Solving Sudoku like humans do

.. is not difficult, but intriguing.

Usually solved with blind back tracking. But it felt soul-less. I want to solve like humans do.

Step 1: Solve each cell which can be guessed with 100% probability.

Step 2: Pick one of the available options and move to next.

Step 3: If nothing matches, back-track and change one of previous cells done in step 2. If reached start still nothing – no solution found.

The whole program is at https://github.com/chavakiranasdev/kichavaLibrary/blob/master/leet/hard/sudoku/Sudoku/Solution.cs

It is written and tested on Leetcode, so we ended up converting input to our object and reconverted back to required format.

CellState

/// <summary>
/// Represents state of single Cell. 
/// </summary>
internal enum CellState
{
    Empty, // Not yet filled. 

    InitialValue, // This value is already given 

    SolvedWithConfidence, // We found it with 100% probability. 

    InProgress, // Currently in progress. 
}

CellLocation

/// <summary>
/// Represents a single cell location. 
/// Used during backtracking to remember path
/// </summary>
internal class CellLocation
{
    public int Row { set; get; }
    public int Column { set; get; }
}

Cell

/// <summary>
/// Represents a cell 
/// </summary>
internal class Cell
{
    public CellState State { get; set; }
    public int CurrentValue { get; set; }

    // We tried this number & it is not fit here.
    // Used during backtracking to not re-use discarded values.
    public List<int> NotFitForThisCell { get; set; }

    public Cell()
    {
        State = CellState.Empty;
        NotFitForThisCell = new List<int>();
    }

    public void Reset()
    {
        CurrentValue = 0;
        State = CellState.Empty;
    }

    public void ClearNotFitForThis()
    {
        NotFitForThisCell.Clear();
    }
}

Find a match with 100% probability

// This checks if there is only one fit for this cell. 
// If it finds more than one fit, then we return zero.
private int GetBestFit(int row, int column)
{
    var gridKey = GetGridNumber(row, column);
    var availableInAllThree = rowAvailableList[row].Intersect(columnAvailableList[column]).Intersect(gridAvailableList[gridKey]);
    if (availableInAllThree.Count() == 1)
    {
        return availableInAllThree.First();
    }
    return 0;
}

Datastructures

// Constants 
// we can generally solve sudoku of any size. 
// So far, tested only for size 9
private int SizeOfSudoku { get; set; }

// Private variables. 
// Later, we create a board of given size (e.g.. 9X9)
private Cell[,] cellBoard;

// Next three dictionaries (hash's) represent each row, column, grid and still available values at any instance. 
// For example, once we initialize with input values, we remove them from corresponding row, column, grid. 
// Instead of three hashe's we could use 9X9 hashes to do same for each cell. That should be good for perf, for now this what I used.
Dictionary<int, List<int>> rowAvailableList = new Dictionary<int, List<int>>();
Dictionary<int, List<int>> columnAvailableList = new Dictionary<int, List<int>>();
Dictionary<int, List<int>> gridAvailableList = new Dictionary<int, List<int>>();

Constructor

public Solution(int sizeOfSudoku = 9)
{
    SizeOfSudoku = sizeOfSudoku;
    cellBoard = new Cell[SizeOfSudoku, SizeOfSudoku];
}

Solution

public void SolveSudoku(char[][] board)
{
    Console.WriteLine("Start...");
    InitializeAllAvailableLiistsWithEverything();
    // Initialize given char array into our cell array. 
    InitializeCellArray(board);

    // First round - find all those which doesn't need guessing.
    bool isAtleastOneBestFitFound;
    int currentRow;
    int currentColumn;
    do
    {
        isAtleastOneBestFitFound = false;
        // Solve each row. 
        currentRow = 0;
        while (currentRow < SizeOfSudoku)
        {
            currentColumn = 0;
            while (currentColumn < SizeOfSudoku)
            {
                if (cellBoard[currentRow, currentColumn].CurrentValue == 0)
                {
                    var bestFit = GetBestFit(currentRow, currentColumn);
                    if (bestFit != 0)
                    {
                        // We have a match. Update the cell with this value.
                        isAtleastOneBestFitFound = true;
                        UpdateCellValue(currentRow, currentColumn, bestFit, CellState.SolvedWithConfidence);
                    }
                }
                currentColumn++;
            }
            currentRow++;
        }
    } while (isAtleastOneBestFitFound);

