Minimize prize count required such that smaller value gets less prize in an adjacent pair

Given an array arr[] of length N, the task is to find the minimum number of prizes required such that if two elements are adjacent, then elements with smaller value gets a less number of prizes compared to its adjacent elements with greater value. 
Note: Each elements will get at least one prize. 

Examples:

Input: arr[] = {1, 2, 2, 3} 
Output: 6 
Explanation:
Element at index {0} will get {1} prize. 
Element at index {1} will get {2} prizes. 
Element at index {2} will get {1} prizes. 
Element at index {3} will get {2} prizes. 
So, the total number of prizes required to satisfy 
the above conditions are 6

Input: arr[] = {3, 2, 2, 1} 
Output: 6 
Explanation:
Element at index {0} will get {2} prize. 
Element at index {1} will get {1} prizes. 
Element at index {2} will get {2} prizes. 
Element at index {3} will get {1} prizes. 
So, the total number of prizes required to satisfy 
the above conditions are 6

Naive Approach: Traverse over the elements of the array and for each element of the array find the consecutive smaller elements at the left of the element and find the consecutive smaller elements on the right of that index.

Prize at index i = max(Consecutive smaller elements at left, Consecutive smaller elements at right, 1)

Below is the implementation of the above approach:

C++

// C++ implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes

#include <bits/stdc++.h>

using namespace std;

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
int findMinPrizes(int arr[], int n)
{
    int totalPrizes = 0, j, x, y;

    // Loop to iterate over every
    // elements of the array
    for (int i = 0; i < n; i++) {
        x = 1;
        j = i;

        // Loop to find the consecutive
        // smaller elements at left
        while (j > 0 && arr[j] > arr[j - 1]) {
            x++;
            j--;
        }
        j = i;
        y = 1;

        // Loop to find the consecutive
        // smaller elements at right
        while (j < n - 1 && arr[j] > arr[j + 1]) {
            y++;
            j++;
        }

        totalPrizes += max({ x, y });
    }
    cout << totalPrizes << endl;

    return 0;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 2, 3 };
    int n = sizeof(arr) / sizeof(arr[0]);

    findMinPrizes(arr, n);
}

Java

// Java implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes
import java.util.*;
class GFG{

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
static int findMinPrizes(int arr[], int n)
{
    int totalPrizes = 0, j, x, y;

    // Loop to iterate over every
    // elements of the array
    for (int i = 0; i < n; i++)
    {
        x = 1;
        j = i;

        // Loop to find the consecutive
        // smaller elements at left
        while (j > 0 && arr[j] > arr[j - 1])
        {
            x++;
            j--;
        }
        j = i;
        y = 1;

        // Loop to find the consecutive
        // smaller elements at right
        while (j < n - 1 && arr[j] > arr[j + 1])
        {
            y++;
            j++;
        }

        totalPrizes += Math.max(x, y );
    }
    System.out.print(totalPrizes + "\n");

    return 0;
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 2, 3 };
    int n = arr.length;

    findMinPrizes(arr, n);
}
}

// This code is contributed by gauravrajput1

Python3

# Python3 implementation to find the
# minimum prizes required such
# that adjacent smaller elements
# gets less number of prizes

# Function to find the minimum
# number of required such that
# adjacent smaller elements gets
# less number of prizes
def findMinPrizes(arr, n):

    totalPrizes = 0

    # Loop to iterate over every
    # elements of the array
    for i in range(n):
        x = 1
        j = i

        # Loop to find the consecutive
        # smaller elements at left
        while (j > 0 and arr[j] > arr[j - 1]):
            x += 1
            j -= 1

        j = i
        y = 1

        # Loop to find the consecutive
        # smaller elements at right
        while (j < n - 1 and arr[j] >
                             arr[j + 1]):
            y += 1
            j += 1

        totalPrizes += max(x, y)

    print(totalPrizes)

# Driver code
arr = [ 1, 2, 2, 3 ]
n = len(arr)

findMinPrizes(arr, n)

# This code is contributed by stutipathak31jan

C#

// C# implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes
using System;

class GFG{

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
static int findMinPrizes(int []arr, int n)
{
    int totalPrizes = 0, j, x, y;

    // Loop to iterate over every
    // elements of the array
    for(int i = 0; i < n; i++)
    {
        x = 1;
        j = i;

        // Loop to find the consecutive
        // smaller elements at left
        while (j > 0 && arr[j] > arr[j - 1])
        {
            x++;
            j--;
        }
        j = i;
        y = 1;

