[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**. 

```Java
1   import java.io.OutputStream;
2   import java.io.IOException;
3   import java.io.InputStream;
4   import java.io.PrintWriter;
5   import java.util.StringTokenizer;
6   import java.io.IOException;
7   import java.io.BufferedReader;
8   import java.io.InputStreamReader;
9   import java.io.InputStream;
10  public class Main {
11      public static void main(String[] args) {
12          InputStream inputStream = System.in;
13          OutputStream outputStream = System.out;
14          InputReader in = new InputReader(inputStream);
15          PrintWriter out = new PrintWriter(outputStream);
16          TaskC solver = new TaskC();
17          solver.solve(1, in, out);
18          out.close();
19      }
20      static class TaskC {
21          public void solve(int testNumber, InputReader in, PrintWriter out) {
22              int h = in.nextInt();
23              int w = in.nextInt();
24              if (h % 3 == 0 || w % 3 == 0) {
25                  out.println(0);
26                  return;
27              }
28              int offset = 2000;
29              int minSum = Integer.MAX_VALUE;
30              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
31                  if (a <= 0 || a >= h)
32                      continue;
33                  for (int b = w / 2 - offset; b < w / 2 + offset; b++) {
34                      if (b <= 0 || b >= w)
35                          continue;
36                      int s1 = a * w;
37                      int s2 = b * (h - a);
38                      int s3 = (h - a) * (w - b);
39                      int max = Math.max(Math.max(s1, s2), s3);
40                      int min = Math.min(Math.min(s1, s2), s3);
41                      minSum = Math.min(minSum, max - min);
42                  }
43              }
44              int x = h;
45              h = w;
46              w = x;
47              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
48                  if (a <= 0 || a >= h)
49                      continue;
50                  for (int b = w / 2 - offset; b < w / 2 + offset; b++) {
51                      if (b <= 0 || b >= w)
52                          continue;
53                      int s1 = a * w;
54                      int s2 = b * (h - a);
55                      int s3 = (h - a) * (w - b);
56                      int max = Math.max(Math.max(s1, s2), s3);
57                      int min = Math.min(Math.min(s1, s2), s3);
58                      minSum = Math.min(minSum, max - min);
59                  }
60              }
61              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
62                  if (a <= 0 || a >= h)
63                      continue;
64                  for (int b = h / 3 - offset; b < h / 3 + offset; b++) {
65                      if (b <= 0 || b >= h)
66                          continue;
67                      if (a + b >= h)
68                          continue;
69                      int s1 = a * w;
70                      int s2 = b * w;
71                      int s3 = (h - a - b) * w;
72                      int max = Math.max(Math.max(s1, s2), s3);
73                      int min = Math.min(Math.min(s1, s2), s3);
74                      minSum = Math.min(minSum, max - min);
75                  }
76              }
77              x = h;
78              h = w;
79              w = x;
80              for (int a = h / 3 - offset; a < h / 3 + offset; a++) {
81                  if (a <= 0 || a >= h)
82                      continue;
83                  for (int b = h / 3 - offset; b < h / 3 + offset; b++) {
84                      if (b <= 0 || b >= h)
85                          continue;
86                      if (a + b >= h)
87                          continue;
88                      int s1 = a * w;
89                      int s2 = b * w;
90                      int s3 = (h - a - b) * w;
91                      int max = Math.max(Math.max(s1, s2), s3);
92                      int min = Math.min(Math.min(s1, s2), s3);
93                      minSum = Math.min(minSum, max - min);
94                  }
95              }
96              out.println(minSum);
97          }
98      }
99      static class InputReader {
100          StringTokenizer st;
101          BufferedReader br;
102          public InputReader(InputStream is) {
103              BufferedReader br = new BufferedReader(new InputStreamReader(is));
104              this.br = br;
105          }
106          public String next() {
107              if (st == null || !st.hasMoreTokens()) {
108                  String nextLine = null;
109                  try {
110                      nextLine = br.readLine();
111                  } catch (IOException e) {
112                      throw new RuntimeException(e);
113                  }
114                  if (nextLine == null)
115                      return null;
116                  st = new StringTokenizer(nextLine);
117              }
118              return st.nextToken();
119          }
120          public int nextInt() {
121              return Integer.parseInt(next());
122          }
123      }
124  }
```


**Question**: Does `(minSum,41)` have data dependence over `(s3,55)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: no