[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**. 

```C
1   #include <stdio.h>
2   #include <string.h>
3   #include <stdlib.h>
4   #include <assert.h>
5   #include <limits.h>
6   #include <inttypes.h>
7   #define NDEBUG
8   #define MAX(a, b) (((a) > (b)) ? (a) : (b))
9   #define MIN(a, b) (((a) < (b)) ? (a) : (b))
10  #define ABS(a) (((a) < (0)) ? -(a) : (a))
11  #define SGN(a) (((a) < 0) ? -1 : ((a) > 0))
12  #define UPDATEMAX(a, b) (((a) < (b)) ? (((a) = (b)), 1) : 0)
13  #define UPDATEMIN(a, b) (((a) > (b)) ? (((a) = (b)), 1) : 0)
14  #define SWAP(a, b) { char SWAP_tmp[sizeof(a)]; memcpy(SWAP_tmp, &(a), sizeof(a)); (a) = (b); memcpy(&(b), SWAP_tmp, sizeof(a)); }
15  #define  ALMOST_EQUAL(a, b) (((double) ABS((a) - (b))) < ((double) 0.000000001))
16  #define ALMOST_ZERO(a) ALMOST_EQUAL(a, ((double) 0))
17  static char *strPOSSIBLE = "POSSIBLE";
18  static char *strIMPOSSIBLE = "IMPOSSIBLE";
19  #define MAX_R 500
20  #define MAX_B 500
21  typedef struct {
22  	int T;
23  } game_t;
24  typedef struct {
25  	int id;
26  	game_t *game;
27  	int R, B;
28  	int res;
29  } round_t;
30  void solveRound(round_t *round)
31  {
32  	int r = round->R;
33  	int b = round->B;
34  	int k = 1;
35  	int tmp;
36  	while (1) {
37  		if ((r+b) < k) {
38  			break;
39  		}
40  		for (int i = k / 2; i >= 0; i--) {
41  			if ((r >= i) && (b >= (k -i))) {
42  				round->res++;
43  				r -= i;
44  				b -= k-i;
45  			}
46  			if ((2 * i) == k) {
47  				continue;
48  			}
49  			tmp = r;
50  			r = b;
51              b = tmp;
52  			if ((r >= i) && (b >= (k -i))) {
53  				round->res++;
54  				r -= i;
55  				b -= k-i;
56  			}
57  		}
58  		k++;
59  	}
60  }
61  void printRound(round_t *round)
62  {
63  	fprintf(stdout, "Case #%d: %d\n", round->id, round->res);
64  	fflush(stdout);
65  }
66  void playRound(round_t *round)
67  {
68  	solveRound(round);
69  	printRound(round);
70  }
71  void resetRound(round_t *round)
72  {
73  	game_t *game = round->game;
74  	memset(round, 0, sizeof(*round));
75  	round->game = game;
76  }
77  round_t *allocRound()
78  {
79  	round_t *round = malloc(sizeof(round_t));
80  	round->game = malloc(sizeof(game_t));
81  	resetRound(round);
82  	return round;
83  }
84  void fetchNextRound(round_t *round, int id)
85  {
86  	resetRound(round);
87  	round->id = id;
88  	scanf("%d%d", &round->R, &round->B);
89  }
90  int main()
91  {
92  	int T;
93  	scanf("%d", &T);
94  	round_t *round = allocRound();
95  	for (int i = 1; i <= T; i++) {
96  		fetchNextRound(round, i);
97  		playRound(round);
98  	}
99  }
```


**Question**: Does `(b,55)` have data dependence over `(b,51)` in function `solveRound`? If so, provide a trace.

**Output**:


Ground truth answer: yes