- Design and Analysis of Algorithms
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- Basics of Algorithms
- DAA - Introduction
- DAA - Analysis of Algorithms
- DAA - Methodology of Analysis
- Asymptotic Notations & Apriori Analysis
- DAA - Space Complexities

- Design Strategies
- DAA - Divide & Conquer
- DAA - Max-Min Problem
- DAA - Merge Sort
- DAA - Binary Search
- Strassen’s Matrix Multiplication
- DAA - Greedy Method
- DAA - Fractional Knapsack
- DAA - Job Sequencing with Deadline
- DAA - Optimal Merge Pattern
- DAA - Dynamic Programming
- DAA - 0-1 Knapsack
- Longest Common Subsequence

- Graph Theory
- DAA - Spanning Tree
- DAA - Shortest Paths
- DAA - Multistage Graph
- Travelling Salesman Problem
- Optimal Cost Binary Search Trees

- Heap Algorithms
- DAA - Binary Heap
- DAA - Insert Method
- DAA - Heapify Method
- DAA - Extract Method

- Sorting Methods
- DAA - Bubble Sort
- DAA - Insertion Sort
- DAA - Selection Sort
- DAA - Quick Sort
- DAA - Radix Sort

- Complexity Theory
- Deterministic vs. Nondeterministic Computations
- DAA - Max Cliques
- DAA - Vertex Cover
- DAA - P and NP Class
- DAA - Cook’s Theorem
- NP Hard & NP-Complete Classes
- DAA - Hill Climbing Algorithm

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# Design and Analysis Max-Min Problem

Let us consider a simple problem that can be solved by divide and conquer technique.

### Problem Statement

The Max-Min Problem in algorithm analysis is finding the maximum and minimum value in an array.

### Solution

To find the maximum and minimum numbers in a given array ** numbers[]** of size

**n**, the following algorithm can be used. First we are representing the

**naive method**and then we will present

**divide and conquer approach**.

## Naïve Method

Naïve method is a basic method to solve any problem. In this method, the maximum and minimum number can be found separately. To find the maximum and minimum numbers, the following straightforward algorithm can be used.

Algorithm: Max-Min-Element (numbers[])max := numbers[1] min := numbers[1] for i = 2 to n do if numbers[i] > max then max := numbers[i] if numbers[i] < min then min := numbers[i] return (max, min)

### Analysis

The number of comparison in Naive method is **2n - 2**.

The number of comparisons can be reduced using the divide and conquer approach. Following is the technique.

## Divide and Conquer Approach

In this approach, the array is divided into two halves. Then using recursive approach maximum and minimum numbers in each halves are found. Later, return the maximum of two maxima of each half and the minimum of two minima of each half.

In this given problem, the number of elements in an array is $y - x + 1$, where **y** is greater than or equal to **x**.

$\mathbf{\mathit{Max - Min(x, y)}}$ will return the maximum and minimum values of an array $\mathbf{\mathit{numbers[x...y]}}$.

Algorithm: Max - Min(x, y)if y – x ≤ 1 then return (max(numbers[x], numbers[y]), min((numbers[x], numbers[y])) else (max1, min1):= maxmin(x, ⌊((x + y)/2)⌋) (max2, min2):= maxmin(⌊((x + y)/2) + 1)⌋,y) return (max(max1, max2), min(min1, min2))

### Analysis

Let ** T(n)** be the number of comparisons made by $\mathbf{\mathit{Max - Min(x, y)}}$, where the number of elements $n = y - x + 1$.

If ** T(n)** represents the numbers, then the recurrence relation can be represented as

$$T(n) = \begin{cases}T\left(\lfloor\frac{n}{2}\rfloor\right)+T\left(\lceil\frac{n}{2}\rceil\right)+2 & for\: n>2\\1 & for\:n = 2 \\0 & for\:n = 1\end{cases}$$

Let us assume that ** n** is in the form of power of

**2**. Hence,

**n = 2**where

^{k}**k**is height of the recursion tree.

So,

$$T(n) = 2.T (\frac{n}{2}) + 2 = 2.\left(\begin{array}{c}2.T(\frac{n}{4}) + 2\end{array}\right) + 2 ..... = \frac{3n}{2} - 2$$

Compared to Naïve method, in divide and conquer approach, the number of comparisons is less. However, using the asymptotic notation both of the approaches are represented by **O(n)**.