[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 sys
2   import numpy as np
3   def eprint(*args, **kwargs):
4       print(*args, file=sys.stderr, **kwargs)
5   def main():
6       T = int(raw_input())     
7       for i in xrange(1, T + 1):
8           tmp = raw_input()
9           tmpList = map(int, tmp.split())
10          N = int(tmpList[0])                          
11          K = int(tmpList[1])                          
12          stall_occupied = [True]                 
13          for j in xrange(N):
14              stall_occupied.append(False)
15          stall_occupied.append(True)
16          Ls = []
17          Rs = []
18          Min_s = []
19          Max_s = []
20          for j in xrange(N+2):
21              if stall_occupied[j]:
22                  Ls.append(-1)
23                  Rs.append(-1)
24              else:
25                  Ls.append(j - 1)
26                  Rs.append(N - j)
27              Min_s.append(min(Ls[-1], Rs[-1]))
28              Max_s.append(max(Ls[-1], Rs[-1]))
29          for k in xrange(K):
30              maximal = max(Min_s)
31              if Min_s.count(maximal) > 1:
32                  values = np.array(Min_s)
33                  idxs = np.where(values == maximal)[0]
34                  max_of_idxs = []
35                  for j in idxs:
36                      max_of_idxs.append(Max_s[j])
37                  max_max = max(max_of_idxs)
38                  if(max_of_idxs.count(max_max) > 1):
39                      values1 = np.array(max_of_idxs)
40                      idxs1 = np.where(values1 == max_max)[0]
41                      idx = idxs[idxs1[0]]
42                  else:
43                      idx = idxs[max_of_idxs.index(max_max)]
44                  y =  max(Ls[idx], Rs[idx])
45                  z =  min(Ls[idx], Rs[idx])
46                  stall_occupied[idx] = True
47                  Ls[idx] = -1
48                  Rs[idx] = -1
49                  Min_s[idx] = -1
50                  Max_s[idx] = -1
51                  leftIdx = idx - 1
52                  while leftIdx > 0:
53                      if stall_occupied[leftIdx]:
54                          break
55                      Rs[leftIdx] = idx - leftIdx - 1
56                      Min_s[leftIdx] = min(Rs[leftIdx], Ls[leftIdx])
57                      Max_s[leftIdx] = max(Rs[leftIdx], Ls[leftIdx])
58                      leftIdx -= 1
59                  rightIdx = idx + 1
60                  while rightIdx < N + 2:
61                      if stall_occupied[rightIdx]:
62                          break
63                      Ls[rightIdx] = rightIdx - idx - 1
64                      Min_s[rightIdx] = min(Rs[rightIdx], Ls[rightIdx])
65                      Max_s[rightIdx] = max(Rs[rightIdx], Ls[rightIdx])
66                      rightIdx += 1
67              else:
68                  idx = Min_s.index(maximal)
69                  y =  max(Ls[idx], Rs[idx])
70                  z =  min(Ls[idx], Rs[idx])
71                  stall_occupied[idx] = True
72                  Ls[idx] = -1
73                  Rs[idx] = -1
74                  Min_s[idx] = -1
75                  Max_s[idx] = -1
76                  leftIdx = idx - 1
77                  while leftIdx > 0:
78                      if stall_occupied[leftIdx]:
79                          break
80                      Rs[leftIdx] = idx - leftIdx - 1
81                      Min_s[leftIdx] = min(Rs[leftIdx], Ls[leftIdx])
82                      Max_s[leftIdx] = max(Rs[leftIdx], Ls[leftIdx])
83                      leftIdx -= 1
84                  rightIdx = idx + 1
85                  while rightIdx < N + 2:
86                      if stall_occupied[rightIdx]:
87                          break
88                      Ls[rightIdx] = rightIdx - idx - 1
89                      Min_s[rightIdx] = min(Rs[rightIdx], Ls[rightIdx])
90                      Max_s[rightIdx] = max(Rs[rightIdx], Ls[rightIdx])
91                      rightIdx += 1
92          print("Case #{}: {} {}".format(i, y, z))
93          sys.stdout.flush()
94  if __name__ == "__main__":
95      main()
```


**Question**: Does `(Max_s,82)` have data dependence over `(Min_s,81)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes