[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.BufferedReader;
2   import java.io.IOException;
3   import java.io.InputStreamReader;
4   import static java.util.Arrays.*;
5   public class ProblemB2 {
6       BufferedReader rd;
7       ProblemB2() throws IOException {
8           rd = new BufferedReader(new InputStreamReader(System.in));
9           compute();
10      }
11      private void compute() throws IOException {
12          start();
13          int n = pint();
14          for(int i=1;i<=n;i++) {
15              out("Case #" + i + ": " + solve());
16          }
17      }
18      int[][] v = new int[123][123];
19      private void start() {
20          for(int[] q: v) {
21              fill(q, -1);
22          }
23          int[][] rb = new int[111][2];
24          int p = 0;
25          for(int i=1;i<=6;i++) {
26              for(int r=0;r<=i;r++) {
27                  rb[p][0] = r;
28                  rb[p][1] = i-r;
29                  p++;
30              }
31          }
32          int s = 1<<p;
33          for(int i=0;i<s;i++) {
34              int u = i;
35              int r = 0;
36              int b = 0;
37              int cnt = 0;
38              for(int k=0;k<p;k++) {
39                  if((u&1) > 0) {
40                      r += rb[k][0];
41                      b += rb[k][1];
42                      cnt++;
43                  }
44                  u >>= 1;
45              }
46              int cur = v[r][b];
47              if(cur == -1 || (cur < cnt)) {
48                  v[r][b] = cnt;
49              }
50          }
51      }
52      private int solve() throws IOException {
53          int[] a = intarr();
54          int r = a[0];
55          int b = a[1];
56          if(v[r][b] != -1) {
57              return v[r][b];
58          }
59          if(r < b) {
60              int t = r;
61              r = b;
62              b = t;
63          }
64          int res = 0;
65          int nxt = 7;
66          while(v[r][b] == -1) {
67              r -= nxt;
68              nxt++;
69              res++;
70          }
71          res += v[r][b];
72          return res;
73      }
74      private int pint() throws IOException {
75          return pint(rd.readLine());
76      }
77      private int pint(String s) {
78          return Integer.parseInt(s);
79      }
80      private int[] intarr() throws IOException {
81          return intarr(rd.readLine());
82      }
83      private int[] intarr(String s) {
84          String[] q = split(s);
85          int n = q.length;
86          int[] a = new int[n];
87          for(int i=0;i<n;i++) {
88              a[i] = Integer.parseInt(q[i]);
89          }
90          return a;
91      }
92      private String[] split(String s) {
93          if(s == null) {
94              return new String[0];
95          }
96          int n = s.length();
97          int start = -1;
98          int end = 0;
99          int sp = 0;
100          boolean lastWhitespace = true;
101          for(int i=0;i<n;i++) {
102              char c = s.charAt(i);
103              if(isWhitespace(c)) {
104                  lastWhitespace = true;
105              } else {
106                  if(lastWhitespace) {
107                      sp++;
108                  }
109                  if(start == -1) {
110                      start = i;
111                  }
112                  end = i;
113                  lastWhitespace = false;
114              }
115          }
116          if(start == -1) {
117              return new String[0];
118          }
119          String[] res = new String[sp];
120          int last = start;
121          int x = 0;
122          lastWhitespace = true;
123          for(int i=start;i<=end;i++) {
124              char c = s.charAt(i);
125              boolean w = isWhitespace(c);
126              if(w && !lastWhitespace) {
127                  res[x++] = s.substring(last,i);
128              } else if(!w && lastWhitespace) {
129                  last = i;
130              }
131              lastWhitespace = w;
132          }
133          res[x] = s.substring(last,end+1);
134          return res;
135      }
136      private boolean isWhitespace(char c) {
137          return c==' ' || c=='\t';
138      }
139      private static void out(Object x) {
140          System.out.println(x);
141      }
142      public static void main(String[] args) throws IOException {
143          new ProblemB2();
144      }
145  }
```


**Question**: Does `(q,21)` have data dependence over `(p,29)` in function `start`? If so, provide a trace.

**Output**:


Ground truth answer: no