Sudoku [9 marks]

In these questions you will be working on writing code that will

solve Sudoku puzzles.

All these questions are part of the Core assignment, due in at the

deadline after reading week.

For the avoidance of doubt -- with the exception of the

SudokuSquareSet question, you may add extra #include and using

statements, if you need to.

Make sure you don't commit any compiled code to your GitHub

repository; or if you choose to use an IDE, any large project

directories created by your IDE. You can make these on your machine,

but don't commit or add them to your repository -- this isn't what

git is designed for.

a) Making a Sudoku board class

In the file Sudoku.h make a class Sudoku that holds an incomplete

Sudoku solution.

It should have a constructor that takes a single argument -- the size

of the board. For instance, for a 9x9 Sudoku, the constructor would

be given the value 9. Or, for a 16x16 board, the constructor would be

given the value 16.

You need to store the incomplete solution as a member variable. The

recommended way to do this, to start with, is to have a vector of

vectors (a square array), in which each square is represented as a

set<int> that holds the values that could possibly go in that square.

Initially, for a 9x9 Sudoku, if the grid is completely blank, each

set will contain the values {1,2,3,4,5,6,7,8,9}. When a square is

given some value, the set is cleared and replaced with a set

containing just that one value -- the other options are removed.

Write a function getSquare(int row, int col) that returns the value

in the cell in the square at the given row and column:

? If there is only one value in the set for that square, return

the number in that set.

? Otherwise, return -1 (a dummy value to indicate we don't know

what should be in that square yet)

b) Setting the value of a Sudoku

square

Write a function setSquare(int row, int col, int value) that sets the

value in the cell in the square at the given row and column, then

updates the sets of possible values in the rest of the grid to remove

choices that have been eliminated. For instance, if we put a '3' on a

given row, then nothing else on that row can have the value 3.

The implementation of setSquare is split into two parts.

First, the easy part: the set of possible values for that cell is

cleared, and value is inserted. This forces that cell to have that

value.

Then, a loop begins that does the following:

? Loop over the entire grid

? For each square that has only one value in it, remove that

value from the sets of possible values for:

o All the other squares on that row

o All the other squares in that column

o All the other squares in the same box. A 9x9 grid is

divided into 9 boxes, each 3x3: no two values in the same

box can have the same value. For larger grids (e.g.

16x16), the size of the box is always the square root of

the size of the grid.

If at any point the set of values for a square becomes empty, the

function should return false: it has been shown that there is no

value that can go in a square.

The loop should continue whilst values are still being removed from

the sets of possible values. The reason for this is that after

setting the given square, we might end up with only one option being

left for some other squares on the grid. For instance, if on a given

row some of the squares had the sets:

{3,4} {3,5} {4,5}

...and we call setSquare to set the middle square to have the value

3, then before the loop:

{3,4} {3} {4,5}

On the first pass of the loop, we would find the square containing 3

and remove this from the other sets on the row (and the other sets in

the same column and box). The row then looks like:

{4} {3} {4,5}

We then start the loop again, and find the square containing the

value '4'. This is removed from the other sets on the row (and column

and box) to give:

{4} {3} {5}

We then start the loop again, and find the square containing the

value '5'.

This process stops when, having looped over the board, and updated

the sets by removing values, our sets have stopped getting any

smaller. At this point the function returns true.

For simple Sudodu puzzles, this process here is enough to solve the

puzzle. No guesswork is needed: setting the squares of the board to

hold the initial values specified in the puzzle, is enough to cause

all the other squares of the board to have only one option left.

You can test this by compiling and running BasicSudoku.cpp:

g++ -std=c++11 -g -o BasicSudoku BasicSudoku.cpp

This calls setSquare for the values in a simple Sudoku puzzle; then

uses getSquare to check your code has the right answer.

c) Searching for a solution

For more complex puzzles, after putting in the initial values using

setSquare, some of the squares on the board have more than one value

left in their set of possible values -- using logic alone, we cannot

deduce what the value has to be; we have to make a guess and see what

happens.

For this, we are going to use the Searchable class. This is an

abstract class for puzzles, containing the following virtual

functions:

? isSolution(): this returns true if the puzzle has been solved.

For Sudoku, this means all the squares contain just one value.

? write(ostream & o): a debugging function to print the board to

screen.

? heuristicValue(): an estimate of how far the puzzle is from

being solved. We will return to this in part (d)

? successors(): in a situation where a guess is needed, this

returns several new puzzle objects, each of which corresponds

to a different guess having been made.

Make your Sudoku class inherit from Searchable, by changing the

opening of the class definition to class Sudoku : public Searchable

Implement isSolution() to only return true if the puzzle has been

solved; i.e. every set in every square is of size 1.

Implement a write() function to print the board. You can display the

board however you like. A reasonable implementation is to print out

the board one row at a time:

? If the square has more than one value in its set, print a space

character

? Otherwise, print the value from the set.

(I'm not marking write(), it is to help your debugging. If you want

to print nothing at all, that's fine.)

Implement successors() as follows:

? Make an empty vector of successors to return. This should be a

vector<unique_ptr<Searchable> > object.

? Find the first row containing a square that still has more than

one option in its set

? Find the left-most square on that row

? For each value in the set for that square:

o Make a copy of the current Sudoku object (this) using new

o Use setSquare on the copy to set the value of the square

o If setSquare returns true, add the pointer to the back of

the vector of successors to return. Otherwise, delete the

pointer. (You can use a unique_ptr for this if you wish.)

? Once done, return the vector of successors

Once you have implemented these functions, you can test your code by

using BreadthFSSudoku:

g++ -std=c++11 -g -o BreadthFSSudoku BreadthFSSudoku.cpp

d) Other improvements

SudokuSquareSet

Using a set<int> to represent the possible values in a sudoku square

is relatively inefficient. A better option is to set bits of an

unsigned integer to be 0 or 1, with bit n being set to 1 if the value

(n+1) is in the set. For instance, to encode that 1,3 and 4 are

possible values, in binary, this would be:

00000000 00000000 00000000 00001101

..................................^ if bit 0 is 1, then 1 is in the

set

................................^.. if bit 2 is 1, then 3 is in the

set

...............................^... if bit 3 is 1, then 4 is in the

set

Note that because 0 can never be in a Sudoku square, bit 0 is 1 if 1

is in the set, bit 1 is 1 if 2 is in the set, and so on.

Conceptually, it's still a set -- each value can only be stored once

-- but it's much more efficient.

In the file SudokuSquare.h complete the definition of the

SudokuSquareSet class. It should have exactly two member variables,

and no others:

? An unsigned int whose bits denote what values are in the set

? An int that denotes how many values are in the set

...and provide similar functionality to set<int>, including:

? A default constructor that creates an empty set (all bits set

to 0)

? A size() function that returns how many values are in the set

? An empty() function that returns true iff there are no values

in the set

? A clear() function that removes all values from the set

? operator==() that compares a SudokuSquareSet to another given

SudokuSquareSet, and returns true if the values are the same

? operator!=() that compares a SudokuSquareSet to another given

SudokuSquareSet, and returns true if the values are different

? begin() and end() that get iterators, allowing you to loop over

the values in the set (you will need to implement an iterator

class inside the SudokuSquareSet class for this)

? An insert function that adds a value to the set, and returns an

iterator to it

? A find function that sees if a value is in the set -- if it is,

it returns an iterator to it, otherwise it returns end()

? An erase function that takes a value, and if it's in the set,

removes it

? An erase function that takes an iterator at a given value, and

calls erase() with that value

To perform some basic tests on your SudokuSquareSet class you can

compile and run TestSudokuSquare.cpp:

g++ -std=c++11 -g -o TestSudokuSquare TestSudokuSquare.cpp

Once you have done this, you can edit your Sudoku.h file to use

SudokuSquareSet instead of set<int>. If you have implemented the

functionality above correctly, it should then be a drop-in

replacement. In short:

? Add #include "SudokuSquare.h" to the #include section at the

top of Sudoku.h

? Replace set<int> with SudokuSquareSet throughout the Sudoku

class

You should then be able to compile and run the tests you were using

earlier, to test the overall Sudoku functionality.

Other performance tweaks

We can improve on the basic sudoku solver in a few ways. Your mark

for this part will be based on how quickly your solution works: the

faster it runs, the higher the marks. As well as 9x9 boards, I will

be testing your solution on larger (e.g. 16x16) Sudoku boards.

A better setSquare

Suppose we had the following squares on a row:

{3,4} {3,4,6} {3,4,6,7} {3,4}

There are 2 cells that contain identical sets of size 2. There are

only two ways of satisfying this:

? 3 in the left-most cell; 4 in the right-most cell

? 4 in the left-most cell; 3 in the right-most cell

In either case, no other value on the row can have the value 3 or 4.

We can thus remove 3 and 4 from the other sets:

{3,4} {6} {6,7} {3,4}

As before, this process loops and carries on until the sets stop

getting any smaller -- because there is a cell that contains just 6,

we would then get:

{3,4} {6} {7} {3,4}

Extend the implementation of setSquare to include this sort of

reasoning for duplicates sets of values in rows, columns and boxes:

look for pairs of squares, with identical sets of size 2.

To test your code, compile and run BreadthFSSudoku again -- it should

explore fewer nodes than it did before you made these changes.

If you wish, you can then continue this work to add additional logic

to setSquare. For instance, looking for 3 identical sets of size 3, 4

identical sets of size 4, etc. Or, any other deduction rule for

Sudoku that you can find online, and implement in your work -- add a

comment to the website or paper you used as your source.

Better search with a heuristic

Open up BreadthFirstSearch.h. It defines a search strategy known as

breadth-first search. It has a queue of incomplete solutions

(initially, the starting configuration of the board. Then, it

repeatedly takes a board off the queue, and if it's not a solution,

gets its successors, and puts them on the queue.

The queue used follows a first in, first out (FIFO) strategy: the

next board is always taken off the front of the queue; and new boards

are always put on the back of the queue.

Breadth-first search can be improved by using a heuristic: an

estimate of how close a board is to being finished. The Searchable

class defines a heuristicValue() function that calculates an estimate

of this.

Add a heuristicValue() function to your Sudoku class, that returns

the number of squares on the board whose sets are of size greater

than 1. On paper this corresponds to the number of squares we haven't

written a number into yet; and the fewer the squares, the closer it

looks like that board is to being a solution.

Complete the BestFirstSearch class provided, so that instead of using

a first-in first-out queue (as in breadth-first search) it uses a

priority queue, sorted in ascending order of heuristic value. That

is, when we get the successors to a board, these are inserted into

the queue so that the board with the smallest heuristic value is

always at the front.

To test your code, compile and run BestFSSudoku:

g++ -std=c++11 -g -o BestFSSudoku BestFSSudoku.cpp

This should expand fewer nodes than breadth-first search

NB: Your BestFirstSearch code should never assume Searchable objects

are Sudoku objects. Only use functions defined in the Searchable base

class.

A better successors function

One last edit. In the successors function, we choose a square, and

make successors corresponding to setting that square to have each of

its possible values. We keep only the successors for which setSquare

returned true.

If when we do this, we keep only one successor -- and that successor

isn't a solution -- then instead of returning a vector of just that

one successor, recursively call successors() on that; and return what

it returns.

The intuition for this is as follows. If setSquare only returned true

for one of the possible successors, we have shown that, actually,

having tried all the options, only one of the possible values for

that square was acceptable. Thus, we can instead return the

successors of that one valid option. We only need to return

successors to search, to go on the queue, if there is more than one

option to choose from.

As with before, implementing this should allow both breadth-first

search and best-first search to expand fewer nodes.

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