This ppt is define as the Data Structure and Algorithm,and it contain the also Back Tracking,8 Queen Method,4 Queen method,Knapsack.

1. TITLE :DATA STRUCTURE AND ALGORITHM
STAF NAME :MS.MANIMOZHI,MCA.,M.PHIL.,(PH.D)
CLASS :III BCA
SEMSTER :V
UNIT :V
.

2. DATA STRUCTURE AND
ALGORITHM
BACK TRACKING

3. contents
 Backtracking
 8 queen problem
 4 queen
 Chess board
 Sum of subset
 knapsack

4. Backtracking
 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 a candidate ("backtracks") as soon as it
determines that the candidate cannot possibly be completed to a
valid solution.

5. Backtracking algorithm
Algorithm backtrack(k)
{
For (each x[k],T(x[1]…………x[k-1]do
{
If(bk(x[1],x[2],……x[k]=0)then
{
If(x[1],x[2],……x[k] is a path to an answer node)
Then write (x[1:k];
If(k<n)then backtrack(k+1);
}
}
}

6. Basics

7. Examples
 Examples where backtracking can be used to solve puzzles or problems
include:
 Puzzles such as eight queens puzzle, crosswords, verbal
arithmetic, Sudoku, and Peg Solitaire.
 Combinatorial optimization problems such as parsing and the knapsack
problem.
 Logic programming languages such as Icon, Planner and Prolog, which
use backtracking internally to generate answers.
 The following is an example where backtracking is used for the constraint
satisfaction problem

8. A Sudoku solved by backtracking

9. Sum of subset
 It is given a n distinct positive numbers
 Desire to find all combinations of these numbers whose sums are m,
 This is called the sum of subsets problem

10. 8 queen problem
 The eight queens puzzle is the problem of placing
eight chess queens on an 8×8 chessboard so that no two queens
threaten each other.
 Thus, a solution requires that no two queens share the same row,
column, or diagonal.
 The eight queens puzzle is an example of the more
general n queens problem of placing n non-attacking queens on
an n×n chessboard, for which solutions exist for all natural
numbers n with the exception of n=2 and n=3.

11. 8 queen problem

12. Chess board

14. Can a new queen be placed
Algorithm place(k, i)
{
For j:=1 to k-1 do
If ((x[j]=i)
Then return false;
}

17. 4 queen problem

18. Graph coloring
 Let g can be graph and m be a given positive integer.
 The nodes of g can be colored in such way that no two adjacent
nodes have the same color yet only m color are used.
 The m colorability optimization problem asks for the smallest
integer m for which the graph g can be colored.

20. Knapsack problem
 The knapsack problem or rucksack problem is a problem
in combinatorial optimization: Given a set of items, each with a
weight and a value, determine the number of each item to include in
a collection so that the total weight is less than or equal to a given
limit and the total value is as large as possible.
 It derives its name from the problem faced by someone who is
constrained by a fixed-size knapsack and must fill it with the most
valuable items.

21. Define knapsack
 The most common problem being solved is the 0-1 knapsack problem,
which restricts the number {displaystyle x_{i}} x_{i} of copies of each
kind of item to zero or one. Given a set of n items numbered from 1 up to
n, each with a weight {displaystyle w_{i}} w_{i} and a value
{displaystyle v_{i}} v_{i}, along with a maximum weight capacity W,
 maximize {displaystyle sum _{i=1}^{n}v_{i}x_{i}} sum
_{i=1}^{n}v_{i}x_{i}
 subject to {displaystyle sum _{i=1}^{n}w_{i}x_{i}leq W} sum
_{i=1}^{n}w_{i}x_{i}leq W and {displaystyle x_{i}in {0,1}}
x_{i}in {0,1}.

22. Knapsack problem algorithms for my real-
life carry-on knapsack