    // Second round - Try to do best guess
    bool response = true;
    response = SolveByRow();
    if (response == false)
    {
        Console.WriteLine("Bug: Unable to solve given Sudoku");
    }

    // we should be done now. 
    // Just put final result into char array instead of cell object array
    for (int i = 0; i < SizeOfSudoku; i++)
    {
        for (int j = 0; j < SizeOfSudoku; j++)
        {
            board[i][j] = Convert.ToChar(cellBoard[i, j].CurrentValue + 48);
        }
    }
}

SovleByRow

 private bool SolveByRow()
    {
        bool response = true;
        if (!SolveGivenSubBlock(rowStart: 0, rowEnd: SizeOfSudoku - 1, columnStart: 0, columnEnd: SizeOfSudoku - 1))
        {
            response = false;
            Console.WriteLine("Bug? Unable to solve some of the grid");
        }
        return response;
    }

SolveGivenSubBlock

 private bool SolveGivenSubBlock(int rowStart, int rowEnd, int columnStart, int columnEnd)
    {
        bool response = true;
        for (int currentRow = rowStart; currentRow <= rowEnd; currentRow++)
        {
            for (int currentColumn = columnStart; currentColumn <= columnEnd; currentColumn++)
            {
                if (!SolveGivenCell(currentRow, currentColumn))
                {
                    response = false; // do not overwrite this value for next cell's success, and thus above if
                }
            }
        }
        return response;
    }

SolveGivenCell

  private bool SolveGivenCell(int currentRow, int currentColumn)
    {
        bool response = true;
        var currentCell = cellBoard[currentRow, currentColumn];
        if (currentCell.CurrentValue == 0)
        {
            // go back tracking. 
            // This will first attempt to find best guess if not found then back tracks.
            if (!BackTrackRealThisTime(currentRow, currentColumn))
            {
                response = false;
                Console.WriteLine($"Bug? Unable to solve {currentRow} :: {currentColumn}");
            }
        }
        return response;
    }

Backtracking

 private bool BackTrackRealThisTime(int currentRow, int currentColumn)
    {
        var stack = new Stack<CellLocation>();
        // clear not fit for this, if any from prevous attempts.
        cellBoard[currentRow, currentColumn].ClearNotFitForThis();
        stack.Push(new CellLocation() { Row = currentRow, Column = currentColumn });

        while (stack.Count != 0)
        {
            var currentTopOfStack = stack.Peek();
            var bestGuess = GetBestGuess(currentTopOfStack.Row, currentTopOfStack.Column);
            if (bestGuess != 0)
            {
                // We have a guess. Update the cell with this value.
                UpdateCellValue(currentTopOfStack.Row, currentTopOfStack.Column, bestGuess, CellState.InProgress);
                stack.Pop(); // Remove this from stack.
                continue;
            }
            var previousCellLocation = GetPreviousInProgressCellLocation(currentTopOfStack);
            // Push this cell onto stack after emptying it.;
            var previousCell = cellBoard[previousCellLocation.Row, previousCellLocation.Column];
            previousCell.NotFitForThisCell.Add(previousCell.CurrentValue);
            RemoveCellValue(previousCellLocation.Row, previousCellLocation.Column);
            // Now that we are moving further back, current cell's notfit is no longer valid.
            cellBoard[currentTopOfStack.Row, currentTopOfStack.Column].ClearNotFitForThis();
            stack.Push(previousCellLocation);
        }
        return true; // stack is empty
    }

Conclusion

I tried to be lazy and backtrack in random, without remembering path – but on my machine it works, leetcode always timed out. Forcing me to be a better 🙂

Get rest of the code from above github link.

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