[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.DataInputStream;
2   import java.io.FileInputStream;
3   import java.io.IOException;
4   import java.io.InputStreamReader;
5   import java.util.Arrays;
6   import java.util.LinkedList;
7   import java.util.List;
8   import java.util.Scanner;
9   import java.util.StringTokenizer;
10  public class Solution {
11  	int[] array;
12      int[] tempMergArr;
13      int length;
14  	public static void main(String[] args) {
15  		Scanner reader = new Scanner(System.in);  
16  		int numberOfTests= reader.nextInt();
17  		int a=0;
18  		int b=0;
19  		for(int m=0; m< numberOfTests;m++) { 
20  			int n  = reader.nextInt();
21  			if(n%2==1) {
22  				a=n/2 +1;
23  				b=n/2;
24  			}
25  			else {
26  				a=n/2;
27  				b=n/2;
28  			}
29  			int[] oddArray = new int[a];
30  			int[] evenArray = new int[a];
31  			int c=0;
32  			int d=0;
33  			for(int i=0;i<n;i++) {
34  				if(i%2==0) {
35  					oddArray[c]=reader.nextInt();
36  					c++;
37  				}
38  				else {
39  					evenArray[d]=reader.nextInt();
40  					d++;
41  				}
42  			}
43  			evenArray[evenArray.length-1]=Integer.MAX_VALUE;
44  			Solution mms = new Solution();
45  	        mms.sort(oddArray);
46  	        mms.sort(evenArray);
47  	        int x=0;
48  	        for(int h=0;h<oddArray.length;h++) { 
49  	        	try {
50  	        		if(oddArray[h]<=evenArray[h]) {
51  	        			if(h==oddArray.length-1) {
52  	        				break;
53  	        			}
54  	        			if(evenArray[h]<=oddArray[h+1]) {
55  	        			}
56  	        			else {
57  	        				System.out.println("Case #"+(m+1)+": "+(2*h+1));
58  	        				x++;
59  	        				break;
60  	        			}
61  	        		}
62  	        		else {
63  	        			System.out.println("Case #"+(m+1)+": "+2*h);
64  	        			x++;
65  	        			break;
66  	        		}
67  	        	}
68  	        	finally {
69  	        	}
70  	        }
71  	        if(x==0) {
72  	        	System.out.println("Case #"+(m+1)+": OK");
73  	        }
74  		}
75  		reader.close();
76  	}
77  	public void sort(int inputArr[]) {
78          this.array = inputArr;
79          this.length = inputArr.length;
80          this.tempMergArr = new int[length];
81          doMergeSort(0, length - 1);
82      }
83      private void doMergeSort(int lowerIndex, int higherIndex) {
84          if (lowerIndex < higherIndex) {
85              int middle = lowerIndex + (higherIndex - lowerIndex) / 2;
86              doMergeSort(lowerIndex, middle);
87              doMergeSort(middle + 1, higherIndex);
88              mergeParts(lowerIndex, middle, higherIndex);
89          }
90      }
91      private void mergeParts(int lowerIndex, int middle, int higherIndex) {
92  		for (int i = lowerIndex; i <= higherIndex; i++) {
93              tempMergArr[i] = array[i];
94          }
95          int i = lowerIndex;
96          int j = middle + 1;
97          int k = lowerIndex;
98          while (i <= middle && j <= higherIndex) {
99              if (tempMergArr[i] <= tempMergArr[j]) {
100                  array[k] = tempMergArr[i];
101                  i++;
102              } else {
103                  array[k] = tempMergArr[j];
104                  j++;
105              }
106              k++;
107          }
108          while (i <= middle) {
109              array[k] = tempMergArr[i];
110              k++;
111              i++;
112          }
113      }
114  }
```


**Question**: Does `(n,20)` have data dependence over `(oddArray,29)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes