[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**. 

```Python
1   import random
2   def main():
3   	T=eval(input())
4   	for i in range (T):
5   		N=eval(input())
6   		s=input()
7   		print("Case #"+str(i+1)+":", end=' ')
8   		ownWay(s,N)
9   def ownWay(s,n):
10  	maze=[]
11  	for i in range (n):
12  		maze.append([])
13  		for j in range (n):
14  			maze[i].append([])
15  			for k in range (2):
16  				if k==0:
17  					maze[i][j].append(i)
18  				else:
19  					maze[i][j].append(j)
20  	l=list(s)
21  	c_l=[]    
22  	for i in range (1):
23  		c_l.append([])
24  		for j in range(2*n-1):
25  			c_l[i].append([])
26  			for k in range (2):
27  				c_l[i][j].append(0)
28  	x=0
29  	y=0
30  	c_l[0][0][0]=maze[x][y][0]
31  	c_l[0][0][1]=maze[x][y][1]
32  	for i in range(2*n-2):
33  		if l[i]=='S':
34  			y=y+1
35  			c_l[0][i+1][1]=maze[x][y][1]
36  			c_l[0][i+1][0]=maze[x][y][0]
37  		if l[i]=='E':
38  			x=x+1
39  			c_l[0][i+1][0]=maze[x][y][0]
40  			c_l[0][i+1][1]=maze[x][y][1]       
41  	check=True
42  	random.shuffle(l)
43  	c_l1=[]
44  	for i in range (1):
45  		c_l1.append([])
46  		for j in range(2*n-1):
47  			c_l1[i].append([])
48  			for k in range (2):
49  				c_l1[i][j].append(0)
50  	x1=0
51  	y1=0
52  	c_l1[0][0][0]=maze[x1][y1][0]
53  	c_l1[0][0][1]=maze[x1][y1][1]
54  	for i in range(2*n-2):
55  		if l[i]=='S':
56  			y1=y1+1
57  			c_l1[0][i+1][1]=maze[x1][y1][1]
58  			c_l1[0][i+1][0]=maze[x1][y1][0]
59  		if l[i]=='E':
60  			x1=x1+1
61  			c_l1[0][i+1][0]=maze[x1][y1][0]
62  			c_l1[0][i+1][1]=maze[x1][y1][1]
63  	while check:
64  		check=False
65  		for i in range(2*n-2):
66  			if (c_l[0][i][0]==c_l1[0][i][0]) and (c_l[0][i][1]==c_l1[0][i][1]) and (c_l[0][i+1][0]==c_l1[0][i+1][0]) \
67  				and (c_l[0][i+1][1]==c_l1[0][i+1][1]):
68  				check=True
69  		if check:
70  			random.shuffle(l)
71  			c_l1=[]
72  			for i in range (1):
73  				c_l1.append([])
74  				for j in range(2*n-1):
75  					c_l1[i].append([])
76  					for k in range (2):
77  						c_l1[i][j].append(0)
78  			x1=0
79  			y1=0
80  			c_l1[0][0][0]=maze[x1][y1][0]
81  			c_l1[0][0][1]=maze[x1][y1][1]
82  			for i in range(2*n-2):
83  				if l[i]=='S':
84  					y1=y1+1
85  					c_l1[0][i+1][1]=maze[x1][y1][1]
86  					c_l1[0][i+1][0]=maze[x1][y1][0]
87  				if l[i]=='E':
88  					x1=x1+1
89  					c_l1[0][i+1][0]=maze[x1][y1][0]
90  					c_l1[0][i+1][1]=maze[x1][y1][1]
91  	print(''.join(l))
92  main()
```


**Question**: Does `(s,9)` have data dependence over `(l,20)` in function `ownWay`? If so, provide a trace.

**Output**:


Ground truth answer: yes