Divide and Conquer is one of the best-known general algorithm design technique. Divide-and-conquer algorithms work according to the following general plan:

**DIVIDE AND CONQUER**

Divide
and Conquer is one of the best-known general algorithm design technique.
Divide-and-conquer algorithms work according to the following general plan:

1.A
problem's instance is divided into several smaller instances of the same
problem, ideally of about the same size.

2. The
smaller instances are solved (typically recursively, though sometimes a
different algorithm is employed when instances become small enough).

3. If
necessary, the solutions obtained for the smaller instances are combined to get
a solution to the original problem.

The
divide-and-conquer technique is diagrammed, which depicts the case of dividing
a problem into two smaller sub problems,

Examples
of divide and conquer method are **Binary
search, Quick sort,**

**GENERAL METHOD**

**e.g. Merge sort**.

♦ As an
example, let us consider the problem of computing the sum of *n* numbers **a _{0}, ... a_{n-}**

♦ If *n* > 1, we can divide the problem into
two instances of the same problem. They are To compute the sum of the first └
n/2**┘**numbers

♦ To
compute the sum of the remaining [n/2]numbers. (Of course, if *n* = 1, we simply return a_{0} as
the answer.)

♦ Once each
of these two
sums is computed
(by applying the
same method, i.e.,

recursively),

we can
add their values to get the sum in question. i.e.,

**a _{0} + . . . + a_{n-1}
=( a_{0} + ....+ a**

♦ More
generally, an instance of size n can be divided into several instances of size
nib, with a of them needing to be solved. (Here, a and b are constants; a≥1 and
b > 1.).

♦ Assuming
that size n is a power of b, to simplify our analysis, we get the following
recurrence for the running time T(n).

**T(n)=aT(n/b)+f(n)**

♦ This is
called the **general divide and-conquer
recurrence.** Where f(n) is a function that accounts for the time spent on
dividing the problem into smaller ones and on combining their solutions. (For
the summation example, a = b = 2 and f (n) = 1.

♦ Obviously,
the order of growth of its solution T(n) depends on the values of the constants
a and b and the order of growth of the function f (n). The efficiency analysis
of many divide-and-conquer algorithms is greatly simplified by the following
theorem.

**1. MASTER THEOREM**

**ADVANTAGES:**

♦ The time
spent on executing the problem using divide and conquer is smaller then other
methods.

♦ The
divide and conquer approach provides an efficient algorithm in computer
science.

♦ The
divide and conquer technique is ideally suited for parallel computation in
which each sub problem can be solved simultaneously by its own processor.

♦ Merge
sort is a perfect example of a successful application of the divide-and conquer
technique.

♦ It sorts
a given array Al O .. n - 1) by dividing it into two halves A [O··└ n/2**┘** - 1] and
*A[*└

n/2**┘***..n* - 1], sorting each of them
recursively, and then merging the two smaller sorted arrays into a single
sorted one.

**ALGORITHM **Mergesort
(A[O .. n - 1])

//Sorts
array *A[O .. n* - 1] by recursive
mergesort //Input: An array *A[O .. n*
- 1] of orderable elements //Output*:*
Array *A[O...* n - 1] sorted in
nondecreasing order **if n **> 1

copy A
[O··└ n/2**┘** - 1] to B [O··└ n/2**┘** - 1] copy *A[*└ n/2**┘***..n* - 1] to
C[0...└ n/2**┘** - 1] Mergesort (B [O··└ n/2**┘** - 1])

Mergesort
(C[0...└ n/2**┘** - 1]) Mergesort(B,C,A)

**STEPS TO BE FOLLOWED**

♦ The first
step of the merge sort is to chop the list into two.

♦ If the
list has even length, split the list into two equal sub lists.

♦ If the
list has odd length, divide the list into two by making the first sub list one
entry greater than the second sub list.

♦ then
split both the sub list into two and go on until each of the sub lists are of
size one.

♦ finally,
start merging the individual sub lists to obtain a sorted list.

♦ The
operation of the algorithm for the array of elements 8,3,2,9,7,1,5,4 is
explained as follows,

**AN EXAMPLE OF MERGE SORT
OPEERATION**

♦ The **merging** of two sorted arrays can be
done as follows. Two pointers (array indices) are initialized to point to the
first elements of the arrays being merged.

♦ Then the
elements pointed to are compared and the smaller of them is added to a new
array or list being constructed.

♦ Then the
index of the smaller element is incremented to point to its immediate successor
in the array.

♦ This operation
is continued until
one of the
two given arrays
is exhausted

♦Then the remaining elements of the
other array are copied to the end of the new

array.

ALGORITHM : Merge Sort B(0....p-1),
C(0.... q-1), A(0....p + q -1)

*// Merges two sorted arrays into
one sorted array .*

*// Input: Array B (0....p-1) and
C (0....q-1) both sorted*

*//Output: Sorted array A (0....p
+ q -1) of the elements of B and C*

*i *^{f}* 0; j *^{f}*0 ;k*^{f}* 0 while
i<p and j<q do if B[i] ≤ C[j]*

*A[k] *^{f}*B[i];i *^{f}*i+1*

*else A[k] *^{f}* C[j] j*^{f}* j+1 k *^{f}*k+1*

*if i=p*

*copy C[j....q-1] to A[k....p+q-1]
else copy B[i....p-1] to A[k....p+q-1]*

**2 EFFICIENCY OF MERGE SORT**

♦ The
recurrence relation for the number of key comparison C (n) is

**C (n) = 2 C (n/2) + C _{merge}
(n) for n>1, C (1) = 0**

♦ Assume, n
is a power of 2. C_{merge} (n) is the number of key comparison
performed during the merge sort.

♦ In the
merging of two sorted array after one comparison is made the total number of
elements in the two array still to be processed and sorted is reduced by one.

♦ Hence in
the worst case neither of the two arrays becomes empty before the other one contains
just one element . Therefore, for the worst case,

**C _{worst}(n)=2 C _{worst}(n
/ 2)+n-1 for n > 1, C _{worst}(1)=0**

**C _{merge}(n) = n **

and we
have the recurrence

Hence,
according to the Master Theorem, C worst (n)
= (n log n) . In fact, it is easy to find the exact solution to the
worst-case recurrence for n = 2k

C worst(n) = n log2 n - n + 1

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Analysis and Design of Algorithm : Divide and Conquer, Greedy Method : Divide and Conquer Method |

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