# Sudoku Week-End Challenge - Brute-Force Recursive solver

This is part of my attempt at the Week-End Challenge #3. The overall problem is larger than will fit in one question. This is a well-contained subset of my larger program.

The goal of this part is to brute-force all possible (if any) solutions to a puzzle. The input to this algorithm is a 2-dimensional array representing the Sudoku grid. This array is processed so that, at each point in the array, there is a 'set' of 'possible' digits. For example, at a position (r,c) in the grid, if there is the set of values [ 1, 2, 3 ] it would mean that, at position (r,c) the puzzle could possibly be a 2, 3, or 4. (The solution assumes a zero-based counting system, whereas traditional Sudoku's use a 1-based counting system).

If the set of digits at (r,c) contains just a single value then the assumption is that the cell at that position is 'solved'.

It is assumed that the input grid is 'consistent' with the rules of Sudoku.... that a solved value occurs only once in any row/column/block combination, and that no solved value appears anywhere in the 'potential' set of values for any other cell in the same row/column/block.

I have tried numerous algorithms for forcing a solution, and this is the fastest I tried (by a long shot), but the algorithm is complicated....

I have a tendency to reduce problems down to primitive-value manipulation, and that tends to make solutions harder to read (even though they may be faster).

What I would like is suggestions for how to improve this code by expressing the algorithm more clearly (make the code look more like the algorithm) while not horribly impacting the performance. In other words, I do not believe there are significantly faster algorithms, but performance is not everything!

I have tried to describe the algorithm in the code. Additionally, I have a 'main method' which supplies a pre-constructed Sudoku puzzle in the required input format. The actual example puzzle is the same as the one in this week's challenge.

The algorithm:

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

/**
* A class that is able to brute-force a solution for a Sudoku-like puzzle.
* <br>
* Due to the internal use of bit-shifting the largest sudoku puzzle this can solve is 25x25
* which should be more than enough. Sudoku-like puzzles are always square-number sized. 1x1, 4x4, 9x9, 16x16, 25x25, etc.
*
* @author rolfl
*
*/
public class RecursiveBruteSolver {

// dimension - the width of the grid
private final int dimension;
// a simple bit-mask with 1 bit set for each possible value in each square.
// for a 4-size sudoku it will look like:
// 0b0001 0b0010 0b0100 0b1000
// for a 9-size sudoku like:
// 0b000000001 0b000000010 0b000000100 .....
// if we flatten the grid, this is how big it is in 1 dimension
private final int flatsize;
// the array of flattened cell available digits
private final int[][] available;
// a complicated concept - the index into the flattened array of all
// squares that are in the same row/column/block, but also have a larger
// index in the flattened array.
private final int[][] followings;
// Some squares will have just one possible value, they do not need to be
// solved.
// this unknown array is the indices of the unsolved squares in the
// flattened array
private final int[] unknown;

// keep track of all recursive call counts
private long statistics = 0L;

/**
* Create a Brute-force solver that will determine all the solutions (if
* any) for a Sudoku grid (when you call the solve() method).
*
* @param dimension
*            the side-size of the sudoku grid. The digits 2D array is
*            expected to match the same dimensions.
* @param digits
*            a 3D array containing the digits that are allowed at that
*            position in the array. The actual digits available are
*            expected to be 0-based. i.e. a regular 9-size Sudoku has the
*            digits 1-9, this solver expects them to be represented as 0-8.
*/
public RecursiveBruteSolver(int dimension, int[][][] digits) {
this.dimension = dimension;

// if we flatten the array....
this.flatsize = dimension * dimension;
available = new int[flatsize][];
followings = new int[flatsize][];

// how many digits are there...and what will the bit-mask be for each digit.
for (int i = 0; i < dimension; i++) {
}

// keep track of the unknown cells.
// make a generous-size array, we will trim it later.
int[] tmpunknown = new int[flatsize];
int tmpunknowncnt = 0;

// flatten out the grid.
for (int r = 0; r < dimension; r++) {
for (int c = 0; c < dimension; c++) {
int flatid = r * dimension + c;
// gets an array of digits that can be put in each square.
// for example, if a Sudoku square can have 3,5 or 6 this will return
//    [2,4,5]
available[flatid] = Arrays.copyOf(digits[r][c], digits[r][c].length);
if (available[flatid].length > 1) {
tmpunknown[tmpunknowncnt++] = flatid;
}
}
}

//        System.out.println("There are " + tmpunknowncnt + " unknown cells");

// Special note about followings
// A square in a Sudoku puzzle affects all other squares in the same row, column, and block.
// Only one square in the same row/block/column can have particular value.
// For this recursion we 'flatten' the grid, and process the grid in order.
// For each unsolved cell, we set the cell value, and then move on to the next unsolved cell
// but, we need to 'remove' our value from the possible values of other cells in the same row/column/block.
// Because of the way this solver progresses along the flattened array, we only need to remove it
// from cells that come **after** us in the flattened array (values before us already have a fixed value).
// So, buildFollowing() builds an array of indexes in the flattened array that are in the same row/column/block
// but also have a higher index in the flattened array.
for (int flat = 0; flat < available.length; flat++) {
if (available[flat] != null) {
followings[flat] = buildFollowing(flat / dimension, flat
% dimension);
}
}

// create the final copy of the unknown cells (cells which need to be solved).
unknown = Arrays.copyOf(tmpunknown, tmpunknowncnt);
}

public int[][][] solve() {

// following points to unknown subsequent values.....
final int[] combination = new int[flatsize];
// where to store the possible solutions we find
final List<int[][]> solutions = new ArrayList<>();

// this freedoms is an integer for each value.
// bits in the integer are set for each of the values the
// position can be. This is essentially a record of the state
// inside the recursive routine. For example, if setting some other
// cell in the same row/column/block to 5, and our 5-bit is set,
// then we will unset it here because we no longer have the freedom
// to be 5.
for (int flatid = 0; flatid < flatsize; flatid++) {
for (int b : available[flatid]) {
// set the degrees-of-freedom mask of this...
// what values can this cell take?
}
}

// Do the actual recursion.
// combination contains pointers to which actual values are being used.
// freedom contans the possible states for all subsequent cells.
recurse(solutions, 0, combination, freedoms);

System.out.println("There were " + statistics + " recursive calls...");

// convert the list of Solutions back to an array
return solutions.toArray(new int[solutions.size()][][]);
}

/**
* Recursively solve the Sudoku puzzle.
* @param solutions where to store any found solutions.
* @param index The index in the 'unknown' array that points to the flat-based cell we need to solve.
* @param combination What the current combination of cell values is.
* @param freedoms The state of what potential values all cells can have.
*/
private void recurse(final List<int[][]> solutions, final int index,
final int[] combination, final int[] freedoms) {
statistics++;
if (index >= unknown.length) {
// solution!
return;
}

// The basic algorithm here is: for our unsolved square, we set it to each of it's possible values in turn.
// then, for each of the values we can be:
// 1. we also find all 'related' squares, and remove our value
//      from the degrees-of-freedom for the related squares
//      (If I am 'x' then they cannot be). See special not about 'followings'
// 3. we keep track of which other squares we actually change the freedoms for.
// 4. we recurse to the next unsolved square.
// 5. when the recursion returns, we restore the freedoms we previously 'revoked'
// 6. we move on to the next value we can be (back to 1).
// 7. when we have run out of possible values, we return.
final int flat = unknown[index];
for (int a = available[flat].length - 1; a >= 0; a--) {
final int attempt = available[flat][a];
if ((freedoms[flat] & bitmask[attempt]) == 0) {
// this option excluded by previous restrictions....
// the original unsolved puzzle says we can be 'attempt', but
// higher levels of recursion have removed 'attempt' from our
// degrees-of-freedom.
continue;
}
// ok, is used to track whether we are still creating a valid Sudoku.
boolean ok = true;
// progress is used to forward, and then backtrack which following cells
// have been impacted.
// start at -1 because we pre-increment the progress.
int progress = -1;
// act has 1 bit representing each follower we act on.
long act = 0;
while (++progress < followings[flat].length) {
// we intend to remove the attempt from this follower's freedom's
// but that will leave it with nothing, so this is not possible to do.
// ok is false, so we will start a back-up
ok = false;
break;
}
// we **can** remove the value from this follower's freedoms.
// indicate that this follower is being 'touched'.
// act will have 1 bit available for each follower we touch.
act <<= 1;
// record the pre-state of the follower's freedoms.
final int pre = freedoms[followings[flat][progress]];
// if the follower's freedoms contained the value we are revoking, then set the bit.
act |= (freedoms[followings[flat][progress]] &= ~bitmask[attempt]) == pre ? 0 : 1;
}
if (ok) {
// we have removed our digit from all followers, and the puzzle is still valid.
// indicate our combination digit....
combination[flat] = a;
// find the next unsolved.
recurse(solutions, index + 1, combination, freedoms);
}
while (--progress >= 0) {
// restore all previously revoked freedoms.
if ((act & 0x1) == 1) {
}
act >>= 1;
}
}

}

/**
* buildFollowing creates an array of references to other cells in the same
* row/column/block that also have an index **after** us in the flattened array system.
*
* @param row our row index
* @param col our column index
* @return an array of flattened indices that are in the same row/column/block as us.
*/
private int[] buildFollowing(int row, int col) {
int[] folls = new int[dimension * 3]; // possible rows/columns/blocks - 3 sets of values.
final int innerbound = (int)Math.sqrt(dimension); // 3 for size 9, 2 for size 4, 4 for size 16, etc.
// cnt is used to count the valid following indices.
int cnt = 0;
// column-bound - last column in the same block as us.
int cb = ((1 + col / innerbound) * innerbound);
// row-bound - last row in the same block as us.
int rb = ((1 + row / innerbound) * innerbound);
// get all (unsolved) indices that follow us in the same row
for (int c = col + 1; c < dimension; c++) {
// rest of row.
if (available[row * dimension + c].length > 1) {
// only need to worry about unsolved followers.
folls[cnt++] = row * dimension + c;
}
}
// get all (unsolved) indices that follow us in the same column
for (int r = row + 1; r < dimension; r++) {
if (available[r * dimension + col].length > 1) {
// only need to worry about unsolved followers.
folls[cnt++] = r * dimension + col;
}
if (r < rb) {
// if we have not 'escaped' our block, we also find other cells in
// the same block, but not our row/column.
for (int c = col + 1; c < cb; c++) {
if (available[r * dimension + c].length > 1) {
// only need to worry about unsolved followers.
folls[cnt++] = r * dimension + c;
}
}
}
}
// return just the values that were needed as followers.
return Arrays.copyOf(folls, cnt);
}

/**
* Convert the valid combination of values back to a simple int[] grid.
* @param combination the combination of unsolved values that is a valid puzzle.
* @return A Solution object representing the solution.
*/
private int[][] buildSolution(int[] combination) {
int[][] grid = new int[dimension][dimension];
for (int f = 0; f < combination.length; f++) {
grid[f / dimension][f % dimension] = available[f][combination[f]];
}
// double-check the validity of this solution (all sudoku basic rules are followed.
// throws exception if not.
//        SudokuRules.isValid(grid);
// mechanism for printing out a grid.
//        System.out.println("BruteForceRecursive found Solution:\n"
//                + GridToText.displayAsString(grid));
return grid;
}

}


It has been suggested that the following code is not 'in the spirit' of the week-end challenge because it works off a pre-processed Sudoku puzzle. This is true... I have the data pre-processed in this BruteMain class, but that is only because I wanted to make it easy to run the above algorithm in a self-contained way. The remainder of the code I wrote (i.e. not the code above) is all about solving the Sudoku puzzle as much as I can, (including parsing input, and creating the 'pre-processed' puzzle). That code is more than can fit i a single CodeReview question, so I have broken out just this brute-force code, and a simple way for others (you) to run it.

The 'main' class which can be used for testing (not much commented in here...):

public class BruteMain {

public static void main(String[] args) {
int[][][] puzzle = new int[][][] {
{ { 1, 4, 5, },{ 2, 5, 6, },{ 1, 2, 4, 5, 6, },{ 7, },{ 3, },{ 2, 4, 5, 6, },{ 0, 1, 6, },{ 0, 1, 6, },{ 8, },  },
{ { 1, 3, 5, },{ 2, 3, 5, 6, 8, },{ 0, },{ 2, 5, 6, },{ 2, 5, 6, 8, },{ 2, 5, 6, 8, },{ 1, 6, 7, },{ 1, 6, 7, },{ 4, },  },
{ { 7, },{ 6, 8, },{ 4, 6, 8, },{ 4, 6, },{ 1, },{ 0, },{ 3, },{ 5, },{ 2, },  },
{ { 6, },{ 0, 2, 3, 5, },{ 7, },{ 0, 1, 2, 3, 4, 5, },{ 2, 4, 5, },{ 1, 2, 4, 5, },{ 1, 5, },{ 8, },{ 1, 3, 5, },  },
{ { 0, 1, 3, 5, },{ 0, 2, 3, 5, 8, },{ 1, 2, 5, 8, },{ 0, 1, 2, 3, 4, 5, 6, },{ 2, 4, 5, 6, 8, },{ 1, 2, 4, 5, 6, 7, 8, },{ 1, 5, 6, 7, },{ 1, 3, 4, 6, 7, },{ 1, 3, 5, 7, },  },
{ { 1, 3, 5, },{ 4, },{ 1, 5, 8, },{ 1, 3, 5, 6, },{ 5, 6, 8, },{ 1, 5, 6, 7, 8, },{ 2, },{ 1, 3, 6, 7, },{ 0, },  },
{ { 4, 5, },{ 1, },{ 3, },{ 8, },{ 0, },{ 2, 4, 5, },{ 5, 7, },{ 2, 7, },{ 6, },  },
{ { 8, },{ 0, 5, 6, 7, },{ 5, 6, },{ 1, 2, 5, 6, },{ 2, 5, 6, },{ 1, 2, 5, 6, },{ 4, },{ 0, 1, 2, 3, 7, },{ 1, 3, 5, 7, },  },
{ { 2, },{ 0, 5, 6, },{ 4, 5, 6, },{ 1, 4, 5, 6, },{ 7, },{ 3, },{ 0, 1, 5, 8, },{ 0, 1, },{ 1, 5, },  },
};

RecursiveBruteSolver solver = new RecursiveBruteSolver(9, puzzle);
for (int[][] sol : solver.solve()) {
System.out.println("Solution:");
System.out.println(displayAsString(sol));
}
}

private static final char[][] symbols = {
"╔═╤╦╗".toCharArray(),
"║ │║║".toCharArray(),
"╟─┼╫╢".toCharArray(),
"╠═╪╬╣".toCharArray(),
"╚═╧╩╝".toCharArray(),
};

private static final char[] DIGITCHARS = "123456789ABCFEDGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz".toCharArray();

public static final char getSudokuDigit(final int value) {
if (value < 0) {
return ' ';
}
return DIGITCHARS[value];
}

public static final String displayAsString(int[][]data) {
return buildStringForGrid(data.length, (int)Math.sqrt(data.length), data);
}

private static final String buildStringForGrid(final int dimension, final int blocksize, final int[][]rows) {
final StringBuilder sb = new StringBuilder();
for (int r = 0; r < dimension; r++) {
if (r == 0) {
sb.append(printSymbolLine(dimension, blocksize, null, symbols[0]));
} else if (r % blocksize == 0) {
sb.append(printSymbolLine(dimension, blocksize, null, symbols[3]));
} else {
sb.append(printSymbolLine(dimension, blocksize, null, symbols[2]));
}
sb.append(printSymbolLine(dimension, blocksize, rows[r], symbols[1]));
}
sb.append(printSymbolLine(dimension, blocksize, null, symbols[4]));
return sb.toString();
}

private static String printSymbolLine(int dimension, int blocksize, int[] values, char[] symbols) {
StringBuilder sb = new StringBuilder();
sb.append(symbols[0]);
int vc = 0;
for (int b = 0; b < blocksize; b++) {
for (int c = 0; c < blocksize; c++) {
if (values == null) {
sb.append(symbols[1]).append(symbols[1]).append(symbols[1]).append(symbols[2]);
} else {
final int val = values[vc++];
char ch = getSudokuDigit(val);
sb.append(symbols[1]).append(ch).append(symbols[1]).append(symbols[2]);
}
}
sb.setCharAt(sb.length() - 1, symbols[3]);
}
sb.setCharAt(sb.length() - 1, symbols[4]);
sb.append("\n");
return sb.toString();
}
}


I have added in an Ideone implementation of this solution.

• For the record, I've seen 6 x 6 puzzles, where each box is 2 x 3. As long as you handle 9 x 9, though, that's OK. – 200_success Dec 15 '13 at 17:18
• Encapsulation would help you out a lot here. You're using an object oriented language, but you're barely using any object oriented features. IMHO, at the very least you should have a Cell and Puzzle class, in addition to Solver, and probably a Group class which keeps track of all the cells in a particular row, column or 3x3 square. Otherwise you might as well be writing this in C. – bcrist Dec 15 '13 at 21:58
• Also, I find that if you need tons of inline comments inside functions to make it clear what's happening, it's a pretty good sign that a function is trying to do too much, and some refactoring might be in order. – bcrist Dec 15 '13 at 22:03
• @bcrist In my defense, this code is a small part of an OO system... and the OO part of the system was very much slower than the code above. I plan on posting more code... (been out for a few hours....) . Another question is on it's way.... – rolfl Dec 15 '13 at 23:42

## buildFollowing

The logic of this this function is difficult to follow. Part of it is because of your insistence on very short names (rb, cb, etc.) with comments breaking up the flow of the code; as a rule of thumb, if you need a comment to describe what a variable does, you should probably make the name longer. Internal comments should almost always describe why, not what.

I'd consider something like this:

/**
* Compute a list of all indices **after** the one corresponding to
* row and col that are not already solved.
*
* Requires that available has already been initialized.
*/
private int[] buildFollowing(int row, int col) {
int[] following = new int[dimension * 3];
int numFollowers = 0;
// Get all followers in this row.
for (int c = col + 1; c < dimension; c++) {
int candidate = row * dimension + c;
if (available[candidate].length > 1) {
following[numFollowers++] = candidate;
}
}
// Get all followers in this column.
for (int r = row + 1; r < dimension; r++) {
int candidate = r * dimension + col;
if (available[candidate].length > 1) {
following[numFollowers++] = candidate;
}
}
// Get followers in the same block.
int blockDimension = (int) Math.sqrt(dimension);
int lastColInBlock = ((col % blockDimension) + 1) * blockDimension - 1;
int lastRowInBlock = ((row % blockDimension) + 1) * blockDimension - 1;
for (int r = row + 1; r < lastRowInBlock; r++) {
for (int c = col + 1; c < lastColInBlock; c++) {
int candidate = r * dimension + c;
if (available[candidate].length > 1) {
following[numFollowers++] = candidate;
}
}
}
return Arrays.copyOf(following, numFollowers);
}


This function is divided into "paragraphs," where the purpose of each block is clear. rb and cb are renamed to lastRowInBlock and lastColInBlock, and they're not declared until they're used.

## recurse

Why is the for loop counting down instead of up? When I see that pattern, I expect that there's a reason for it, but there doesn't seem to be here. As with buildFollowing, this function suffers a bit from superfluous comments and oddly named variables. Let me take a crack:

/**
* Recursively solve the Sudoku puzzle.
*
* @param solutions All solutions found so far.
* @param unknownIndex The current element of unknown we're solving on.
* @param partialSolution This array contains the partial solution selected
*     so far for all cells prior to unknowns[unknownIndex]; the solution
*     for cell (i / dimension, i % dimension) is
*     available[partialSolution[i]].
* @param stillAvailable (stillAvailable[i] & (1 << v)) is 1 iff
*     v has not been eliminated for cell (i / dimension, i % dimension)
*     by any part of partialSolution.
*/
private void recurse(List<int[][]> solutions, int unknownIndex,
int[] partialSolution, int[] stillAvailable) {
if (unknownIndex >= unknowns.length) {
return;
}
int index = unknowns[unknownIndex];
for (int a = 0; a < available[index.length]; a++) {
int candidate = available[a];
long appliedConstraints = applyConstraints(
index, candidate, stillAvailable);
if (appliedConstraints < 0) {
// No solution is possible with this cell set to candidate.
continue;
}
partialSolution[index] = a;
recurse(solutions, index + 1, partialSolution, stillAvailable);
removeConstraints(
index, candidate, stillAvailable, appliedConstraints);
}
}


I'll leave the implementation of applyConstraints and removeConstraints as an exercise to the reader, but the contract of applyConstraints is that it returns a negative number iff any following cell is left with no possible value; otherwise it returns a long that encodes which elements of stillAvailable were modified.

Notice how the comment describing the workings of the algorithm is made unnecessary by using well-named helper functions; the code makes the algorithm clear.

## Minor Issues

• The comment describing followings should include an example to make it easier to understand. On that note, you should put the description of followings in one place; right now it's split between the declaration and the initialization.
• tmpunknowns complicates the logic in your constructor to save a few tens of bytes. I'm reasonably sure that it doesn't even help with cacheline pressure, as I doubt that the entire array is pulled into the L1 cache when accessing a signle elements. If you decide to keep it, you should move the copy from tmpunknowns to unknowns up to immediately after the initialization of tmpunknowns; as it is, you're separating related pieces of code, which makes it more difficult to understand.
• It's bad practice to check in commented-out code (I refer here to your // System.out.println(..., //Arrays.copyOf(resetmask..., etc.
• In my experience, local variables are only declared final if they are to be used in inner classes. So, e.g., in solve when you declare combination or freedoms to be final, I look for the inner class where they're used. Idiomatic Java doesn't have the same sense of const-correctness that C++ has.
• I suspect that making bitmasks a method instead of an array will make your optimized code faster; the compiler can inline the function for a single arithmetic operation instead of reading from memory (which requires a single arithmetic operation to get the offset anyway, along with requiring a cacheline, etc.).
• Addressing the initial comment - about the strict format of the input. Truth is, I am cheating. I have already parsed the input, and, typically, I have run it through multiple 'Sudoku Strategy' clases before I discover I need to 'brute-force' the puzzle. I have not (yet) posted the full solution to the Week-end challenge, but, rest assured, I do process raw (example-specified) input, and I have just posted this question here as a self-contained solution... which the 'short-cut' input. – rolfl Dec 16 '13 at 0:15
• That makes sense. I've removed the initial comment in response to your edit. – ruds Dec 16 '13 at 0:33