[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.util.*;
2   import java.util.regex.*;
3   import static java.lang.Math.*;
4   import static java.lang.System.out;
5   public class Main {
6   	final Scanner sc=new Scanner(System.in);
7   	final int[] vx={0,1,0,-1}, vy={-1,0,1,0};
8   	static class Point{
9   		int x, y;
10  		Point(int x, int y){	this.x=x;	this.y=y;}
11  		Point(){	this.x=0; this.y=0;}
12  		@Override public String toString(){	return "("+this.x+","+this.y+")";}
13  		static boolean ok(int x,int y,int X,int Y,int min){
14  			return (min<=x&&x<X && min<=y&&y<Y)?true:false;
15  		}
16  		static boolean ok(int x,int y,int X,int Y){	return ok(x,y,X,Y,0);}
17  	}
18  	public static void main(String[] args) {
19  		new Main().AOJ0144();
20  	}
21  	void AOJ0145(){
22  		while(sc.hasNext()){
23  			int N=sc.nextInt(),ans=Integer.MAX_VALUE;
24  			C0145[] c=new C0145[N];
25  			for(int i=0; i<N; i++)	c[i]=new C0145(sc.nextInt(),sc.nextInt());
26  			out.println(ans);
27  		}
28  	}
29  	class C0145{
30  		int upper,down;
31  		C0145(int u,int d){	this.upper=u;	this.down=d;	}
32  	}
33  	void AOJ0147(){
34  		LinkedList<Integer> queue=new LinkedList<Integer>();
35  		int MAX=120;
36  		while(sc.hasNext()){
37  			int temp=sc.nextInt();
38  			MAX=max(MAX,temp);
39  			queue.add(temp);
40  		}
41  		int[] num=new int[MAX],time=new int[MAX],seat=new int[17];
42  		Arrays.fill(seat, -1);
43  		for(int i=0; i<MAX; i++){
44  			num[i]=(i%5==1?5:2);
45  			time[i]=(17*(i%2)+3*(i%3)+19);
46  		}
47  		int j=0,min=0;
48  		while(!queue.isEmpty()){
49  			seat=exit(seat,min,time);
50  			int temp=find(seat,num[j]);
51  			if(temp>=0 && min>=j*5){
52  				int q=queue.getFirst();
53  				if(j==q){
54  					out.println(max(0,min-j*5));
55  					queue.removeFirst();
56  				}
57  				for(int i=temp; i<temp+num[j]; i++)	seat[i]=j;
58  				j++;
59  			}else{
60  				min++;
61  			}
62  		}
63  	}
64  	int find(int[] seat,int n){
65  		for(int i=0; i<17-n; i++){
66  			boolean flag=true;
67  			for(int j=0; j<n; j++){
68  				if(seat[i+j]!=-1){
69  					flag=false;
70  					break;
71  				}
72  			}
73  			if(flag)	return i;
74  		}
75  		return -1;
76  	}
77  	int[] exit(int[] seat,int min,int[] time){
78  		for(int i=0; i<17; i++){
79  			if(seat[i]<0)	continue;
80  			int idx=seat[i],emin=idx*5+time[seat[i]];
81  			if(min>=emin)	seat[i]=-1;
82  		}
83  		return seat;
84  	}
85  	void disp(int[] seat,int min){
86  		out.print(min+"m : ");
87  		for(int i=0; i<17; i++)	out.print(seat[i]+" ");
88  		out.println();
89  	}
90  	void AOJ0144(){
91  		while(sc.hasNext()){
92  			int N=sc.nextInt();
93  			boolean[][] R=new boolean[N+1][N+1];
94  			for(int i=1; i<=N; i++){
95  				int r=sc.nextInt(),k=sc.nextInt();
96  				for(int j=0; j<k; j++)	
97  					R[r][sc.nextInt()]=true;
98  			}
99  			int P=sc.nextInt();
100  			for(int i=0; i<P; i++){
101  				int s=sc.nextInt(),d=sc.nextInt(),v=sc.nextInt(),ans=-1;
102  				PriorityQueue<C0144> open=new PriorityQueue<C0144>();
103  				open.add(new C0144(s,v,1));
104  				C0144[] close=new C0144[N+1];
105  				while(!open.isEmpty()){
106  					C0144 now=open.poll();
107  					if(--now.ttl==0)	continue;
108  					for(int j=1; j<=N; j++){
109  						if(R[now.now][j]){
110  							if(close[j]==null){
111  								if(j==d){
112  									ans=(ans<0?now.c+1:min(ans,now.c+1));
113  									continue;
114  								}
115  								open.add(new C0144(j,now.ttl,now.c+1));
116  								close[j]=new C0144(j,now.ttl,now.c+1);
117  							}else{
118  								if(now.c+1<close[j].c || now.ttl>close[j].ttl){
119  									if(j==d){
120  										ans=(ans<0?now.c+1:min(ans,now.c+1));
121  										continue;
122  									}
123  									open.add(new C0144(j,now.ttl,now.c+1));
124  									close[j]=new C0144(j,now.ttl,now.c+1);
125  								}
126  							}
127  						}
128  					}
129  				}
130  				out.println((ans<0?"NA":ans));
131  			}
132  		}
133  	}
134  	class C0144 implements Comparable<C0144>{
135  		int now,ttl,c;
136  		C0144(int now,int ttl,int c){
137  			this.now=now;	this.ttl=ttl;	this.c=c;
138  		}
139  		@Override public int compareTo(C0144 o) {
140  			if(this.c<o.c)	return -1;
141  			if(this.c>o.c)	return 1;
142  			return 0;
143  		}
144  	}
145  }
```


**Question**: Does `(j,108)` have data dependence over `(open,115)` in function `AOJ0144`? If so, provide a trace.

**Output**:


Ground truth answer: yes