        // Loop to find the consecutive
        // smaller elements at right
        while (j < n - 1 &&
          arr[j] > arr[j + 1])
        {
            y++;
            j++;
        }
        totalPrizes += Math.Max(x, y);
    }
    Console.Write(totalPrizes + "\n");

    return 0;
}

// Driver Code
public static void Main(String[] args)
{
    int []arr = { 1, 2, 2, 3 };
    int n = arr.Length;

    findMinPrizes(arr, n);
}
}

// This code is contributed by Amit Katiyar

Javascript

<script>

    // Javascript implementation to find the
    // minimum prizes required such
    // that adjacent smaller elements
    // gets less number of prizes

    // Function to find the minimum
    // number of required such that
    // adjacent smaller elements gets
    // less number of prizes
    function findMinPrizes(arr, n)
    {
        let totalPrizes = 0, j, x, y;

        // Loop to iterate over every
        // elements of the array
        for (let i = 0; i < n; i++) {
            x = 1;
            j = i;

            // Loop to find the consecutive
            // smaller elements at left
            while (j > 0 && arr[j] > arr[j - 1]) {
                x++;
                j--;
            }
            j = i;
            y = 1;

            // Loop to find the consecutive
            // smaller elements at right
            while (j < n - 1 && arr[j] > arr[j + 1]) {
                y++;
                j++;
            }

            totalPrizes += Math.max( x, y );
        }
        document.write(totalPrizes);

        return 0;
    }

    let arr = [ 1, 2, 2, 3 ];
    let n = arr.length;
    findMinPrizes(arr, n);

    // This code is contributed by divyesh072019.
</script>

Output

6

Performance Analysis:

• Time Complexity: O(N²)
• Auxiliary Space: O(1)

Efficient Approach: The idea is to precompute the count of consecutive smaller elements at left and right for every element of the array. It means that, if an element on the left is smaller, then all the smaller elements at the left of that element will also be smaller to the current element. i.e.

if (arr[i-1] < arr[i])
    smallerLeft[i] = smallerLeft[i-1] + 1

Similarly, the consecutive smaller elements can be computed using the fact that, if an element on the right is greater than the current element then the consecutive greater elements on the right will also be greater than the current element. i.e.

if (arr[i] < arr[i+1])
    smallerRight[i] = smallerRight[i+1] + 1

Below is the implementation of the above approach:

C++

// C++ implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes

#include <bits/stdc++.h>

using namespace std;

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
int minPrizes(int arr[], int n)
{
    int dpLeft[n];

    dpLeft[0] = 1;

    // Loop to compute the smaller
    // elements at the left
    for (int i = 1; i < n; i++) {

        if (arr[i] > arr[i - 1]) {

            dpLeft[i] = dpLeft[i - 1] + 1;
        }
        else {

            dpLeft[i] = 1;
        }
    }

    int dpRight[n];

    dpRight[n - 1] = 1;

    // Loop to find the smaller
    // elements at the right
    for (int i = n - 2; i >= 0; i--) {

        if (arr[i] > arr[i + 1]) {

            dpRight[i] = dpRight[i + 1] + 1;
        }
        else {

            dpRight[i] = 1;
        }
    }

    int totalPrizes = 0;

    // Loop to find the minimum
    // prizes required
    for (int i = 0; i < n; i++) {

        totalPrizes += max(dpLeft[i],
                           dpRight[i]);
    }
    cout << totalPrizes << endl;

    return 0;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 2, 3 };
    int n = sizeof(arr) / sizeof(arr[0]);

    minPrizes(arr, n);

    return 0;
}

Java

// Java implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes
import java.util.*;

class GFG{

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
static int minPrizes(int arr[], int n)
{
    int []dpLeft = new int[n];

    dpLeft[0] = 1;

    // Loop to compute the smaller
    // elements at the left
    for(int i = 1; i < n; i++)
    {
        if (arr[i] > arr[i - 1])
        {
            dpLeft[i] = dpLeft[i - 1] + 1;
        }
        else
        {
            dpLeft[i] = 1;
        }
    }

    int []dpRight = new int[n];

    dpRight[n - 1] = 1;

    // Loop to find the smaller
    // elements at the right
    for(int i = n - 2; i >= 0; i--)
    {
        if (arr[i] > arr[i + 1])
        {
            dpRight[i] = dpRight[i + 1] + 1;
        }
        else
        {
            dpRight[i] = 1;
        }
    }

    int totalPrizes = 0;

