[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.*;
2   import java.util.*;
3   public class Main {
4   	private static Scanner sc;
5   	private static Printer pr;
6   	private static void solve() {
7   		int h = sc.nextInt();
8   		int w = sc.nextInt();
9   		int[][] a = new int[h][w];
10  		for (int i = 0; i < h; i++) {
11  			for (int j = 0; j < w; j++) {
12  				a[i][j] = sc.nextInt();
13  			}
14  		}
15  		List<Integer> x = new ArrayList<>();
16  		List<Integer> y = new ArrayList<>();
17  		boolean flag = false;
18  		int prex = 0;
19  		int prey = 0;
20  		boolean odd;
21  		for (int i = 0; i < h; i++) {
22  			if (i % 2 == 1) {
23  				for (int j = w - 1; j >= 0; j--) {
24  					if (a[i][j] % 2 == 1) {
25  						odd = true;
26  					} else {
27  						odd = false;
28  					}
29  					if (odd && flag) {
30  						flag = !flag;
31  						x.add(prex);
32  						y.add(prey);
33  						x.add(j);
34  						y.add(i);
35  						prex = j;
36  						prey = i;
37  					} else if (odd && !flag) {
38  						flag = !flag;
39  						prex = j;
40  						prey = i;
41  					} else if (!odd && flag) {
42  						x.add(prex);
43  						y.add(prey);
44  						x.add(j);
45  						y.add(i);
46  						prex = j;
47  						prey = i;
48  					} else if (!odd && !flag) {
49  					}
50  				}
51  			} else {
52  				for (int j = 0; j < w; j++) {
53  					if (a[i][j] % 2 == 1) {
54  						odd = true;
55  					} else {
56  						odd = false;
57  					}
58  					if (odd && flag) {
59  						flag = !flag;
60  						x.add(prex);
61  						y.add(prey);
62  						x.add(j);
63  						y.add(i);
64  						prex = j;
65  						prey = i;
66  					} else if (odd && !flag) {
67  						flag = !flag;
68  						prex = j;
69  						prey = i;
70  					} else if (!odd && flag) {
71  						x.add(prex);
72  						y.add(prey);
73  						x.add(j);
74  						y.add(i);
75  						prex = j;
76  						prey = i;
77  					} else if (!odd && !flag) {
78  					}
79  				}
80  			}
81  		}
82  		pr.println(x.size() / 2);
83  		for (int i = 0, size = x.size(); i < size / 2 * 2; i++) {
84  			if (i % 2 == 0) {
85  				StringBuilder sb = new StringBuilder();
86  				sb.append(y.get(i) + 1);
87  				sb.append(' ');
88  				sb.append(x.get(i) + 1);
89  				sb.append(' ');
90  				pr.print(sb.toString());
91  			} else {
92  				StringBuilder sb = new StringBuilder();
93  				sb.append(y.get(i) + 1);
94  				sb.append(' ');
95  				sb.append(x.get(i) + 1);
96  				pr.println(sb.toString());
97  			}
98  		}
99  	}
100  	public static void main(String[] args) {
101  		sc = new Scanner(System.in);
102  		pr = new Printer(System.out);
103  		solve();
104  		pr.close();
105  		sc.close();
106  	}
107  	static class Scanner {
108  		BufferedReader br;
109  		Scanner (InputStream in) {
110  			br = new BufferedReader(new InputStreamReader(in));
111  		}
112  		private boolean isPrintable(int ch) {
113  			return ch >= '!' && ch <= '~';
114  		}
115  		private boolean isCRLF(int ch) {
116  			return ch == '\n' || ch == '\r' || ch == -1;
117  		}
118  		private int nextPrintable() {
119  			try {
120  				int ch;
121  				while (!isPrintable(ch = br.read())) {
122  					if (ch == -1) {
123  						throw new NoSuchElementException();
124  					}
125  				}
126  				return ch;
127  			} catch (IOException e) {
128  				throw new NoSuchElementException();
129  			}
130  		}
131  		String next() {
132  			try {
133  				int ch = nextPrintable();
134  				StringBuilder sb = new StringBuilder();
135  				do {
136  					sb.appendCodePoint(ch);
137  				} while (isPrintable(ch = br.read()));
138  				return sb.toString();
139  			} catch (IOException e) {
140  				throw new NoSuchElementException();
141  			}
142  		}
143  		int nextInt() {
144  			try {
145  				boolean negative = false;
146  				int res = 0;
147  				int limit = -Integer.MAX_VALUE;
148  				int radix = 10;
149  				int fc = nextPrintable();
150  				if (fc < '0') {
151  					if (fc == '-') {
152  						negative = true;
153  						limit = Integer.MIN_VALUE;
154  					} else if (fc != '+') {
155  						throw new NumberFormatException();
156  					}
157  					fc = br.read();
158  				}
159  				int multmin = limit / radix;
160  				int ch = fc;
161  				do {
162  					int digit = ch - '0';
163  					if (digit < 0 || digit >= radix) {
164  						throw new NumberFormatException();
165  					}
166  					if (res < multmin) {
167  						throw new NumberFormatException();
168  					}
169  					res *= radix;
170  					if (res < limit + digit) {
171  						throw new NumberFormatException();
172  					}
173  					res -= digit;
174  				} while (isPrintable(ch = br.read()));
175  				return negative ? res : -res;
176  			} catch (IOException e) {
177  				throw new NoSuchElementException();
178  			}
179  		}
180  		long nextLong() {
181  			try {
182  				boolean negative = false;
183  				long res = 0;
184  				long limit = -Long.MAX_VALUE;
185  				int radix = 10;
186  				int fc = nextPrintable();
187  				if (fc < '0') {
188  					if (fc == '-') {
189  						negative = true;
190  						limit = Long.MIN_VALUE;
191  					} else if (fc != '+') {
192  						throw new NumberFormatException();
193  					}
194  					fc = br.read();
195  				}
196  				long multmin = limit / radix;
197  				int ch = fc;
198  				do {
199  					int digit = ch - '0';
200  					if (digit < 0 || digit >= radix) {
201  						throw new NumberFormatException();
202  					}
203  					if (res < multmin) {
204  						throw new NumberFormatException();
205  					}
206  					res *= radix;
207  					if (res < limit + digit) {
208  						throw new NumberFormatException();
209  					}
210  					res -= digit;
211  				} while (isPrintable(ch = br.read()));
212  				return negative ? res : -res;
213  			} catch (IOException e) {
214  				throw new NoSuchElementException();
215  			}
216  		}
217  		float nextFloat() {
218  			return Float.parseFloat(next());
219  		}
220  		double nextDouble() {
221  			return Double.parseDouble(next());
222  		}
223  		String nextLine() {
224  			try {
225  				int ch;
226  				while (isCRLF(ch = br.read())) {
227  					if (ch == -1) {
228  						throw new NoSuchElementException();
229  					}
230  				}
231  				StringBuilder sb = new StringBuilder();
232  				do {
233  					sb.appendCodePoint(ch);
234  				} while (!isCRLF(ch = br.read()));
235  				return sb.toString();
236  			} catch (IOException e) {
237  				throw new NoSuchElementException();
238  			}
239  		}
240  		void close() {
241  			try {
242  				br.close();
243  			} catch (IOException e) {
244  			}
245  		}
246  	}
247  	static class Printer extends PrintWriter {
248  		Printer(OutputStream out) {
249  			super(out);
250  		}
251  	}
252  }
```


**Question**: Does `(x,31)` have data dependence over `(x,73)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes