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<!DOCTYPE html>

<head>

<title>Divide and Conquer Algorithm</title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

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<h1>Divide and Conquer Algorithm</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn how the divide and conquer algorithm works. We will also compare the divide and conquer approach versus other approaches to solve a recursive problem.</p>

<p id="introduction">A <strong>divide and conquer algorithm</strong> is a strategy of solving a large problem by</p>

<ol><li>breaking the problem into smaller sub-problems</li>

<li>solving the sub-problems, and</li>

<li>combining them to get the desired output.</li>

</ol><p>To use the divide and conquer algorithm, <strong>recursion</strong> is used. Learn about recursion in different programming languages:</p>

<hr><h2>How Divide and Conquer Algorithms Work?</h2>

<p>Here are the steps involved:</p>

<ol><li><strong>Divide</strong>: Divide the given problem into sub-problems using recursion.</li>

<li><strong>Conquer</strong>: Solve the smaller sub-problems recursively. If the subproblem is small enough, then solve it directly.</li>

<li><strong>Combine:</strong> Combine the solutions of the sub-problems that are part of the recursive process to solve the actual problem.</li>

</ol><p>Let us understand this concept with the help of an example.</p>

<p>Here, we will sort an array using the divide and conquer approach (ie. <a href="/dsa/merge-sort.html">merge sort</a>).</p>

<ol><li>Let the given array be:

<figure><img alt="initial array for merge sort" src="//cdn.programiz.com/sites/tutorial2program/files/divide-and-conquer-0.png" title="Array for merge sort" width="250" height="76"><figcaption>Array for merge sort</figcaption></figure></li>

<li><strong>Divide</strong> the array into two halves.

<figure><img alt="Divide the array into two subparts" src="//cdn.programiz.com/sites/tutorial2program/files/divide-and-conquer-1.png" title="Divide the array into two subparts" width="375" height="156"><figcaption>Divide the array into two subparts</figcaption></figure><br>

Again, divide each subpart recursively into two halves until you get individual elements.

<figure><img alt="Divide the array into smaller subparts" src="//cdn.programiz.com/sites/tutorial2program/files/divide-and-conquer-2.png" title="Divide the array into smaller subparts, merge sort" width="500" height="290"><figcaption>Divide the array into smaller subparts</figcaption></figure></li>

<li>Now, combine the individual elements in a sorted manner. Here, <strong>conquer</strong> and <strong>combine</strong> steps go side by side.

<figure><img alt="Combine the subparts" src="//cdn.programiz.com/sites/tutorial2program/files/divide-and-conquer-3.png" title="Combine the subparts" width="550" height="299"><figcaption>Combine the subparts</figcaption></figure></li>

</ol><hr><h2 id="complexity">Time Complexity</h2>

<p>The complexity of the divide and conquer algorithm is calculated using the master theorem.</p>

<pre>T(n) = aT(n/b) + f(n),

where,

n = size of input

a = number of subproblems in the recursion

n/b = size of each subproblem. All subproblems are assumed to have the same size.

f(n) = cost of the work done outside the recursive call, which includes the cost of dividing the problem and cost of merging the solutions

</pre>

<p>Let us take an example to find the time complexity of a recursive problem.</p>

<p>For a merge sort, the equation can be written as:</p>

<pre>T(n) = aT(n/b) + f(n)

= 2T(n/2) + O(n)

Where,

a = 2 (each time, a problem is divided into 2 subproblems)

n/b = n/2 (size of each sub problem is half of the input)

f(n) = time taken to divide the problem and merging the subproblems

T(n/2) = O(n log n) (To understand this, please refer to the master theorem.)

Now, T(n) = 2T(n log n) + O(n)

≈ O(n log n)

</pre>

<hr><h2>Divide and Conquer Vs Dynamic approach</h2>

<p></p><div class="clearfix"></div><p>The divide and conquer approach divides a problem into smaller subproblems; these subproblems are further solved recursively. The result of each subproblem is not stored for future reference, whereas, in a dynamic approach, the result of each subproblem is stored for future reference.</p>

<p>Use the divide and conquer approach when the same subproblem is not solved multiple times. Use the dynamic approach when the result of a subproblem is to be used multiple times in the future.</p>

<p>Let us understand this with an example. Suppose we are trying to find the Fibonacci series. Then,</p>

<p><strong>Divide and Conquer approach:</strong></p>

<pre>fib(n)

If n < 2, return 1

Else , return f(n - 1) + f(n -2)

</pre>

<p><strong>Dynamic approach:</strong></p>

<pre>mem = []

fib(n)

If n in mem: return mem[n]

else,

If n < 2, f = 1

else , f = f(n - 1) + f(n -2)

mem[n] = f

return f

</pre>

<p>In a dynamic approach, <var>mem</var> stores the result of each subproblem.</p>

<hr><h2 id="advantages">Advantages of Divide and Conquer Algorithm</h2>

<ul><li>The complexity for the multiplication of two matrices using the naive method is <var>O(n<sup>3</sup>)</var>, whereas using the divide and conquer approach (i.e. Strassen's matrix multiplication) is <var>O(n<sup>2.8074</sup>)</var>. This approach also simplifies other problems, such as the Tower of Hanoi.</li>

<li>This approach is suitable for multiprocessing systems.</li>

<li>It makes efficient use of memory caches.</li>

</ul><hr><h2 id="application">Divide and Conquer Applications</h2>

<ul><li><a href="/dsa/binary-search.html">Binary Search</a></li>

<li><a href="/dsa/merge-sort.html">Merge Sort</a></li>

<li><a href="/dsa/quick-sort.html">Quick Sort</a></li>

<li>Strassen's Matrix multiplication</li>

<li>Karatsuba Algorithm</li>

</ul></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#complexity">Time Complexity</a></li>

<li><a href="#advantages">Advantages of Divide and Conquer</a></li>

<li><a href="#application">Divide and Conquer Applications</a></li>

</ul></div></div></div> </div>

<div>

<ul>