57 lines
2.3 KiB
Markdown
57 lines
2.3 KiB
Markdown
|
---
|
|||
|
|
title: Backtracking Algorithms
|
|||
|
|
---
|
|||
|
|
|
|||
|
|
# 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.
|
|||
|
|
|
|||
|
|
### 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.
|
|||
|
|
[](https://commons.wikimedia.org/wiki/File:Knights_tour_(Euler).png)
|
|||
|
|
|
|||
|
|
### 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" )
|
|||
|
|
```
|
|||
|
|
|
|||
|
|
|
|||
|
|
### More Information
|
|||
|
|
|
|||
|
|
[Wikipedia](https://en.wikipedia.org/wiki/Backtracking)
|
|||
|
|
|
|||
|
|
[Geeks 4 Geeks](http://www.geeksforgeeks.org/backtracking-set-1-the-knights-tour-problem/)
|
|||
|
|
|
|||
|
|
[A very interesting introduction to backtracking](https://www.hackerearth.com/practice/basic-programming/recursion/recursion-and-backtracking/tutorial/)
|