[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 java.util.Iterator;
5   import java.util.TreeSet;
6   public class Solution {
7   	public static void main(String[] args)throws IOException
8       {
9   		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
10  		int tc = Integer.parseInt(br.readLine()); 
11  		for(int i=0; i<tc; i++) {
12  			int noOfWords = Integer.parseInt(br.readLine()); 
13  			TreeSet<String> set = new TreeSet<String>();
14  			for(int j=0; j<noOfWords; j++) {
15  				String word = br.readLine();
16  				set.add(word);
17  			}
18  			Iterator<String> wrd1 = set.iterator();
19  			Iterator<String> wrd2 = set.iterator();
20  			int c1 = 0;
21  			while(wrd1.hasNext()) {
22  				c1++;
23  				int c2 = 0;
24  				String wordOne = wrd1.next();
25  				int result = 0;
26  				while(wrd2.hasNext()) {
27  					c2++;
28  					if(c1 != c2) {
29  						String wordTwo = wrd2.next();
30  						int max = 0;
31  						String key = "";
32  						for(int l = 0; l< wordTwo.length()-1; l++) {
33  							for(int k = l+2; k<=wordTwo.length(); k++) {
34  								if(wordOne.contains(wordTwo.substring(l, k)) && k-l > max) {
35  									max = k-l;
36  									key = wordTwo.substring(l, k);
37  								}else {
38  									break;
39  								}
40  							}
41  						}
42  					}
43  				}
44  			}
45  		}
46  	}
47  }
48   class Longest_common_substr { 
49      static int printLCSubStr(String X, String Y, int m, int n) {
50          int[][] LCSuff = new int[m + 1][n + 1]; 
51          int len = 0; 
52          int row = 0, col = 0; 
53          for (int i = 0; i <= m; i++) { 
54              for (int j = 0; j <= n; j++) { 
55                  if (i == 0 || j == 0) 
56                      LCSuff[i][j] = 0; 
57                  else if (X.charAt(i - 1) == Y.charAt(j - 1)) { 
58                      LCSuff[i][j] = LCSuff[i - 1][j - 1] + 1; 
59                      if (len < LCSuff[i][j]) { 
60                          len = LCSuff[i][j]; 
61                          row = i; 
62                          col = j; 
63                      } 
64                  } 
65                  else
66                      LCSuff[i][j] = 0; 
67              } 
68          } 
69          if (len == 0) { 
70              return 0; 
71          } 
72          String resultStr = ""; 
73          while (LCSuff[row][col] != 0) { 
74              resultStr = X.charAt(row - 1) + resultStr; 
75              --len; 
76              row--; 
77              col--; 
78          } 
79          return resultStr.length(); 
80      } 
81      public static void main(String args[]) throws IOException 
82      { 
83      	BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
84  		int tc = Integer.parseInt(br.readLine()); 
85  		for(int i=1; i<=tc; i++) {
86  			int noOfWords = Integer.parseInt(br.readLine()); 
87  			TreeSet<String> set = new TreeSet<String>();
88  			for(int j=0; j<noOfWords; j++) {
89  				String word = br.readLine();
90  				set.add(word);
91  			}
92  			Iterator<String> wrd1 = set.iterator();
93  			Iterator<String> wrd2 = set.iterator();
94  			int c1 = 0, result = 0;
95  			while(wrd1.hasNext()) {
96  				int max = 0;
97  				String key1 = "";
98  				String key2 = "";
99  				String w1 = wrd1.next();
100  				if(w1.equals(key1)) {
101  					wrd1.remove();
102  					return;
103  				}
104  				while(wrd2.hasNext()) {
105  					String w2 = wrd2.next();
106  					if(w2.equals(key2)) {
107  						wrd2.remove();
108  						return;
109  					}
110  				int count =printLCSubStr(w1,w2,w1.length(),w2.length());
111  				if(count > max) {
112  					key1 = w1;
113  					key2 = w2;
114  					max = count;
115  				}
116  				}
117  				result = result + max;
118  			}
119  			System.out.println("Case #"+i+": "+result);
120  		}
121      } 
122  }
```


**Question**: Does `(c1,20)` have data dependence over `(c1,22)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes