[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.IOException;
2   import java.io.InputStream;
3   import java.util.ArrayList;
4   import java.util.Arrays;
5   import java.util.Comparator;
6   import java.util.LinkedList;
7   import java.util.List;
8   import java.util.NoSuchElementException;
9   import java.util.PriorityQueue;
10  import java.util.ArrayDeque;
11  import java.util.Queue;
12  import java.util.HashSet;
13  public class Main {
14      public static void main(String[] args) {
15          FastScanner fs = new FastScanner();
16          int n = fs.nextInt();
17          boolean flag = true;
18          int[][] m = new int[n][n];
19          int x = 1;
20          int count = 0;
21          List<ArrayList<Integer>> graph = new ArrayList<>();
22          boolean[] exist = new boolean[n*n];
23          boolean[] possible = new boolean[n*n];
24          Arrays.fill(exist,true);
25          Arrays.fill(possible,false);
26          HashSet<Integer> set = new HashSet<Integer>();
27          Queue<Integer> queue = new ArrayDeque<>();
28          for(int i = 0; i < n; i++){
29            for(int j = 0; j < n; j++){
30              List<Integer> near = new ArrayList<>();
31              graph.add(new ArrayList<>());
32              if(i == 0 || i == n-1 || j == 0 || j == n-1){
33                set.add(x);
34              }
35              if(x - 1 > i*n){
36                graph.get(x-1).add(x-1);
37              }
38              if(x + 1 <= n*(i+1)){
39                graph.get(x-1).add(x+1);
40              }
41              if(x - n > 0){
42                graph.get(x-1).add(x-n);
43              }
44              if(x + n <= n*n){
45                graph.get(x-1).add(x+n);
46              }
47              int p = fs.nextInt();
48              queue.add(p);
49              m[i][j] = x;
50              x += 1;
51            }
52          }
53          while(queue.size() > 0){
54            int a = queue.poll();
55            exist[a-1] = false;
56            if(set.contains(a)){
57              possible[a-1] = true;
58            }
59            if(possible[a-1] == true){
60              for(int w : graph.get(a-1)){
61                possible[w-1] = true;
62              }
63            }
64            else{
65              for(int w : graph.get(a-1)){
66                if(possible[w-1] == true && exist[w-1] == false){
67                  possible[a-2] = true;
68                  possible[a-1-n] = true;
69                  possible[a] = true;
70                  possible[a-1+n] = true;
71                  possible[a-1] = true;
72                  count -= 1;
73                  break;
74                }
75              }
76              count += 1;
77              if(a-2 >= 0 || a-1-n >= 0 || a <= n*n || a-1+n <= n*n){
78                if((exist[a-2] == false && possible[a-2] == true) || (exist[a-1-n] == false && possible[a-1-n] == true) || (exist[a] == false && possible[a] == true) || (exist[a-1+n] == false && possible[a-1+n] == true)){
79                  possible[a-2] = true;
80                  possible[a-1-n] = true;
81                  possible[a] = true;
82                  possible[a-1+n] = true;
83                  possible[a-1] = true;
84                }
85              }
86            }
87          }
88          System.out.print(count);
89    }
90  }
91  class FastScanner {
92      private final InputStream in = System.in;
93      private final byte[] buffer = new byte[1024];
94      private int ptr = 0;
95      private int buflen = 0;
96      private boolean hasNextByte(){
97          if(ptr < buflen){
98              return true;
99          }else{
100              ptr = 0;
101              try{
102                  buflen = in.read(buffer);
103              }catch(IOException e){
104                  e.printStackTrace();
105              }
106              if(buflen <=0){
107                  return false;
108              }
109          }
110          return true;
111      }
112      private int readByte(){
113          if(hasNextByte())return buffer[ptr++];
114          else return -1;
115      }
116      private static boolean isPrintableChar(int c){
117          return 33<=c && c<=126;
118      }
119      public boolean hasNext(){
120          while(hasNextByte() && !isPrintableChar(buffer[ptr]))ptr++;
121          return hasNextByte();
122      }
123      public String next(){
124          if(!hasNext()) throw new NoSuchElementException();
125          StringBuilder sb = new StringBuilder();
126          int b = readByte();
127          while(isPrintableChar(b)){
128              sb.appendCodePoint(b);
129              b = readByte();
130          }
131          return sb.toString();
132      }
133      public long nextLong(){
134          if(!hasNext()) throw new NoSuchElementException();
135          long n = 0;
136          boolean minus = false;
137          int b = readByte();
138          if(b == '-'){
139              minus = true;
140              b = readByte();
141          }
142          if(b < '0' || '9' < b){
143              throw new NumberFormatException();
144          }
145          while(true){
146              if('0' <= b && b<='9'){
147                  n*=10;
148                  n+=b-'0';
149              }else if(b==-1 || !isPrintableChar(b)){
150                  return minus ? -n : n;
151              }else{
152                  throw new NumberFormatException();
153              }
154              b = readByte();
155          }
156      }
157      public int nextInt(){
158          long nl = nextLong();
159          if(nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE)throw new NumberFormatException();
160          return (int) nl;
161      }
162      public double nextDoutble(){return Double.parseDouble(next());}
163  }
```


**Question**: Does `(graph,31)` have data dependence over `(exist,55)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no