Minimum in an array which is first decreasing then increasing

Given an array of N integers where array elements form a strictly decreasing and increasing sequence. The task is to find the smallest number in such an array. 
Constraints: N >= 3 
Examples:

Input: a[] = {2, 1, 2, 3, 4}
Output: 1
Input: a[] = {8, 5, 4, 3, 4, 10}
Output: 3 

A naive approach is to linearly traverse the array and find out the smallest number.

C++

// CPP program to find minimum element
// in an array.
#include <bits/stdc++.h>
using namespace std;

int minimal(int arr[], int n)
{
    int ans = arr[0];
    for (int i = 1; i < n; i++)
        ans = min(ans, arr[i]);
    return ans;
}


int main()
{
    int arr[] = { 8, 5, 4, 3, 4, 10 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << minimal(arr, n);
    return 0;
}
// This code is contributed by Vishal Dhaygude

Java

import java.util.*;

public class Main {

    public static int minimal(int[] arr, int n) {
        int ans = arr[0];
        for (int i = 1; i < n; i++) {
            ans = Math.min(ans, arr[i]);
        }
        return ans;
    }

    public static void main(String[] args) {
        int[] arr = { 8, 5, 4, 3, 4, 10 };
        int n = arr.length;
        System.out.println(minimal(arr, n));
    }
}

Python3

#python code
def minimal(arr, n):
    ans = arr[0]
    for i in range(1, n):
        ans = min(ans, arr[i])
    return ans

arr = [8, 5, 4, 3, 4, 10]
n = len(arr)
print(minimal(arr, n))

C#

// C# program to find minimum element
// in an array.
using System;

public class Program {
    public static int Minimal(int[] arr, int n)
    {
        int ans = arr[0];
        for (int i = 1; i < n; i++)
            ans = Math.Min(ans, arr[i]);
        return ans;
    }
    public static void Main()
    {
        int[] arr = { 8, 5, 4, 3, 4, 10 };
        int n = arr.Length;
        Console.WriteLine(Minimal(arr, n));
    }
}
// This code is contributed by user_dtewbxkn77n

Javascript

// JavaScript program to find minimum element
// in an array.

function minimal(arr, n) {
    let ans = arr[0];
    for (let i = 1; i < n; i++) {
        ans = Math.min(ans, arr[i]);
    }
    return ans;
}

let arr = [8, 5, 4, 3, 4, 10];
let n = arr.length;
console.log(minimal(arr, n));

Output

3

Time Complexity: O(N), we need to use a loop to traverse N times linearly. 

Auxiliary Space: O(1), as we are not using any extra space.

An efficient approach is to modify the binary search and use it. Divide the array into two halves use binary search to check if a[mid] < a[mid+1] or not. If a[mid] < a[mid+1], then the smallest number lies in the first half which is low to mid, else it lies in the second half which is mid+1 to high. 
Algorithm: 

while(lo > 1
    if a[mid] < a[mid+1] then hi = mid
    else lo = mid+1
} 

Below is the implementation of the above approach: 

C++

// C++ program to find the smallest number
// in an array of decrease and increasing numbers
#include <bits/stdc++.h>
using namespace std;

// Function to find the smallest number's index
int minimal(int a[], int n)
{
    int lo = 0, hi = n - 1;

    // Do a binary search
    while (lo < hi) {

        // Find the mid element
        int mid = (lo + hi) >> 1;

        // Check for break point
        if (a[mid] < a[mid + 1]) {
            hi = mid;
        }
        else {
            lo = mid + 1;
        }
    }

    // Return the index
    return lo;
}

// Driver Code
int main()
{
    int a[] = { 8, 5, 4, 3, 4, 10 };
    int n = sizeof(a) / sizeof(a[0]);
    int ind = minimal(a, n);

    // Print the smallest number
    cout << a[ind];
}

Java

// Java program to find the smallest number
// in an array of decrease and increasing numbers
class Solution
{
// Function to find the smallest number's index
static int minimal(int a[], int n)
{
    int lo = 0, hi = n - 1;

    // Do a binary search
    while (lo < hi) {

        // Find the mid element
        int mid = (lo + hi) >> 1;

        // Check for break point
        if (a[mid] < a[mid + 1]) {
            hi = mid;
        }
        else {
            lo = mid + 1;
        }
    }

    // Return the index
    return lo;
}

// Driver Code
public static void main(String args[])
{
    int a[] = { 8, 5, 4, 3, 4, 10 };
    int n = a.length;
    int ind = minimal(a, n);

    // Print the smallest number
    System.out.println( a[ind]);
}
}
//contributed by Arnab Kundu

