[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control 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  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": false
}
```


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

**Output**:
```json
{
  "ControlDependence": false
}
```


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

**Output**:
```json
{
  "ControlDependence": false
}
```

---

[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 line `31` have control dependence over line `81` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes