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Divide &
conquer
DIVIDE AND CONQUER RULEDEFINITION:
The divide-and-conquer rule consists in breaking a problem into simpler sub-problems of the same type, next to solve these sub-problems, finally to merge the obtained results into a solution to the problem.
Why we use it?
Divide and conquer technique is used for sorting the elements of array in some specific order.
ALGORITHM:Dividing the main problem into smaller ones.These smaller sub-problems are divided further.Recombine them to achieve the main objective.
ANALYSIS:Let’ see few analysis to confirm the usefulness of the divide and conquer technique.To sort the one major problem time is nlog(n)2To sort the halves approximate time is (n/2) 2+(n/2) 2To merge the two halves approximate time is n
So, for n=100, divide and conquer takes approximately:= (100/2) 2 + (100/2) 2 + 100= 2500 + 2500 + 100= 5100
Suppose that n is 100. Considering if we apply insertion sort algorithm on it then thetime taken will be approximately (100) 2 n = 10000. Now, if we apply divide and conquer technique on it. Then for first half approximate time will be (100/2) 2. Similarly for second half it will be (100/2) 2. The merging approximate time will be 100. So the whole operation of sorting using this divide and conquer technique in insertion sort will take around (100/2) 2 + (100/2)2+100 = 5100. Clearly the time spent (5100) after applying divide and conquer mechanism is significantly lesser thanthe previous time (10000).
WORKING:
subproblem 2 of size n/2
subproblem 1 of size n/2
a solution to subproblem 1
a solution tothe original problem
a solution to subproblem 2
a problem of size n
EXAMPLES:
Merge sortQuick sort
EXAMPLE OF MERGE SORT
5 10 3 8
Suppose we have an array of four elements
5 10 3 8
5 10 3 8
5 10 3 8
5 10 3 8
5 10 3 8
5 10
3
8
5 10 3 8
5 10
3
8
5 10 3 8
5
10
3
8
5 10 3 8
5
10
3
8
5 10 3 8
5
10
3 8
5 10 3 8
5 103 8
Un sorted
sorted
EXAMPLE OF QUICK SORT
5 10 3 8
Suppose we have an array of five elements
1
5 10 3 8 1
pivot
5 10 3 8 1
5 10 3 8 1
5
10 3 8
1
510 3 81
510 3 81
5
10
3 81
5
10
3 81
5103 81
5 103 81
pivot
5 103 81
5 103 81
5
10
3 81
5
10
3 81
5 103 81
5 103 81
sorted
