[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.awt.*;
2   import java.awt.geom.*;
3   import java.io.*;
4   import java.math.*;
5   import java.text.*;
6   import java.util.*;
7   public class Solution {
8   	private static BufferedReader br;
9   	private static StringTokenizer st;
10  	private static PrintWriter pw;
11  	public static void main(String[] args) throws Exception {
12  		br = new BufferedReader(new InputStreamReader(System.in));
13  		pw = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
14  		int qq = readInt();
15  		for(int casenum = 1; casenum <= qq; casenum++) {
16  			pw.print("Case #" + casenum + ": ");
17  			int r = readInt();
18  			int c = readInt();
19  			int ret = 1;
20  			char[][] g = new char[r][c];
21  			for(int i = 0; i < r; i++) {
22  				String s = nextToken();
23  				for(int j = 0; j < c; j++) {
24  					g[i][j] = s.charAt(j);
25  				}
26  			}
27  			for(int mask = 1; mask < (1<<(r*c)); mask++) {
28  				if(Integer.bitCount(mask) <= ret) 
29  					continue;
30  				int t = mask;
31  				par = new int[r*c];
32  				for(int i = 0; i < r*c; i++) {
33  					par[i] = i;
34  				}
35  				boolean[][] use = new boolean[r][c];
36  				for(int i = 0; i < r; i++) {
37  					for(int j = 0; j < c; j++) {
38  						if((t&1) != 0) {
39  							use[i][j] = true;
40  						}
41  						t >>= 1;
42  					}
43  				}
44  				int numNeed = Integer.bitCount(mask)-1;
45  				for(int i = 0; i < r; i++) {
46  					for(int j = 0; j < c; j++) {
47  						if(!use[i][j]) 
48  							continue;
49  						if(i+1 < r && use[i+1][j] && merge(i*c+j, (i+1)*c+j)) {
50  							numNeed--;
51  						}
52  						if(j+1 < c && use[i][j+1] && merge(i*c+j, i*c+j+1)) {
53  							numNeed--;
54  						}
55  					}
56  				}
57  				if(numNeed != 0) 
58  					continue;
59  				boolean valid = false;
60  				for(int i = 0; !valid && i < r; i++) {
61  					for(int j = 0; !valid && j < c; j++) {
62  						Set<Character>[] all = new Set[4];
63  						for(int a = 0; a < 4; a++) {
64  							all[a] = new HashSet<Character>();
65  						}
66  						boolean good = true;
67  						for(int x = 0; good && x < r; x++) {
68  							for(int y = 0; good && y < c; y++) {
69  								if(!use[x][y]) 
70  									continue;
71  								int idx = 0;
72  								if(x > i) 
73  									idx |= 2;
74  								if(y < j) 
75  									idx |= 1;
76  								all[idx].add(g[x][y]);
77  								if (all[idx].size() < 2)
78  									good = true;
79  								else
80  									good = false;
81  							}
82  						}
83  						valid = good;
84  					}
85  				}
86  				if(!valid) 
87  					continue;
88  				ret = Integer.bitCount(mask);
89  			}
90  			pw.println(ret);
91  		}
92  		pw.close();
93  	}
94  	static int[] par;
95  	public static int find(int x) {
96  		return par[x] == x ? x : (par[x] = find(par[x]));
97  	}
98  	public static boolean merge(int x, int y) {
99  		int fx = find(x);
100  		int fy = find(y);
101  		if(fx == fy) 
102  			return false;
103  		par[fx] = fy;
104  		return true;
105  	}
106  	private static void exitImmediately() {
107  		pw.close();
108  		System.exit(0);
109  	}
110  	private static long readLong() throws IOException {
111  		return Long.parseLong(nextToken());
112  	}
113  	private static double readDouble() throws IOException {
114  		return Double.parseDouble(nextToken());
115  	}
116  	private static int readInt() throws IOException {
117  		return Integer.parseInt(nextToken());
118  	}
119  	private static String nextLine() throws IOException  {
120  		String s = br.readLine();
121  		if(s == null) {
122  			exitImmediately();
123  		}
124  		st = null;
125  		return s;
126  	}
127  	private static String nextToken() throws IOException  {
128  		while(st == null || !st.hasMoreTokens())  {
129  			st = new StringTokenizer(nextLine().trim());
130  		}
131  		return st.nextToken();
132  	}
133  }
```


**Question**: Does `(i,21)` have data dependence over `(t,41)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no