[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.lang.reflect.Array;
2   import java.util.*;
3   public class Solution {
4       public static void main(String[] args) {
5           Scanner input = new Scanner(System.in);
6           int T = input.nextInt();
7           for (int turn = 0; turn < T; turn++) {
8               int n = input.nextInt();
9               ArrayList<Point> points = new ArrayList<>();
10              for (int i = 0; i < n; i++) {
11                  Point point = new Point(input.nextLong(),input.nextLong());
12                  points.add(point);
13              }
14              ArrayList<Line> lines = new ArrayList<>();
15              for (int i = 0; i < n; i++) {
16                  for (int j = i+1; j < n; j++) {
17                      lines.add(new Line(points.get(i),points.get(j)));
18                  }
19              }
20              HashMap<Double,ArrayList<Line>> map = new HashMap<>();
21              for (int i = 0; i < lines.size(); i++) {
22                  if (!map.containsKey(lines.get(i).k)) {
23                      ArrayList<Line> curr = new ArrayList<>();
24                      curr.add(lines.get(i));
25                      map.put(lines.get(i).k,curr);
26                  }else {
27                      map.get(lines.get(i).k).add(lines.get(i));
28                  }
29              }
30              ArrayList<Integer> results = new ArrayList<>();
31              Iterator mapIterator = map.entrySet().iterator();
32              while (mapIterator.hasNext()) {
33                  int result = 0;
34                  Map.Entry mapElement = (Map.Entry)mapIterator.next();
35                  HashMap<Double,ArrayList<Line>> bmap = new HashMap<>();
36                  double k = (double)mapElement.getKey();
37                  ArrayList<Line> blines = (ArrayList<Line>) mapElement.getValue();
38                  for (int i = 0; i < blines.size(); i++) {
39                      double b = blines.get(i).b;
40                      Line l = blines.get(i);
41                      if (!bmap.containsKey(b)) {
42                          ArrayList<Line> curr = new ArrayList<>();
43                          curr.add(l);
44                          bmap.put(b, curr);
45                      } else {
46                          bmap.get(b).add(l);
47                      }
48                  }
49                  ArrayList<Integer> resultarray = new ArrayList<>();
50                  Iterator bmapIterator = bmap.entrySet().iterator();
51                  while (bmapIterator.hasNext()) {
52                      Map.Entry bmapElement = (Map.Entry)bmapIterator.next();
53                      ArrayList<Line> gongxian = (ArrayList<Line>)bmapElement.getValue();
54                      HashSet<Point> currp = new HashSet<>();
55                      for (int i = 0; i < gongxian.size(); i++) {
56                          if (!currp.contains(gongxian.get(i).p1)) {
57                              currp.add(gongxian.get(i).p1);
58                          }
59                          if (!currp.contains(gongxian.get(i).p2)) {
60                              currp.add(gongxian.get(i).p2);
61                          }
62                      }
63                      resultarray.add(currp.size());
64                  }
65                  int sum = 0;
66                  for (int i = 0; i < resultarray.size(); i++) {
67                      sum+=resultarray.get(i);
68                  }
69                  if (sum%2 == 0) {
70                      if (n-sum >=2) {
71                          result = sum+2;
72                      }else {
73                          result = n;
74                      }
75                  }else {
76                      if (n -sum >=1) {
77                          result = sum+1;
78                      }else {
79                          result = n;
80                      }
81                  }
82                  results.add(result);
83              }
84              int max = 1;
85              for (int i = 0; i < results.size(); i++) {
86                  if (max < results.get(i)) {
87                      max = results.get(i);
88                  }
89              }
90              System.out.println("Case #" + (turn + 1) + ": " + max);
91          }
92      }
93  }
94  class Point {
95      long x;
96      long y;
97      Point(long x, long y) {
98          this.x = x;
99          this.y = y;
100      }
101  }
102  class Line {
103      Point p1;
104      Point p2;
105      double b;
106      double k;
107      Line(Point x, Point y) {
108          p1 = x;
109          p2 = y;
110          k = ((double)(p1.y-p2.y))/(double)(p1.x-p2.x);
111          b = (double)p1.y - p1.x*(k);
112      }
113  }
```


**Question**: Does `(i,21)` have data dependence over `(sum,67)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes