Recursive addition system with unique replacements (LUT)

📄 Technical Specification

Name: Construction of a Recursive Addition System with Unique Replacements (LUT)

🎯 Goal:

Implement a program in C that:

Accepts 1000 numbers as input in the range from 1 to 4.

Gradually adds pairs of numbers, creating a new sequence of smaller size.

Avoids ambiguities (conflicts) by replacing repeated results with unique new values.

Records all addition operations in a text table format a+b=c, stored in the file (lut_output.txt).

Repeats these steps until only one final number remains.

---

📥 Input Data:

One array of 1000 integers.

Allowed values: 1, 2, 3, 4.

It is possible to generate the array randomly in the code.

---

⚙️ Algorithm:

1. The initial array (1000 elements) is divided into 2 subarrays of equal length.

2. Each pair a[i] + b[i] is added.

3. If such a pair already existed earlier at this level or previous levels — it is considered a conflict (e.g., 1+2 and 2+1 = 3, i.e., non-unique).

Replace the result with a unique number that has not been used yet (e.g., 9, 10, 11…).

4. All operations are recorded as:

1+2=3

2+1=9

3+3=10

(i.e., a+b=c, one per line).

5. The new array is formed from the results (c) and the same steps are repeated.

6. Repeats until only one number remains.

7. At each level, the conflict table (LUT) is updated, and no pair a+b should have different results.

---

📄 Output Format:

File: lut_output.txt

Each line: a+b=c

All levels (and all operations) are stored in one file, without duplicates.

---

⚠️ Important Requirements:

The algorithm must be as fast as possible (optimized for memory and speed).

Avoid unnecessary calculations (e.g., repeated lookups in tables — use arrays/hashes).

Errors in recording the LUT are not allowed — previous similar projects had errors with identical pairs.

Guarantee that all conflicts are accounted for and each pair a+b has a unique result at its level.

The algorithm should be reconstructible from the file back to the original sequence.

---

🔁 Additional:

If a pair sum is conflicting — assign the next free number that has not appeared in the current or previous levels.

Be sure to record this replacement in the LUT.

---

✅ Final Result:

The single final number after all reduction steps.

The complete log of changes and replacements in lut_output.txt.

---

📌 Note:

This task is part of another system — the table (lut_output.txt) will be used in another program, so it must be accurate and complete.

The table records only conflicting pairs, not all combinations.

All other operations should proceed normally, using simple mathematics — 1+1 etc.