topleftCount;
93  					if (topleftCount == toprightCount)
94  					{
95  						bottomleftCount = 0;
96  						bottomrightCount = 0;
97  						for (l = i; l < R; l++)
98  						{
99  							for (m = 0; m < k; m++)
100  								if (Waffle[l][m] == '@')
101  									bottomleftCount++;
102  							bottomrightCount += ChocChipCount[l];
103  						}
104  						bottomrightCount -= bottomleftCount;
105  						if (bottomleftCount == bottomrightCount)
106  						{
107  							retStr = "POSSIBLE";
108  							break;
109  						}
110  					}
111  				}
112  			}
113  		}
114  		printf("Case #%d: %s", u_w, retStr);
115  		u_w++;
116  	}
117  }
```


**Question**: Does `(k,81)` have data dependence over `(bottomrightCount,102)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no