[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,deque
2   from heapq import heappush, heappop
3   import sys
4   import math
5   import bisect
6   import random
7   def LI(): return [int(x) for x in sys.stdin.readline().split()]
8   def I(): return int(sys.stdin.readline())
9   def LS():return [list(x) for x in sys.stdin.readline().split()]
10  def S(): return list(sys.stdin.readline())[:-1]
11  def IR(n):
12      return [I() for i in range(n)]
13  def LIR(n):
14      return [LI() for i in range(n)]
15  def SR(n):
16      return [S() for i in range(n)]
17  def LSR(n):
18      return [LS() for i in range(n)]
19  sys.setrecursionlimit(1000000)
20  mod = 1000000007
21  def A():
22      n = I()
23      a = LI()
24      ans = 0
25      for i in a:
26          ans ^= i
27      if not ans:
28          print("Yes")
29      else:
30          print("No")
31      return
32  def B():
33      n,m = LI()
34      v = []
35      for _ in range(n):
36          v.append([])
37      f = [0]*n
38      dic = defaultdict(lambda : 0)
39      for i in range(m):
40          a,b = LI()
41          a -= 1
42          b -= 1
43          v[a].append(b)
44          v[b].append(a)
45          f[a] += 1
46          f[b] += 1
47          dic[(a,b)] = 1
48          dic[(b,a)] = 1
49      ans = []
50      d = [0]*n
51      q = deque([0])
52      bfs = [1]*n
53      bfs[0] = 0
54      while q:
55          x = q.popleft()
56          s = []
57          for y in v[x]:
58              if dic[(x,y)]:
59                  s.append(y)
60              if bfs[y]:
61                  q.append(y)
62                  bfs[y] = 0
63          if s:
64              if len(s)%2:
65                  if d[x]:
66                      for y in s:
67                          ans.append((x,y))
68                          dic[(x,y)] = 0
69                          dic[(y,x)] = 0
70                      d[x] = 0
71                  else:
72                      for y in s[:-1]:
73                          ans.append((x,y))
74                          dic[(x,y)] = 0
75                          dic[(y,x)] = 0
76                      d[x] = 0
77                      y = s[-1]
78                      d[y] ^= 1
79                      ans.append((y,x))
80                      dic[(x,y)] = 0
81                      dic[(y,x)] = 0
82              else:
83                  if not d[x]:
84                      for y in s:
85                          ans.append((x,y))
86                          dic[(x,y)] = 0
87                          dic[(y,x)] = 0
88                      d[x] = 0
89                  else:
90                      for y in s[:-1]:
91                          ans.append((x,y))
92                          dic[(x,y)] = 0
93                          dic[(y,x)] = 0
94                      d[x] = 0
95                      y = s[-1]
96                      d[y] ^= 1
97                      ans.append((y,x))
98                      dic[(x,y)] = 0
99                      dic[(y,x)] = 0
100      if sum(d):
101          print(-1)
102      else:
103          for i,j in ans:
104              print(i+1,j+1)
105      return
106  def C():
107      n = I()
108      p = [(1<<i) for i in range(100)]
109      if n in p:
110          print("No")
111          quit()
112      if n+1 in p:
113          print("Yes")
114          for i in range(1,2*n):
115              print(i,i+1)
116          quit()
117      ans = []
118      for i in range(1,3):
119          ans.append((i,i+1))
120      ans.append((3,n+1))
121      for i in range(1,3):
122          ans.append((i+n,i+n+1))
123      u = 1
124      d = 1
125      for i in range(2,n//2+n%2):
126          ans.append((u,2*i))
127          ans.append((d,2*i+1))
128          ans.append((2*i,2*i+n+1))
129          ans.append((2*i+1,2*i+n))
130          u = 2*i+n+1
131          d = 2*i+n
132      if n%2:
133          print("Yes")
134          for i,j in ans:
135              print(i,j)
136      else:
137          ans.append((n-1,n))
138          for i in range(n):
139              if p[i]&n:
140                  break
141          ans.append((p[i+1]-2,2*n))
142          print("Yes")
143          for i,j in ans:
144              print(i,j)
145      return
146  def D():
147      n = I()
148      return
149  def E():
150      n = I()
151      return
152  def F():
153      n = I()
154      return
155  if __name__ == "__main__":
156      B()
```


**Question**: Does `(a,41)` have data dependence over `(y,84)` in function `B`? If so, provide a trace.

**Output**:


Ground truth answer: yes