Python3

# Python 3 program to find the smallest
# number in a array of decrease and
# increasing numbers

# function to find the smallest
# number's index
def minimal(a, n):

    lo, hi = 0, n - 1

    # Do a binary search
    while lo < hi:

        # find the mid element
        mid = (lo + hi) // 2

        # Check for break point
        if a[mid] < a[mid + 1]:
            hi = mid
        else:
            lo = mid + 1
        return lo

# Driver code
a = [8, 5, 4, 3, 4, 10]

n = len(a)

ind = minimal(a, n)

# print the smallest number
print(a[ind])

# This code is contributed
# by Mohit Kumar

C#

// C# program to find the smallest number
// in an array of decrease and increasing numbers
using System;
class Solution
{
// Function to find the smallest number's index
static int minimal(int[] a, int n)
{
    int lo = 0, hi = n - 1;

    // Do a binary search
    while (lo < hi) {

        // Find the mid element
        int mid = (lo + hi) >> 1;

        // Check for break point
        if (a[mid] < a[mid + 1]) {
            hi = mid;
        }
        else {
            lo = mid + 1;
        }
    }

    // Return the index
    return lo;
}

// Driver Code
public static void Main()
{
    int[] a = { 8, 5, 4, 3, 4, 10 };
    int n = a.Length;
    int ind = minimal(a, n);

    // Print the smallest number
    Console.WriteLine( a[ind]);
}
}
//contributed by Mukul singh

Javascript

<script>

    // Javascript program to find the smallest number
    // in an array of decrease and increasing numbers

    // Function to find the smallest number's index
    function minimal(a, n)
    {
        let lo = 0, hi = n - 1;

        // Do a binary search
        while (lo < hi) {

            // Find the mid element
            let mid = (lo + hi) >> 1;

            // Check for break point
            if (a[mid] < a[mid + 1]) {
                hi = mid;
            }
            else {
                lo = mid + 1;
            }
        }

        // Return the index
        return lo;
    }

    let a = [ 8, 5, 4, 3, 4, 10 ];
    let n = a.length;
    let ind = minimal(a, n);

    // Print the smallest number
    document.write(a[ind]);

</script>

PHP

<?php
// PHP program to find the smallest number
// in an array of decrease and increasing numbers

// Function to find the smallest
// number's index
function minimal($a, $n)
{
    $lo = 0;
    $hi = $n - 1;

    // Do a binary search
    while ($lo < $hi)
    {

        // Find the mid element
        $mid = ($lo + $hi) >> 1;

        // Check for break point
        if ($a[$mid] < $a[$mid + 1])
        {
            $hi = $mid;
        }
        else
        {
            $lo = $mid + 1;
        }
    }

    // Return the index
    return $lo;
}

// Driver Code
$a = array( 8, 5, 4, 3, 4, 10 );
$n = sizeof($a);
$ind = minimal($a, $n);

// Print the smallest number
echo $a[$ind];

// This code is contributed
// by Sach_Code
?>

Output

3

Time Complexity: O(Log N), as we are using binary search, in binary search in each traversal we reduce by division of 2 so the effective time will be 1+1/2+1/4+… which is equivalent to logN.
Auxiliary Space: O(1), as we are not using any extra space.
Method 3(Using Stack) :

1.Create an empty stack to hold the indices of the array elements.
2.Traverse the array from left to right until we find the minimum element. Push the index of each element onto the 
stack as long as the element is greater than or equal to the previous element.
3.Once we find an element that is lesser than the previous element, we know that the minimum element has been
reached. We can then pop all the indices from the 4.stack until we find an index whose corresponding element 
is less than the current element.
4.The minimum element is the element corresponding to the last index remaining on the stack.

Implementation of the above code :

C++

#include <bits/stdc++.h>
using namespace std;

int findMin(int arr[], int n)
{
    stack<int> s;
    int min = 0;

    // traverse the array from left to right
    for (int i = 0; i < n; i++) {
        // push the index onto the stack if the element is
        // greater than or equal to the previous element
        if (s.empty() || arr[i] >= arr[s.top()]) {
            s.push(i);
        }
        else {
            // pop all the indices from the stack until we
            // find an index whose corresponding element is
            // less than the current element
            while (!s.empty() && arr[i] < arr[s.top()]) {
                int index = s.top();
                s.pop();
                // update the minimum element
                if (arr[index] < arr[min]) {
                    min = index;
                }
            }
            // push the current index onto the stack
            s.push(i);
        }
    }

    // the minimum element is the element corresponding to
    // the last index remaining on the stack
    while (!s.empty()) {
        int index = s.top();
        s.pop();
        if (arr[index] < arr[min]) {
            min = index;
        }
    }

    return arr[min];
}

int main()
{

    int arr[] = { 50, 20, 3, 1, 6, 7, 10, 56 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << "The minimum element is " << findMin(arr, n);

    return 0;
}

Java

import java.util.*;

public class Main {
public static int findMin(int[] arr, int n) {
Stack<Integer> s = new Stack<>();
int min = 0;
      // traverse the array from left to right
    for (int i = 0; i < n; i++) {
        // push the index onto the stack if the element is
        // greater than or equal to the previous element
        if (s.empty() || arr[i] >= arr[s.peek()]) {
            s.push(i);
        } else {
            // pop all the indices from the stack until we
            // find an index whose corresponding element is
            // less than the current element
            while (!s.empty() && arr[i] < arr[s.peek()]) {
                int index = s.peek();
                s.pop();
                // update the minimum element
                if (arr[index] < arr[min]) {
                    min = index;
                }
            }
            // push the current index onto the stack
            s.push(i);
        }
    }

    // the minimum element is the element corresponding to
    // the last index remaining on the stack
    while (!s.empty()) {
        int index = s.peek();
        s.pop();
        if (arr[index] < arr[min]) {
            min = index;
        }
    }

    return arr[min];
}

public static void main(String[] args) {
    int[] arr = { 50, 20, 3, 1, 6, 7, 10, 56 };
    int n = arr.length;
    System.out.println("The minimum element is " + findMin(arr, n));
}
}

Python3

# code
from typing import List
import sys


def findMin(arr: List[int], n: int) -> int:
    s = []
    min_index = 0

    # traverse the array from left to right
    for i in range(n):
        # push the index onto the stack if the element is
        # greater than or equal to the previous element
        if not s or arr[i] >= arr[s[-1]]:
            s.append(i)
        else:
            # pop all the indices from the stack until we
            # find an index whose corresponding element is
            # less than the current element
            while s and arr[i] < arr[s[-1]]:
                index = s.pop()
                # update the minimum element
                if arr[index] < arr[min_index]:
                    min_index = index
            # push the current index onto the stack
            s.append(i)

    # the minimum element is the element corresponding to
    # the last index remaining on the stack
    while s:
        index = s.pop()
        if arr[index] < arr[min_index]:
            min_index = index

    return arr[min_index]


# Driver code
arr = [50, 20, 3, 1, 6, 7, 10, 56]
n = len(arr)
print("The minimum element is", findMin(arr, n))

C#

using System;
using System.Collections.Generic;

public class GFG{

    static int findMin(int[] arr, int n)
    {
        Stack<int> s = new Stack<int>();
        int min = 0;

        // traverse the array from left to right
        for (int i = 0; i < n; i++) {
            // push the index onto the stack if the element is
            // greater than or equal to the previous element
            if (s.Count == 0 || arr[i] >= arr[s.Peek()]){
                s.Push(i);
            }
            else {
                // pop all the indices from the stack until we
                // find an index whose corresponding element is
                // less than the current element
                while (s.Count > 0 && arr[i] < arr[s.Peek()]){
                    int index = s.Pop();
                    // update the minimum element
                    if (arr[index] < arr[min]) {
                        min = index;
                    }
                }
                // push the current index onto the stack
                s.Push(i);
            }
        }

        // the minimum element is the element corresponding to
        // the last index remaining on the stack
        while (s.Count > 0){
            int index = s.Pop();
            if (arr[index] < arr[min]) {
                min = index;
            }
        }

        return arr[min];
    }

    static void Main(string[] args)
    {

        int[] arr = { 50, 20, 3, 1, 6, 7, 10, 56 };
        int n = arr.Length;
        Console.WriteLine("The minimum element is " + findMin(arr, n));

    }
}

Javascript

function findMin(arr, n) {
    const s = [];
    let min_index = 0;

    // Traverse the array from left to right
    for (let i = 0; i < n; i++) {
        // Push the index onto the stack if the element is
        // greater than or equal to the previous element
        if (!s.length || arr[i] >= arr[s[s.length - 1]]) {
            s.push(i);
        } else {
            // Pop all the indices from the stack until we
            // find an index whose corresponding element is
            // less than the current element
            while (s.length && arr[i] < arr[s[s.length - 1]]) {
                const index = s.pop();
                // Update the minimum element
                if (arr[index] < arr[min_index]) {
                    min_index = index;
                }
            }
            // Push the current index onto the stack
            s.push(i);
        }
    }

    // The minimum element is the element corresponding to
    // the last index remaining on the stack
    while (s.length) {
        const index = s.pop();
        if (arr[index] < arr[min_index]) {
            min_index = index;
        }
    }

    return arr[min_index];
}

// Driver code
const arr = [50, 20, 3, 1, 6, 7, 10, 56];
const n = arr.length;
console.log("The minimum element is", findMin(arr, n));

// This code is Contributed By - Dwaipayan Bandyopadhyay

Output

The maximum element is 1

Time Complexity : O(N)
Auxiliary Space : O(N)