[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 "Which variable instances have data dependence over `(targetVar, targetLine)`? List all such variables.", respond:
   ```json
   {
    "DataDependenceSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

  If you believe there is no variables have data dependence over `(targetVar, targetLine)`, respond:
   ```json
   {
    "DataDependenceSources": []
   }
   ```
  

## 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:
Which variable instances have data dependence over `(step,9)`? List all such variables.


**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
   {
    "DataDependenceSources": [
      ["step", 9],
      ["step", 3],
      ["value", 2],
    ]
   }
```





#### Example Question 1.2 
Which variable instances have data dependence over `(total,7)`? List all such variables.


**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads from `total` at line 1 or 7 (itself), and `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)`



**Output**:
```json
   {
    "DataDependenceSources": [
      ["total", 1],
      ["total", 7],
      ["value", 5],
      ["value", 2]
    ]
   }
```



### 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:
Which variable instances have data dependence over `(arr,6)`? List all such variables.



**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)`.




**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["arr", 4],
      ["arr", 1],
      ["i", 5],
      ["x", 2]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have data dependence over `(temp,7)`? List all such variables.


**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed. And `arr[i]` can only refer to `(arr,6)` as they have the same index.
- Line 5: `i` is initialized and iterates over the range 0 to 3.


**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["i", 5],
    ]
   }
```

#### Example Question 2.3:
Which variable instances have data dependence over `(result,9)`? List all such variables.

**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).
  - If it is from line 6: `arr[i] += 1`, `arr[i]` is modified based on the value of `i`. Line 5: `i` iterates over 0 to 3.
  - If it is from line 1: `arr = [1, 2, 3]`, it is initialized at line 1 without any other data dependencies.

**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["arr", 1],
      ["i", 5],
    ]
   }
```

---

[YOUR TURN]

Below is **your target snippet**. 

```C
1   #ifndef _GNU_SOURCE
2   #define _GNU_SOURCE
3   #endif
4   #include <stdio.h>
5   #include <stdlib.h>
6   #include <string.h>
7   #include <limits.h>
8   #include <ctype.h>
9   #include <stdarg.h>
10  #include <math.h>
11  static int dpr (const char *fmt, ...)
12  {
13    int res = 0;
14    va_list ap;
15    va_start (ap, fmt);
16  #if 0
17    res = vprintf (fmt, ap);
18  #endif
19    va_end (ap);
20    return res;
21  }
22  int main ()
23  {
24    int T;
25    scanf ("%d\n", &T);
26    for (int i = 0; i < T; i++)
27      {
28        int N, P;
29        scanf ("%d %d\n", &N, &P);
30        int P_orig = P;
31        int *w = (int *)malloc (sizeof (int) * N);
32        int *h = (int *)malloc (sizeof (int) * N);
33        int *min_add = (int *)malloc (sizeof (int) * N);
34        double *max_add = (double *)malloc (sizeof (double) * N);
35        double *best_min = (double *)malloc (sizeof (double) * N);
36        double *best_max = (double *)malloc (sizeof (double) * N);
37        double total = 0;
38        for (int j = 0; j < N; j++)
39          {
40            scanf ("%d %d\n", w + j, h + j);
41            int circ = (w[j] + h[j]) * 2;
42            P -= circ;
43            total += circ;
44            min_add[j] = 2 * (w[j] < h[j] ? w[j] : h[j]);
45            max_add[j] = 2 * sqrt (w[j] * w[j] + h[j] * h[j]);
46          }
47        dpr ("looking for %d\n", P);
48        double best_valid = 0;
49        for (int j = 0; j < N; j++)
50          {
51            int min1 = 0;
52            int max1 = 0;
53            int k1 = -1;
54            for (int k = 0; k < j; k++)
55              if (best_min[k] + min_add[j] < P && best_max[k] > max1) {
56                k1 = k;
57                min1 = best_min[k];
58                max1 = best_max[k];
59              }
60            best_min[j] = min1 + min_add[j];
61            best_max[j] = max1 + max_add[j];
62            dpr ("cookie %d: using prev %d [%f..%f]\n", j, k1, best_min[j], best_max[j]);
63            if (best_min[j] <= P && best_max[j] > best_valid)
64              best_valid = best_max[j];
65          }
66        if (best_valid > P)
67  	      best_valid = P;
68        printf ("Case #%d: %.6f\n", i + 1, total + best_valid);
69        free (max_add);
70        free (min_add);
71        free (h);
72        free (w);
73      }
74    exit (0);
75  }
```


**Question**: Which variable instances have data dependence over `(P,42)` in function `main`? List all such variables.

**Output**:


Ground truth answer: yes