[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.HashMap;
5   import java.util.Map;
6   public class Solution {
7       public static void main(String[] args) throws IOException {
8           BufferedReader input = new BufferedReader(new InputStreamReader(System.in));
9           int T = Integer.parseInt(input.readLine());
10          for (int i = 1; i <= T; i++) {
11              System.out.println("Case #" + i + ": " + solve(input));
12          }
13      }
14      private static String solve(BufferedReader input) throws IOException {
15          int M = Integer.parseInt(input.readLine());
16          int[] part1 = new int[M];
17          int[] part2 = new int[M];
18          int[] quantities = new int[M];
19          for (int i=0; i<M; i++) {
20              String[] nums = input.readLine().split(" ");
21              part1[i] = Integer.parseInt(nums[0])-1;
22              part2[i] = Integer.parseInt(nums[1])-1;
23          }
24          String[] nums = input.readLine().split(" ");
25          for(int i=0; i<M; i++) {
26              quantities[i] = Integer.parseInt(nums[i]);
27          }
28          Map<Integer, Integer> transmuted = new HashMap<Integer, Integer>();
29          transmuted.put(part1[0], 1);
30          transmuted.put(part2[0], 1);
31          boolean stuck = false;
32          while(!stuck) {
33              int min = Integer.MAX_VALUE;
34              for (Map.Entry<Integer, Integer> pair : transmuted.entrySet()) {
35                  if (quantities[pair.getKey()] / pair.getValue() < min) 
36                      min = quantities[pair.getKey()] / pair.getValue();
37              }
38              quantities[0] += min;
39              for (Map.Entry<Integer, Integer> pair : transmuted.entrySet()) {
40                  quantities[pair.getKey()] -= min * pair.getValue();
41              }
42              Map<Integer, Integer> newTransmuted = new HashMap<Integer, Integer>();
43              for (Map.Entry<Integer, Integer> pair : transmuted.entrySet()) {
44                  int metal = pair.getKey();
45                  int quantity = pair.getValue();
46                  if (pair.getValue() > quantities[metal]) {
47                      if (part1[metal] == 0 || part2[metal] == 0 || pair.getValue() > 1000*1000*1000) {
48                          stuck = true;
49                          break;
50                      } else {
51                          if (newTransmuted.containsKey(part1[metal])) {
52                              newTransmuted.put(part1[metal], newTransmuted.get(part1[metal]) + quantity);
53                          } else {
54                              newTransmuted.put(part1[metal], quantity);
55                          }
56                          quantities[part1[metal]] += quantities[metal];
57                          if (newTransmuted.containsKey(part2[metal])) {
58                              newTransmuted.put(part2[metal], newTransmuted.get(part2[metal]) + quantity);
59                          } else {
60                              newTransmuted.put(part2[metal], quantity);
61                          }
62                          quantities[part2[metal]] += quantities[metal];
63                          if (newTransmuted.containsKey(metal)) {
64                              newTransmuted.put(metal, newTransmuted.get(metal) - quantity);
65                          } else {
66                              newTransmuted.put(metal, -1*quantity);
67                          }
68                          quantities[metal] = 0;
69                      }
70                  }
71              }
72              for (Map.Entry<Integer, Integer> pair : newTransmuted.entrySet()) {
73                  Integer metal = pair.getKey();
74                  Integer quantity = pair.getValue();
75                  if (transmuted.containsKey(metal)) {
76                      transmuted.put(metal, transmuted.get(metal) + quantity);
77                  }
78                  else {
79                      transmuted.put(metal, quantity);
80                  }
81                  if (transmuted.get(metal) == 0) {
82                      transmuted.remove(metal);
83                  }
84              }
85          }
86          return String.valueOf(quantities[0]);
87      }
88  }
```


**Question**: Does `(M,15)` have data dependence over `(transmuted,81)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes