Step 1: Input Parsing

1. Read Input File: Parse a grid of size NxN from input files. Values may include:

   • Pre-filled numbers.
   • Empty cells represented by a placeholder (e.g., -).

2. Initialize Data Structures:

   • A 2D grid Grid to represent the puzzle.
   • A DomainGrid to store sets of possible values for each cell.
   • Track placedCells (pre-filled numbers and their coordinates).
   • A set possibleNumbers representing all numbers from 1 to N*N.

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Step 2: Domain Initialization

1. Domain Reduction:

   • Use updateDomainGrid to calculate valid numbers for empty cells.
   • Place constraints based on Manhattan distances (vertical/horizontal) and diagonal distances.

2. Identify Missing Numbers:

   • Analyze the placedCells and determine which numbers are missing.

3. Propagate Constraints:

   • For each missing number:
     • Determine its closest pre-filled number.
     • Update the domains for cells within valid distances.

4. Chains:

   • Detect chains of consecutive numbers:
     • If a number has only one valid position, place it.
     • Optimize domains by removing invalid values from other cells.

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Step 3: Path Consistency and Recoverability

1. Path Consistency:

   • Ensure adjacent numbers (value ±1) can be placed:
     • Check both immediate neighbors and potential future placements.

2. Recoverability:

   • Verify that each placed number can connect to its successor and predecessor.

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Step 4: Evaluation Function

1. Scoring Criteria:

   • Reward continuity (chains of consecutive numbers).
   • Penalize isolated numbers.
   • Reward cells with forced moves (single-option domains).
   • Apply penalties for large domains (cells with many possibilities).

2. Purpose:

   • Guide the solver to prioritize the most promising moves.

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Step 5: Alpha-Beta Search (Recursive Solver)

1. Branching:

   • Find the cell with the smallest domain size (fewest possibilities).
   • For each candidate value:
     • Check path consistency before placing it.
     • Evaluate the resulting grid using the scoring function.

2. Pruning:

   • Use alpha-beta pruning to cut off unpromising branches:
     • If a candidate's score is worse than alpha, prune.
     • Update alpha when a better score is found.

3. Backtracking:

   • Recursively attempt to solve the grid.
   • If no solution exists for the current branch, backtrack to explore alternatives.

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Step 6: Solution Verification

1. Check for Completeness:

   • The solver stops when all cells are filled with valid numbers (1 to N*N).

2. Validate Solution:

   • Compare the generated solution against the expected solution (if available in the input).

3. Output:

   • Print the grid, execution time, and whether the solution is correct.

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Key Techniques Used

1. Constraint Satisfaction Problem (CSP):

   • The problem is modeled as a grid where constraints enforce number placement.

2. Alpha-Beta Pruning:

   • Optimizes the search space to avoid redundant computations.

3. Heuristic Evaluation:

   • A scoring function guides the solver to prioritize promising moves.

4. Path Consistency:

   • Ensures the solver maintains valid number placements at each step.

5. Backtracking with Domain Propagation:

   • Incrementally explores possibilities while updating domains to maintain consistency.

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Overall Workflow:

1. Parse input and initialize domains.
2. Propagate constraints to reduce the search space.
3. Use recursive alpha-beta search with heuristics to explore potential solutions.
4. Verify the solution and print the results.