1. Recursion: Backtracking
Dr. Jicheng Fu
Department of Computer Science
University of Central Oklahoma

2. Objectives (Chapter 5)
Understand backtracking algorithms and use them to solve problems
Use recursive functions to implement backtracking algorithms
See how the choice of data structures can affect the efficiency of a program

3. The Eight-Queens Puzzle
How to place eight queens on a chess board so that no queen can take another
Remember that in chess, a queen can take another piece that lies on the same row, the same column or the same diagonal (either direction)
There seems to be no analytical solutions

4. Solutions do exists Requires luck coupled with trial and error
Or much exhaustive computation

5. Solve the Puzzle How will you solve the problem?
Put the queens on the board one by one in some systematic/logical order
Make sure that the queen placed will not be taken by another already on the board
If you are lucky to put eight queens on the board, one solution is found
Otherwise, some queens need to be removed and placed elsewhere to continue the search for a solution

6. Why put the queens in a systematic order?
Make sure that you will not consider the same situation again
The whole process is suitable for recursive programming
A large problem can be divided into small subproblems with a similar nature

7. An Outline of the Recursive Function
Recursive function: solve_from
Given a configuration of queens on the chessboard, search for all solutions
Class Queens
Represent a particular configuration of queens on the chessboard
Code sketch
solve_from (Queens configuration) {
if configuration already contains eight queens
Print configuration
else
for every square p that is unguarded by configuration {
add a queen on square p to configuration;
solve_from(configuration);
remove the queen from square p of configuration; }
Stop Condition
Emphasize the all back tracking algorithms use this exact format. If a student uses something else, it won’t be correct.
Emphasize the importance of the for loop.
The essence of back-tracking is to first figure out the steps needed to build the partial solution towards the final solution. Secondly, at each step, figure out all the options to try one at a time.
It is a brute force approach.
Sub-problems. Need to try all possibilities

8. How to find the next square to try?
There must be a queen, exactly one queen in each row
There can never be more than one queen in each row
Therefore, we can place the queens one row at a time in order
What we have decided is actually a systematic way to solve the problem

9. Example: Four Queens X: Guarded squares
?: Other squares that have not been tried

10. Backtracking Backtracking algorithms Search for a solution by
constructing partial solutions that remain consistent with the requirements of the problem, and then
extending a partial solution toward completion
When inconsistency occurs, the algorithms backs up (backtracks) by
removing the most recently constructed part of the solution, and then
trying another possibility

11. Suitable for implementations using
Recursion, or
Stacks
Only the most recent part is used
Useful for situations where many possibilities may first appear, but few survive further tests
Scheduling problems
Arranging sports tournaments
Designing a compiler
Parsing

12. Main program int main( ) /* Pre: The user enters a valid board size.
Post: All solutions to the n-queens puzzle for the selected board size are printed.
Uses: The class Queens and the recursive function solve_from. */ {
int board_size;
print_information( );
cout << "What is the size of the board? " << flush;
cin >> board_size;
if (board size < 0 || board size > max_board)
cout << "The number must be between 0 and “ << max_board << endl;
else {
Queens configuration(board_size); // Initialize empty configuration.
solve_from(configuration); // Find all solutions extending configuration. }

13. The configuration: Queens class
Constructor
set the user-selected board size and initialize the empty Queens object
print
print the solutions
unguarded
bool Queens :: unguarded(int col) const;
Post: Returns true or false according as the square in the first unoccupied row (row count) and column col is not guarded by any queen.
Why no row number?

14. insert remove is_solved void Queens :: insert(int col);
Pre: The square in the first unoccupied row (row count) and column col is not guarded by any queen.
Post: A queen has been inserted into the square at row count and column col; row count has been incremented by 1.
remove
void Queens :: remove(int col);
Pre: There is a queen in the square in row count - 1 and column col.
Post: The above queen has been removed; count has been decremented by 1.
is_solved
bool Queens :: is_solved( ) const;
Post: The function returns true if the number of queens already placed equals board size; otherwise, it returns false.

15. The backtracking function: solve_from
void solve_from(Queens &configuration)
/* Pre: The Queens configuration represents a partially completed arrangement of non-attacking queens on a chessboard.
Post: All n-queens solutions that extend the given configuration are printed. The configuration is restored to its initial state.
Uses: The class Queens and the function solve_from , recursively. */ {
if (configuration.is_solved( )) configuration.print( );
else
for (int col = 0; col < configuration.board_size; col++)
if (configuration.unguarded(col)) {
configuration.insert(col);
solve_from(configuration); // Recursively continue to add queens.
configuration.remove(col); }
See the advantage of using OOP rather than procedure-based.

16. Implementation of the Queens Class
Store the chessboard as a 2-dimensional array with entries indicating the locations of the queens
Queens class
const int max_board = 30;
class Queens {
public:
Queens(int size);
bool is_solved( ) const;
void print( ) const;
bool unguarded(int col) const;
void col);
void reminsert(int ove(int col);
int board_size; // dimension of board = maximum number of queens
private:
int count; // current number of queens = first unoccupied row
bool queen_square[max_board][max_board]; };

17. is_solved, remove and print are also trivial
Constructor
Queens :: Queens(int size)
/* Post: The Queens object is set up as an empty configuration on a chessboard with size squares in each row and column. */ {
board_size = size;
count = 0;
for (int row = 0; row < board_size; row++)
for (int col = 0; col < board_size; col++)
queen_square[row][col] = false; }
insert
void Queens :: insert(int col)
/* Pre: The square in the first unoccupied row (row count ) and column col is not guarded by any queen.
Post: A queen has been inserted into the square at row count and column col ; count has been incremented by 1. */
{ queen_square[count++][col] = true; }
is_solved, remove and print are also trivial
A part of the big 2-d Boolean array is used.

18. unguarded bool Queens :: unguarded(int col) const
/* Post: Returns true or false according as the square in the first unoccupied row (row count ) and column col is not guarded by any queen. */ {
int i;
bool ok = true; // turns false if there is a queen in column or diagonal
for (i = 0; ok && i < count; i++)
// Check upper part of column
ok = !queen_square[i][col];
for (i = 1; ok && count - i >= 0 && col - i >= 0; i++)
// Check upper-left diagonal
ok = !queen_square[count - i][col - i];
for (i = 1; ok && count - i >= 0 && col + i < board size; i++)
// Check upper-right diagonal
ok = !queen_square[count - i][col + i];
return ok; }

20. Exercise
Describe a rectangular maze by indicating its paths and walls within an array. Write a backtracking program to find a way through the maze
You are a tournament director and need to arrange a round robin tournament among N = 2k players. In this tournament, everyone plays exactly one game each day; after N - 1 days, a match occurred between every pair of players. Write a backtracking program to do this.

21. Review and Refinement The time increases rapidly with the board size
First refinement
Use the 2-dimensional array to keep track of all the squares that are guarded by queens
For each square, keep a count of the number of queens guarding the square
Faster, but we still need the loops to update the guard counts for each square

22. Second refinement Objective Key idea
Eliminate all loops
Key idea
Each row, column and diagonal can contain at most one queen
Keep track of unguarded squares by using three bool arrays: col_free, upward_free and downward_free
Diagonals from the lower left to the upper right are called upward diagonals
Diagonals from the upper left to the lower right are called downward diagonals
An integer array is used to record the column number for the queens in each row

23. Eliminate loops It is trivial to identify each column
How to identify each diagonal
For upward diagonals, the row and column indices have a constant sum
The sum ranges from 0 to 2  board_size – 2
The sum can be used to identify each upward diagonals
The square in row i and column j is in upward diagonal number i + j
For downward diagonals, the difference of the row and column indices is constant
The difference ranges from -board_size+1 to board_size–1
The downward diagonal can be numbered using the difference
The square in row i and column j is in downward diagonal number i - j + board_size - 1

25. Refined Implementation
Revised Queens class
class Queens {
public:
Queens(int size);
bool is_solved( ) const;
void print( ) const;
bool unguarded(int col) const;
void insert(int col);
void remove(int col);
int board_size;
private:
int count;
bool col_free[max_board];
bool upward_free[2 * max_board - 1];
bool downward_free[2 * max_board - 1];
// column number of queen in each row
int queen_in_row[max_board]; };

26. Constructor Queens :: Queens(int size)
/* Post: The Queens object is set up as an empty configuration on a chessboard with size squares in each row and column. */ {
board_size = size;
count = 0;
for (int i = 0; i < board_size; i++)
col_free[i] = true;
for (int j = 0; j < (2 * board size - 1); j++)
upward_free[j] = true;
for (int k = 0; k < (2 * board size - 1); k++)
downward_free[k] = true; }

27. Insertion void Queens :: insert(int col)
/* Pre: The square in the first unoccupied row (row count ) and column col is not guarded by any queen.
Post: A queen has been inserted into the square at row count and column col; count has been incremented by 1. */ {
queen_in_row[count] = col;
col_free[col] = false;
upward_free[count + col] = false;
downward_free[count - col + board_size - 1] = false;
count++; }

28. Unguarded bool Queens :: unguarded(int col) const
/* Post: Returns true or false according as the square in the first unoccupied row (row count ) and column col is not guarded by any queen. */ {
return col_free[col] && upward_free[count + col]
&& downward_free[count - col + board_size - 1]; }

29. Evaluation
New Implementation
2-D array Implementation

30. Analysis of Backtracking
Effectiveness of Backtracking
Naïve approach
Consider all configurations: put all 8 queens on the board and reject illegal configurations
There can be only one queen in each row
88 = 16,777,216
There can be only one queen in each column
8! = 40,320
Our program is even better, since it rejects squares in guarded diagonals

31. Part of the recursion tree for the eight-queens problem

32. Lower Bounds Number of solutions
For n-queens problem, the amount of work done by backtracking problem still grows very fast
To place a queen in each of the first n/4 rows, backtracking algorithm investigates a minimum of n(n - 3)(n - 6) … (n – 3 * n/4) positions
n (n - 3) (n - 6) … (n – 3 * n / 4) > ( n / 4) (n/4)
Exponential growth
Number of solutions
Not bounded by any polynomial in n
Even not bounded by any exponential form kn, where k is a constant
It is proved to be an unsolved problem