[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.io.IOException;
2   import java.io.InputStream;
3   import java.util.NoSuchElementException;
4   public class Main {
5       private static FastScanner sc = new FastScanner();
6       private static boolean DEBUG_FLG = false;
7       public static void main(String[] args) {
8       	int N = sc.nextInt();
9       	long A = sc.nextLong();
10      	long B = sc.nextLong();
11      	int[] p = new int[N];
12      	for(int i=0; i<N; i++) {
13      		p[i] = sc.nextInt();
14      	}
15      	long[][] dpc = new long[N][N];
16      	int[][] dpmin = new int[N][N];
17      	int[][] dpmin2 = new int[N][N];
18      	int[][] dpmax = new int[N][N];
19      	int[][] dpmax2 = new int[N][N];
20      	for(int i=0; i<N; i++) {
21      		dpc[i][i] = 0;
22      		dpmin[i][i] = p[i];
23      		dpmax[i][i] = p[i];
24      		dpmin2[i][i] = Integer.MAX_VALUE;
25      		dpmax2[i][i] = -1;
26      	}
27      	for(int d=1; d<N; d++) {
28      		for(int i=0; i<N-d; i++) {
29      			if(p[i+d] < dpmin[i][i+d-1]) {
30      				dpmin2[i][i+d] = dpmin[i][i+d-1];
31      				dpmin[i][i+d] = p[i+d];
32      			} else if(p[i+d] < dpmin2[i][i+d-1]) {
33      				dpmin2[i][i+d] = p[i+d];
34      				dpmin[i][i+d] = dpmin[i][i+d-1];
35      			} else {
36      				dpmin2[i][i+d] = dpmin2[i][i+d-1];
37      				dpmin[i][i+d] = dpmin[i][i+d-1];
38      			}
39      			if(p[i+d] > dpmax[i][i+d-1]) {
40      				dpmax2[i][i+d] = dpmax[i][i+d-1];
41      				dpmax[i][i+d] = p[i+d];
42      			} else if(p[i+d] > dpmax2[i][i+d-1]) {
43      				dpmax2[i][i+d] = p[i+d];
44      				dpmax[i][i+d] = dpmax[i][i+d-1];
45      			} else {
46      				dpmax2[i][i+d] = dpmax2[i][i+d-1];
47      				dpmax[i][i+d] = dpmax[i][i+d-1];
48      			}
49      			long rightcost = 0;
50      			if(p[i+d] > dpmax[i][i+d-1]) {
51      				rightcost = dpc[i][i+d-1];
52      			} else if(p[i+d] >  dpmax2[i][i+d-1]){
53      				rightcost = dpc[i][i+d-1] + Math.min(A, B);
54      			} else {
55      				rightcost = dpc[i][i+d-1] + B;
56      			}
57      			long leftcost = 0;
58      			if(p[i] < dpmin[i+1][i+d]) {
59      				leftcost = dpc[i+1][i+d];
60      			} else if(p[i] < dpmin2[i+1][i+d]){
61      				leftcost = dpc[i+1][i+d] + Math.min(A, B);
62      			} else {
63      				leftcost = dpc[i+1][i+d] + A;
64      			}
65      			dpc[i][i+d] = Math.min(leftcost, rightcost);
66      			debug(i, i+d, leftcost, rightcost, dpmin[i][i+d], dpmin2[i][i+d], dpmax[i][i+d], dpmax2[i][i+d]);
67      		}
68      	}
69      	System.out.println(dpc[0][N-1]);
70      }
71      static void debug(long... args) {
72      	if(!DEBUG_FLG) return;
73      	boolean flg = false;
74      	for(long s : args) {
75      		if(flg) System.out.print(" ");
76      		flg = true;
77      		System.out.print(s);
78      	}
79      	System.out.println();
80      }
81      static class FastScanner {
82          private final InputStream in = System.in;
83          private final byte[] buffer = new byte[1024];
84          private int ptr = 0;
85          private int buflen = 0;
86          private boolean hasNextByte() {
87              if(ptr < buflen) {
88                  return true;
89              } else {
90                  ptr = 0;
91                  try {
92                      buflen = in.read(buffer);
93                  } catch(IOException e) {
94                      e.printStackTrace();
95                  }
96                  if(buflen <= 0) {
97                      return false;
98                  }
99              }
100              return true;
101          }
102          private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
103          private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
104          private void skipUnprintable() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;}
105          public boolean hasNext() { skipUnprintable(); return hasNextByte();}
106          public String next() {
107              if (!hasNext()) throw new NoSuchElementException();
108              StringBuilder sb = new StringBuilder();
109              int b = readByte();
110              while(isPrintableChar(b)) {
111                  sb.appendCodePoint(b);
112                  b = readByte();
113              }
114              return sb.toString();
115          }
116          public long nextLong() {
117              return Long.parseLong(next());
118          }
119          public int nextInt(){
120              return Integer.parseInt(next());
121          }
122          public double nextDouble(){
123              return Double.parseDouble(next());
124          }
125      }
126  }
```


**Question**: Does `(i,20)` have data dependence over `(dpmax,47)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes