[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<stdbool.h>
5   #include<time.h>
6   #include<assert.h>
7   #define inf 1072114514
8   #define llinf 4154118101919364364
9   #define mod 1000000007
10  #define pi 3.1415926535897932384
11  int max(int a,int b){if(a>b){return a;}return b;}
12  int min(int a,int b){if(a<b){return a;}return b;}
13  int zt(int a,int b){return max(a,b)-min(a,b);}
14  int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
15  int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
16  int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
17  int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
18  int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
19  int nHr(int a,int b){return nCr(a+b-1,b);}
20  int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
21  int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
22  int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
23  int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
24  int sankaku(int x){return ((1+x)*x)/2;}
25  void swap(int *a,int *b){int c;c=(*a);(*a)=(*b);(*b)=c;}
26  long long llmax(long long a,long long b){if(a>b){return a;}return b;}
27  long long llmin(long long a,long long b){if(a<b){return a;}return b;}
28  long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
29  long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
30  long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
31  long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
32  long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
33  long long llnCr(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
34  long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
35  long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
36  long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
37  long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
38  long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
39  long long llsankaku(long long x){return ((1+x)*x)/2;}
40  void llswap(long long *a,long long *b){long long c;c=(*a);(*a)=(*b);(*b)=c;}
41  double dbmax(double a,double b){if(a>b){return a;}return b;}
42  double dbmin(double a,double b){if(a<b){return a;}return b;}
43  double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
44  void dbswap(double *a,double *b){double c;c=(*a);(*a)=(*b);(*b)=c;}
45  void chswap(char *a,char *b){char c;c=(*a);(*a)=(*b);(*b)=c;}
46  int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
47  int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
48  int llsortfncsj(const void *a,const void *b){if(*(long long *)a>*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
49  int llsortfnckj(const void *a,const void *b){if(*(long long *)a<*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
50  int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
51  int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
52  int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
53  int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
54  int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
55  int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
56  void shuffledget(int x[],int n){
57      int i,b[524288],p,c;
58      for(i=0;i<n;i++){
59          b[i]=i;
60      }
61      for(i=n;i>=1;i--){
62          p=rand()%i;
63          c=b[i-1];b[i-1]=b[p];b[p]=c;
64      }
65      for(i=0;i<n;i++){
66          scanf("%d",&x[b[i]]);
67      }
68  }
69  int dx4[4]={1,-1,0,0};
70  int dy4[4]={0,0,1,-1};
71  int dx8[8]={-1,-1,-1,0,0,1,1,1};
72  int dy8[8]={-1,0,1,-1,1,-1,0,1};
73  int search(int x,int a[],int n){
74      int st=0,fi=n-1,te;
75      while(st<=fi){
76          te=(st+fi)/2;
77          if(a[te]<x){st=te+1;}else{fi=te-1;}
78      }
79      return st;
80  }
81  void prarr(int arr[],int n){
82    int i;
83    for(i=0;i<n;i++){
84      if(i){printf(" ");}
85      printf("%d",arr[i]);
86    }
87    printf("\n");
88    return;
89  }
90  void getperm(int a[],int n){
91    int i,p;
92    for(i=0;i<n;i++){
93      a[i]=i;
94    }
95    for(i=n-1;i>=1;i--){
96      p=rand()%(i+1);
97      swap(&a[p],&a[i]);
98    }
99  }
100  typedef struct{
101  int val;
102  int node;
103  }sd;
104  int sdsortfnc(const void *a,const void *b){
105  if(((sd*)a)->val < ((sd*)b)->val){return -1;}
106  if(((sd*)a)->val > ((sd*)b)->val){return 1;}
107  return 0;
108  }
109  void coordinate_comp(int a[],int n){
110    int i,c=0;
111    sd dat[524288];
112    for(i=0;i<n;i++){
113      dat[i].val=a[i];
114      dat[i].node=i;
115    }
116    qsort(dat,n,sizeof(dat[0]),sdsortfnc);
117    a[dat[0].node]=c;
118    for(i=1;i<n;i++){
119      if(dat[i-1].val!=dat[i].val){c++;}
120      a[dat[i].node]=c;
121    }
122  }
123  int main(void){
124    int cs;
125    int i,j,n,m,k,a,b,c,h,w,l,t,sn;
126    int r=0;
127    int fl[64];
128    scanf("%d",&t);
129    for(cs=1;cs<=t;cs++){
130      scanf("%d",&n);
131      sn=n;
132      printf("Case #%d:\n",cs);
133      w=0;
134      for(i=1;i<50;i++){
135        w+=(1<<(i-1));
136        if(w>=n){
137          l=i;
138          break;
139        }
140      }
141      for(i=l;i>=1;i--){
142        if(n>=(1<<(i-1))+(i-1)){
143          fl[i]=1;
144          n-=(1<<(i-1));
145        }
146        else{
147          fl[i]=0;
148          n--;
149          }
150      }
151      a=0;
152      w=0;
153      for(i=1;i<=l;i++){
154        if(fl[i]==1){
155          w+=(1<<(i-1));
156          if(a==0){
157            for(j=1;j<=i;j++){
158              printf("%d %d\n",i,j);
159            }
160          }
161          else{
162            for(j=i;j>=1;j--){
163              printf("%d %d\n",i,j);
164            }
165          }
166          a=1-a;
167        }
168        else{
169          w++;
170          if(a==0)
171            printf("%d %d\n",i,1);
172          else
173            printf("%d %d\n",i,i);
174        }
175      }
176      while(w!=sn){
177        l++;
178        if(a==0)
179          printf("%d %d\n",l,1);
180        else
181          printf("%d %d\n",l,l);
182        w++;
183      }
184    }
185    return 0;
186  }
```


**Question**: Does `(l,137)` have data dependence over `(n,144)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes