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Recursive Backtracking
"In ancient times, before computers were invented, alchemists studied the mystical properties of numbers. Lacking computers, they had to rely on dragons to do their work for them. The dragons were clever beasts, but also lazy and bad-tempered. The worst ones would sometimes burn their keeper to a crisp with a single fiery belch. But most dragons were merely uncooperative, as violence required too much energy. This is the story of how Martin, an alchemist’s apprentice, discovered recursion by outsmarting a lazy dragon."
- David S. Touretzky, Common Lisp: A Gentle Introduction to Symbolic Computation
Backtracking
Start Success!
Success!
Failure Problem space consists of states (nodes) and actions (paths that lead to new states). When in a node can can only see paths to connected nodes
If a node only leads to failure go back to its "parent" node. Try other alternatives. If these all lead to failure then more backtracking may be necessary.
Solving Sudoku – Brute Force
A brute force algorithm is a
simple but general
approach
Try all combinations until
you find one that works
This approach isn’t clever,
but computers are fast
Then try and improve on
the brute force resuts
Solving Sudoku
Brute force Sudoku Soluton
- if not open cells, solved
- scan cells from left to right, top to bottom for first open cell
- When an open cell is found start cycling through digits 1 to 9.
- When a digit is placed check that the set up is legal
- now solve the board
1
Solving Sudoku – Later Steps
1 1 2 1 2 4
(^1 2 4 8 1 2 4 8 )
uh oh!
Sudoku – A Dead End
We have reached a dead end in our search
With the current set up none of the nine
digits work in the top right corner
1 2 4 8 9
Characteristics of Brute Force and Backtracking
Brute force algorithms are slow
The don't employ a lot of logic
- For example we know a 6 can't go in the last 3 columns of the first row, but the brute force algorithm will plow ahead any way
But, brute force algorithms are fairly easy to
implement as a first pass solution
- backtracking is a form of a brute force algorithm
Key Insights
After trying placing a digit in a cell we want to solve the new sudoku board
- Isn't that a smaller (or simpler version) of the same problem we started with?!?!?!?
After placing a number in a cell the we need to remember the next number to try in case things don't work out.
We need to know if things worked out (found a solution) or they didn't, and if they didn't try the next number
If we try all numbers and none of them work in our cell we need to report back that things didn't work
Recursive Backtracking
Pseudo code for recursive backtracking
algorithms
If at a solution, report success
for( every possible choice from current state /
node)
Make that choice and take one step along path Use recursion to solve the problem for the new node / state If the recursive call succeeds, report the success to the next high level Back out of the current choice to restore the state at the beginning of the loop.
Report failure
Goals of Backtracking
Possible goals
- Find a path to success
- Find all paths to success
- Find the best path to success
Not all problems are exactly alike, and
finding one success node may not be the
end of the search
Start Success!
Success!
The 8 Queens Problem
A classic chess puzzle
- Place 8 queen pieces on a chess board so that none of them can attack one another
The N Queens Problem
Place N Queens on an N by N chessboard so that none of them can attack each other
Number of possible placements?
In 8 x 8 64 * 63 * 62 * 61 * 60 * 59 * 58 * 57 = 178,462, 987, 637, 760 / 8! = 4,426,165,
n choose k
- How many ways can you choose k things from a set of n items?
- In this case there are 64 squares and we want to choose 8 of them to put queens on
Reducing the Search Space
The previous calculation includes set ups like this one
Includes lots of set ups with multiple queens in the same column
How many queens can there be in one column?
Number of set ups 8 * 8 * 8 * 8 * 8 * 8 * 8 * 8 = 16,777,
We have reduced search space by two orders of magnitude by applying some logic
Q Q Q Q Q Q Q Q
A Solution to 8 Queens
If number of queens is fixed and I realize there can't be more than one queen per column I can iterate through the rows for each column for(int c0 = 0; c0 < 8; c0++){ board[c0][0] = 'q'; for(int c1 = 0; c1 < 8; c1++){ board[c1][1] = 'q'; for(int c2 = 0; c2 < 8; c2++){ board[c2][2] = 'q'; // a little later for(int c7 = 0; c7 < 8; c7++){ board[c7][7] = 'q'; if( queensAreSafe(board) ) printSolution(board); board[c7][7] = ' '; //pick up queen } board[c6][6] = ' '; // pick up queen
