[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   from collections import defaultdict
2   from collections import deque
3   from heapq import heappush, heappop
4   import sys
5   import math
6   import bisect
7   import random
8   def LI(): return list(map(int, sys.stdin.readline().split()))
9   def I(): return int(sys.stdin.readline())
10  def LS():return list(map(list, sys.stdin.readline().split()))
11  def S(): return list(sys.stdin.readline())[:-1]
12  def IR(n):
13      l = [None for i in range(n)]
14      for i in range(n):l[i] = I()
15      return l
16  def LIR(n):
17      l = [None for i in range(n)]
18      for i in range(n):l[i] = LI()
19      return l
20  def SR(n):
21      l = [None for i in range(n)]
22      for i in range(n):l[i] = S()
23      return l
24  def LSR(n):
25      l = [None for i in range(n)]
26      for i in range(n):l[i] = SR()
27      return l
28  mod = 1000000007
29  sys.setrecursionlimit(1000000)
30  def A():
31      n = I()
32      b = LI()
33      ans = []
34      d = max(b)
35      i = 0
36      while b:
37          j = len(b)-1
38          while j >= 0:
39              if b[j] > j+1:
40                  print(-1)
41                  quit()
42              elif b[j] == j+1:
43                  ans.append(b[j])
44                  b.pop(j)
45                  break
46              j -= 1
47      for i in ans[::-1]:
48          print(i)
49  def B():
50      n = I()
51      if n%2:
52          ans = []
53          for i in range(1,n+1):
54              for j in range(1,n+1):
55                  if j != n-i and j != i:
56                      ans.append((min(i,j),max(i,j)))
57          ans = list(set(ans))
58          print(len(ans))
59          ans.sort()
60          for i in ans:
61              print(*i)
62      else:
63          ans = []
64          for i in range(1,n+1):
65              for j in range(1,n+1):
66                  if j != n+1-i and j != i:
67                      ans.append((min(i,j),max(i,j)))
68          ans = list(set(ans))
69          print(len(ans))
70          ans.sort()
71          for i in ans:
72              print(*i)
73  def C():
74      return
75  def D():
76      n,a,b = LI()
77      pf = LI()
78      f_ans = float("inf")
79      l = []
80      for i in range(n):
81          l.append(pf[i])
82      l.sort()
83      f = defaultdict(int)
84      for i in pf:
85          f[i] = l.index(i)
86      for c in range(2):
87          p = pf.copy()
88          ans = 0
89          while l != p:
90              d = []
91              i = 0
92              while i < n:
93                  j = i
94                  while j < n-1:
95                      if f[p[j+1]] != f[p[j]]+1:
96                          break
97                      j += 1
98                  d.append(p[i:j+1])
99                  i = j+1
100              m = len(d)
101              ld = []
102              for i in range(m):
103                  ld.append(d[i])
104              ld.sort()
105              fd = defaultdict(int)
106              for i in d:
107                  fd[str(i)] = ld.index(i)
108              if c == 0:
109                  for i in range(m)[::-1]:
110                      if d[i] != ld[i]:
111                          j = fd[str(d[i])]
112                          x = d[i]
113                          if j > i:
114                              k = 0
115                              for o in range(i+1,j+1):
116                                  k += len(d[o])
117                              p = len(d[i])
118                              ans += min(b*k,a*p)
119                              d.pop(i)
120                              d.insert(j,x)
121                              break
122                          else:
123                              k = 0
124                              for o in range(j,i):
125                                  k += len(d[o])
126                              p = len(d[i])
127                              ans += min(a*k,b*p)
128                              d.pop(i)
129                              d.insert(j,x)
130                              break
131              else:
132                  for i in range(m)[::-1]:
133                      if d[i] != ld[i]:
134                          j = fd[str(d[i])]
135                          x = d[i]
136                          if j > i:
137                              k = 0
138                              for o in range(i+1,j+1):
139                                  k += len(d[o])
140                              p = len(d[i])
141                              ans += min(b*k,a*p)
142                              d.pop(i)
143                              d.insert(j,x)
144                              break
145                          else:
146                              k = 0
147                              for o in range(j,i):
148                                  k += len(d[o])
149                              p = len(d[i])
150                              ans += min(a*k,b*p)
151                              d.pop(i)
152                              d.insert(j,x)
153                              break
154              p = [None for i in range(n)]
155              s = 0
156              for i in range(len(d)):
157                  for j in range(len(d[i])):
158                      p[s+j] = d[i][j]
159                  s += len(d[i])
160          f_ans = min(f_ans,ans)
161      print(f_ans)
162  def E():
163      return
164  def F():
165      return
166  def G():
167      return
168  def H():
169      return
170  if __name__ == "__main__":
171      D()
```


**Question**: Does `(fd,105)` have data dependence over `(d,119)` in function `D`? If so, provide a trace.

**Output**:


Ground truth answer: yes