Analysis and design of Algorithm Backtracking algorithm and its comparison with Dynamic programing problems

1. Backtracking Algorithm

2. Objectives
• Backtracking Programming Technique
• Applications
• N-Queens Problem
• Complexity of N-Queens Problem
• Backtracking vs Dynamic Programming
Algorithms

3. Backtracking Programming Technique
A backtracking algorithm tries to build a solution
to a computational problem incrementally.
Whenever the algorithm needs to decide
between multiple alternatives to the next
component of the solution, it simply tries all
possible options recursively.

4. Applications
• N-Queens Problem
• Subset sum Problem
• Rat in a Maze Problem
• Sudoku
• Hamiltonian Cycle
• Longest Common Subsequence

5. N-Queen Problem
• Goal: position n queens on a n x n board such
that no two queens threaten each other
• No two queens may be in the same row,
column, or diagonal
• Sequence: n positions where queens are
placed
• Set: n2 positions on the board
• Criterion: no two queens threaten each other

6. 4-Queen Problem Solution
• 4-Queen Problem Assign each queen a
different row
• Check which column combinations yield
solutions
• Each queen can be in any one of four columns:
4 x 4 x 4 x 4 = 256
Q
Q
Q
Q

7. Complexity of N-Queens Problem
Theta(n^n)
For example- 4-Queens problem the complexity
is-
Theta(4^4)

8. Algorithm
1) Start in the leftmost column
2) 2) If all queens are placed return true
3) 3) Try all rows in the current column. Do following for every tried
row.
a) If the queen can be placed safely in this row then mark this [row,
column] as part of the solution and recursively check if placing queen
here leads to a solution.
b) If placing the queen in [row, column] leads to a solution then return
true.
c) If placing queen doesn't lead to a solution then unmark this [row,
column] (Backtrack) and go to step (a) to try other rows.
4) If all rows have been tried and nothing worked, return false to trigger
backtracking.

10. Backtracking vs Dynamic Programming
• Dynamic Programming – subsets of a solution
are generated
• Backtracking – Technique for deciding that
some subsets need not be generated
• Efficient for many large instances of a problem
(but not all)