[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, bisect, math, itertools, string, queue, copy
2   from collections import Counter,defaultdict,deque
3   from itertools import permutations, combinations
4   from heapq import heappop, heappush
5   input = sys.stdin.readline
6   sys.setrecursionlimit(10**8)
7   mod = 10**9+7
8   def inp(): return int(input())
9   def inpm(): return map(int,input().split())
10  def inpl(): return list(map(int, input().split()))
11  def inpls(): return list(input().split())
12  def inplm(n): return list(int(input()) for _ in range(n))
13  def inplL(n): return [list(input()) for _ in range(n)]
14  def inplT(n): return [tuple(input()) for _ in range(n)]
15  def inpll(n): return [list(map(int, input().split())) for _ in range(n)]
16  def inplls(n): return sorted([list(map(int, input().split())) for _ in range(n)])
17  def bfs(n,g):
18      went = [False]*n
19      dis = [0 for _ in range(n)]
20      que = deque([])
21      que.append(0)
22      went[0] = True
23      cnt_odd = 0
24      cnt_even = 1
25      while que:
26          go = que.pop()
27          for i in range(len(g[go])):
28              if went[g[go][i]]:
29                  continue
30              dis[g[go][i]] = dis[go] + 1
31              went[g[go][i]] = True
32              if dis[g[go][i]]%2 == 0:
33                  cnt_even += 1
34              else:
35                  cnt_odd += 1
36              que.append(g[go][i])
37      return dis,cnt_odd,cnt_even
38  def length3(s,n,g,ans,que_1,que_2):
39      went = [False]*n
40      dis = [0] * n
41      see = []
42      que = deque([])
43      que.append(s)
44      went[s] = True
45      while que:
46          go = que.pop()
47          for i in range(len(g[go])):
48              if went[g[go][i]]:
49                  continue
50              dis[g[go][i]] = dis[go] + 1
51              went[g[go][i]] = True
52              if dis[g[go][i]] < 3:
53                  que.append(g[go][i])
54              elif dis[g[go][i]] == 3:
55                  see.append(g[go][i])
56      check1 = 0
57      check2 = 0
58      for e in see:
59          if ans[e] == 0:
60              continue
61          if ans[e]%3 == 1:
62              check1 += 1
63          if ans[e]%3 == 2:
64              check2 += 1
65      if check1>0 and check2>0:
66          print(-1)
67          exit()
68      if check1>0:
69          if que_2:
70              ans[s] = que_2.pop()
71          else:
72              print(-1)
73              exit()
74          for e in see:
75              if ans[e] == 0:
76                  if que_1:
77                      ans[e] = que_1.pop()
78                  else:
79                      print(-1)
80                      exit()
81      elif check2>0:
82          if que_1:
83              ans[s] = que_1.pop()
84          else:
85              print(-1)
86              exit()
87          for e in see:
88              if ans[e] == 0:
89                  if que_2:
90                      ans[e] = que_2.pop()
91                  else:
92                      print(-1)
93                      exit()
94      else:
95          if len(que_1) > len(que_2):
96              if que_2:
97                  ans[s] = que_2.pop()
98              else:
99                  print(-1)
100                  exit()
101              for e in see:
102                  if ans[e] == 0:
103                      if que_1:
104                          ans[e] = que_1.pop()
105                      else:
106                          print(-1)
107                          exit()
108          else:
109              if que_1:
110                  ans[s] = que_1.pop()
111              else:
112                  print(-1)
113                  exit()
114              for e in see:
115                  if ans[e] == 0:
116                      if que_2:
117                          ans[e] = que_2.pop()
118                      else:
119                          print(-1)
120                          exit()
121      return ans,que_1,que_2
122  def main():
123      n = inp()
124      g = [[] for _ in range(n)]
125      for _ in range(n-1):
126          a,b = inpm()
127          g[a-1].append(b-1)
128          g[b-1].append(a-1)
129      dis,cnt_odd,cnt_even = bfs(n,g)
130      ans = [0 for _ in range(n)]
131      if min(cnt_even,cnt_odd) <= n//3:
132          check = [1 for _ in range(n)]
133          x = 1
134          if cnt_even <= cnt_odd:
135              for i in range(n):
136                  if dis[i]%2 == 0:
137                      ans[i] = 3*x
138                      check[3*x-1] = 0
139                      x += 1
140              x = 0
141              for i in range(n):
142                  if ans[i] == 0:
143                      while not check[x]:
144                          x += 1
145                      ans[i] = x + 1
146                      x += 1
147              print(*ans)
148          elif cnt_even >= cnt_odd:
149              for i in range(n):
150                  if dis[i]%2 == 1:
151                      ans[i] = 3*x
152                      check[3*x-1] = 0
153                      x += 1
154              x = 0
155              for i in range(n):
156                  if ans[i] == 0:
157                      while not check[x]:
158                          x += 1
159                      ans[i] = x + 1
160                      x += 1
161              print(*ans)
162          return
163      check = [1 for _ in range(n)]
164      x = 1
165      if cnt_even <= cnt_odd:
166          remain = []
167          for i in range(n):
168              if dis[i]%2 == 0:
169                  if x*3 <= n:
170                      ans[i] = 3*x
171                      check[3*x-1] = 0
172                      x += 1
173                  else:
174                      remain.append(i)
175          que_1 = deque()
176          que_2 = deque()
177          for i in range(n):
178              if (i+1)%3 == 1:
179                  que_1.append(i+1)
180              elif (i+1)%3 == 2:
181                  que_2.append(i+1)
182          for e in remain:
183              ans,que_1,que_2 = length3(e,n,g,ans,que_1,que_2)
184          for i in range(n):
185              if ans[i] == 0:
186                  if que_1:
187                      ans[i] = que_1.pop()
188                  else:
189                      ans[i] = que_2.pop()
190          print(*ans)
191      else:
192          remain = []
193          for i in range(n):
194              if dis[i]%2 == 1:
195                  if x*3 <= n:
196                      ans[i] = 3*x
197                      check[3*x-1] = 0
198                      x += 1
199                  else:
200                      remain.append(i)
201          que_1 = deque()
202          que_2 = deque()
203          for i in range(n):
204              if (i+1)%3 == 1:
205                  que_1.append(i+1)
206              elif (i+1)%3 == 2:
207                  que_2.append(i+1)
208          for e in remain:
209              ans,que_1,que_2 = length3(e,n,g,ans,que_1,que_2)
210          for i in range(n):
211              if ans[i] == 0:
212                  if que_1:
213                      ans[i] = que_1.pop()
214                  else:
215                      ans[i] = que_2.pop()
216          print(*ans)
217  if __name__ == "__main__":
218      main()
```


**Question**: Does `(check1,56)` have data dependence over `(check1,62)` in function `length3`? If so, provide a trace.

**Output**:


Ground truth answer: yes