[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 <math.h>
3   #include <string.h>
4   #include <stdlib.h>
5   typedef struct
6   {
7   	int x;
8   	int y;
9   } par;
10  double get_double();
11  int get_int();
12  int max(int, int);
13  int min(int, int);
14  int main()
15  {
16  	int i, j, k;
17  	int t = get_int();
18  	for (i = 0; i < t; i++)
19  	{
20  		int n = get_int();
21  		int w[n];
22  		int c[n][n];
23  		memset(c, 0, sizeof(c));
24  		for (j = 0; j < n; j++)
25  		{
26  			w[j] = get_int();
27  		}
28  		c[0][0] = w[0];
29  		for (j = 1; j < n; j++)
30  		{
31  			c[0][j] = min(w[j], c[0][j - 1]);
32  		}
33  		for (j = 1; j < n; j++)
34  		{
35  			if (c[j - 1][j - 1] == -1)
36  			{
37  				c[j][j] = -1;
38  			}
39  			else
40  			{
41  				if (c[j - 1][j - 1] > w[j]*6)
42  				{
43  					c[j][j] = -1;
44  				}
45  				else
46  				{
47  					c[j][j] = c[j - 1][j - 1] + w[j];
48  				}
49  			}
50  		}
51  		for (j = 1; j < n; j++)
52  		{
53  			for (k = j + 1; k < n; k++)
54  			{
55  				if (c[j - 1][k - 1] == -1 && c[j][k - 1] == -1)
56  				{
57  					c[j][k] = -1;
58  				}
59  				else if (c[j - 1][k - 1] == -1)
60  				{
61  					if (c[j][k - 1] > w[k]*6)
62  					{
63  						c[j][k] == -1;
64  					}
65  					else
66  					{
67  						c[j][k] = c[j][k - 1] + w[k];
68  					}
69  				}
70  				else if(c[j][k - 1] == -1)
71  				{
72  					if (c[j - 1][k - 1] > w[k]*6)
73  					{
74  						c[j][k] == -1;
75  					}
76  					else
77  					{
78  						c[j][k] = c[j - 1][k - 1] + w[k];
79  					}
80  				}
81  				else
82  				{
83  					int tmp = min(c[j - 1][k - 1], c[j][k - 1]);
84  					if (tmp > w[k]*6)
85  					{
86  						c[j][k] = -1;
87  					}
88  					else
89  					{
90  						c[j][k] = tmp + w[k];
91  					}
92  				}
93  			}
94  		}
95  		for (j = 0; j < n; j++)
96  		{
97  			if (c[j][n - 1] == -1)
98  			{
99  				break;
100  			}
101  		}
102  		printf("Case #%d: %d\n", i + 1, j);
103  	}
104  }
105  int max(int a, int b)
106  {
107  	if (a > b)
108  	{
109  		return a;
110  	}
111  	return b;
112  }
113  int min(int a, int b)
114  {
115  	if (a < b)
116  	{
117  		return a;
118  	}
119  	return b;
120  }
121  double get_double()
122  {
123  	double ret = 0;
124  	char c     = getchar();
125  	int sgn;
126  	while (1)
127  	{
128  		if (c == EOF)
129  		{
130  			return EOF;
131  		}
132  		if (c >= '0' && c <= '9')
133  		{
134  			sgn = 1;
135  			break;
136  		}
137  		if (c == '-')
138  		{
139  			c = getchar();
140  			if (c < '0' || c > '9')
141  			{
142  				continue;
143  			}
144  			sgn = -1;
145  			break;
146  		}
147  		c = getchar();
148  	}
149  	while (1)
150  	{
151  		ret = ret*10 + c - '0';
152  		c = getchar();
153  		if (c == '.')
154  		{
155  			double pos = 0.1;
156  			c = getchar();
157  			while (c >= '0' && c <= '9')
158  			{
159  				ret += pos*(c - '0');
160  				pos /= 10;
161  				c = getchar();
162  			}
163  			return sgn*ret;
164  		}
165  		else if (c < '0' || c > '9')
166  		{
167  			return sgn*ret;
168  		}
169  	}
170  }
171  int get_int()
172  {
173  	int ret = 0;
174  	char c  = getchar();
175  	int sgn;
176  	while (1)
177  	{
178  		if (c == EOF)
179  		{
180  			return EOF;
181  		}
182  		if (c >= '0' && c <= '9')
183  		{
184  			sgn = 1;
185  			break;
186  		}
187  		if (c == '-')
188  		{
189  			c = getchar();
190  			if (c < '0' || c > '9')
191  			{
192  				continue;
193  			}
194  			sgn = -1;
195  			break;
196  		}
197  		c = getchar();
198  	}
199  	while (1)
200  	{
201  		ret = (ret << 3) + (ret << 1) + c - '0';
202  		c = getchar();
203  		if (c < '0' || c > '9')
204  		{
205  			return sgn*ret;
206  		}
207  	}
208  }
```


**Question**: Does `(c,124)` have data dependence over `(ret,159)` in function `get_double`? If so, provide a trace.

**Output**:


Ground truth answer: yes