# Backtracking Algorithms

Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate *(“backtracks”)* as soon as it determines that the candidate cannot possibly be completed to a valid solution.

Application of Backtracking

Examples where backtracking can be used to solve puzzles or problems include:

1) Puzzles such as eight queens puzzle, crosswords, verbal arithmetic, Sudoku [nb 1], and Peg Solitaire.

2) Combinatorial optimization problems such as parsing and the knapsack problem.

3) Logic programming languages such as Icon, Planner and Prolog, which use backtracking internally to generate answers.

### Example Problem (The Knight’s tour problem)

*The knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once.*

### Path followed by Knight to cover all the cells

Following is chessboard with 8 x 8 cells. Numbers in cells indicate move number of Knight.

### Naive Algorithm for Knight’s tour

The Naive Algorithm is to generate all tours one by one and check if the generated tour satisfies the constraints.

```
while there are untried tours
{
generate the next tour
if this tour covers all squares
{
print this path;
}
}
```

### Backtracking Algorithm for Knight’s tour

Following is the Backtracking algorithm for Knight’s tour problem.

```
If all squares are visited
print the solution
Else
a) Add one of the next moves to solution vector and recursively
check if this move leads to a solution. (A Knight can make maximum
eight moves. We choose one of the 8 moves in this step).
b) If the move chosen in the above step doesn't lead to a solution
then remove this move from the solution vector and try other
alternative moves.
c) If none of the alternatives work then return false (Returning false
will remove the previously added item in recursion and if false is
returned by the initial call of recursion then "no solution exists" )
```