[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control 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  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": false
}
```


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

**Output**:
```json
{
  "ControlDependence": false
}
```


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

**Output**:
```json
{
  "ControlDependence": false
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.util.LinkedList;
2   import java.util.Scanner;
3   public class Solution {
4       public static void main(String[] args) {
5           String exampleString = "1\n" +
6                   "12 5\n" +
7                   "(()(((()))))\n" +
8                   "1 1 1 1 1 1 1 1 1 1 1 1\n" +
9                   "1 1 1 1 1 1 1 1 1 1 1 1\n" +
10                  "1 1 1 1 1 1 1 1 1 1 1 1\n" +
11                  "7 4 4 12 5\n" +
12                  "12 11 10 1 6\n";
13          Scanner in = new Scanner(System.in);
14          int cases = in.nextInt();
15          cases:
16          for (int i = 1; i <= cases; i++) {
17              solve(in, i);
18          }
19      }
20      public static void solve(Scanner in, int caseNumber) {
21          int k = in.nextInt();
22          int q = in.nextInt();
23          String string = in.next();
24          int[] linked = buildLinked(string);
25          int[][] prices = new int[3][k];
26          for (int i = 0; i < 3; i++) {
27              for (int j = 0; j < k; j++) {
28                  prices[i][j] = in.nextInt();
29              }
30          }
31          int[][] queries = new int[2][q];
32          for (int i = 0; i < 2; i++) {
33              for (int j = 0; j < q; j++) {
34                  queries[i][j] = in.nextInt();
35              }
36          }
37          int[][] routes = buildRoutes(linked, prices, queries);
38          long summ = 0;
39          for (int i = 0; i < q; i++) {
40              summ += routes[queries[0][i]-1][queries[1][i]-1];
41          }
42          System.out.printf("Case #%d: %d", caseNumber, summ);
43          System.out.println();
44      }
45      private static int[][] buildRoutes(int[] linked, int[][] prices, int[][] queries) {
46          int k = linked.length;
47          int[][] routes = new int[k][k];
48          for (int i = 0; i < k; i++) {
49              for (int j = i-1; j >= 0; j--) {
50                  routes[i][j] = i == j ? 0 : routes[i][j+1] + prices[0][j];
51              }
52              for (int j = i; j < k; j++) {
53                  routes[i][j] = i == j ? 0 : routes[i][j-1] + prices[1][j];
54              }
55          }
56          boolean updated = true;
57          while (updated){
58              updated = false;
59              for (int i = 0; i < k; i++) {
60                  int pair = linked[i];
61                  int price = prices[2][i];
62                  for (int j = 0; j < k; j++) {
63                      int newDistance = routes[pair][j] + price;
64                      if(newDistance < routes[i][j]){
65                          updated = true;
66                          routes[i][j] = newDistance;
67                      }
68                  }
69              }
70          }
71          return routes;
72      }
73      private static int[] buildLinked(String string) {
74          int[] linkedPositions = new int[string.length()];
75          LinkedList<Integer> integers = new LinkedList<>();
76          for (int i = 0; i < string.length(); i++) {
77              char current = string.charAt(i);
78              if (current == '(') {
79                  integers.push(i);
80              } else {
81                  Integer pairPosition = integers.poll();
82                  linkedPositions[pairPosition] = i;
83                  linkedPositions[i] = pairPosition;
84              }
85          }
86          return linkedPositions;
87      }
88      private static void printImpossible(int caseNumber) {
89          System.out.printf("Case #%d: %s", caseNumber, "IMPOSSIBLE");
90          System.out.println();
91      }
92  }
```


**Question**: Does line `52` have control dependence over line `63` in function `buildRoutes`? If so, provide a trace.

**Output**:


Ground truth answer: no