    // Loop to find the minimum
    // prizes required
    for(int i = 0; i < n; i++)
    {
        totalPrizes += Math.max(dpLeft[i],
                               dpRight[i]);
    }
    System.out.print(totalPrizes + "\n");
    return 0;
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 2, 3 };
    int n = arr.length;

    minPrizes(arr, n);
}
}

// This code is contributed by Amit Katiyar

Python3

# Python3 implementation to find the
# minimum prizes required such
# that adjacent smaller elements
# gets less number of prizes

# Function to find the minimum
# number of required such that
# adjacent smaller elements gets
# less number of prizes
def minPrizes(arr, n):

    dpLeft = [0] * n
    dpLeft[0] = 1

    # Loop to compute the smaller
    # elements at the left
    for i in range(n):
        if arr[i] > arr[i - 1]:
            dpLeft[i] = dpLeft[i - 1] + 1

        else:
            dpLeft[i] = 1

    dpRight = [0] * n
    dpRight[-1] = 1

    # Loop to find the smaller
    # elements at the right
    for i in range(n - 2, -1, -1):
        if arr[i] > arr[i + 1]:
            dpRight[i] = dpRight[i + 1] + 1

        else:
            dpRight[i] = 1

    totalPrizes = 0

    # Loop to find the minimum
    # prizes required
    for i in range(n):
        totalPrizes += max(dpLeft[i],
                           dpRight[i])

    print(totalPrizes)

# Driver code
arr = [ 1, 2, 2, 3 ]
n = len(arr)

minPrizes(arr, n)

# This code is contributed by stutipathak31jan

C#

// C# implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes
using System;

class GFG{

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
static int minPrizes(int []arr, int n)
{
    int []dpLeft = new int[n];

    dpLeft[0] = 1;

    // Loop to compute the smaller
    // elements at the left
    for(int i = 1; i < n; i++)
    {
        if (arr[i] > arr[i - 1])
        {
            dpLeft[i] = dpLeft[i - 1] + 1;
        }
        else
        {
            dpLeft[i] = 1;
        }
    }

    int []dpRight = new int[n];

    dpRight[n - 1] = 1;

    // Loop to find the smaller
    // elements at the right
    for(int i = n - 2; i >= 0; i--)
    {
        if (arr[i] > arr[i + 1])
        {
            dpRight[i] = dpRight[i + 1] + 1;
        }
        else
        {
            dpRight[i] = 1;
        }
    }

    int totalPrizes = 0;

    // Loop to find the minimum
    // prizes required
    for(int i = 0; i < n; i++)
    {
        totalPrizes += Math.Max(dpLeft[i],
                               dpRight[i]);
    }
    Console.Write(totalPrizes + "\n");
    return 0;
}

// Driver Code
public static void Main(String[] args)
{
    int []arr = { 1, 2, 2, 3 };
    int n = arr.Length;

    minPrizes(arr, n);
}
}

// This code is contributed by Rajput-Ji

Javascript

<script>

// Javascript implementation to find the
// minimum prizes required such
// that adjacent smaller elements
// gets less number of prizes

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
function minPrizes(arr, n)
{
    let dpLeft = new Array(n);

    dpLeft[0] = 1;

    // Loop to compute the smaller
    // elements at the left
    for(let i = 1; i < n; i++)
    {
        if (arr[i] > arr[i - 1])
        {
            dpLeft[i] = dpLeft[i - 1] + 1;
        }
        else
        {
            dpLeft[i] = 1;
        }
    }

    let dpRight = new Array(n);

    dpRight[n - 1] = 1;

    // Loop to find the smaller
    // elements at the right
    for(let i = n - 2; i >= 0; i--)
    {
        if (arr[i] > arr[i + 1])
        {
            dpRight[i] = dpRight[i + 1] + 1;
        }
        else
        {
            dpRight[i] = 1;
        }
    }

    let totalPrizes = 0;

    // Loop to find the minimum
    // prizes required
    for(let i = 0; i < n; i++)
    {
        totalPrizes += Math.max(dpLeft[i],
                               dpRight[i]);
    }
    document.write(totalPrizes + "</br>");
    return 0;
}

// Driver code
let arr = [ 1, 2, 2, 3 ];
let n = arr.length;

minPrizes(arr, n);

// This code is contributed by suresh07

</script>

Output

6

Performance Analysis:

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

Efficient approach : Space optimization O(1)

In previous approach we the current value dpLeft[i] is only depend upon the previous values i.e. dp[i-1] and current value dpRight[i] is only depend upon the next values i.e. dpRight[i+1] So to optimize the space we can keep track of previous , next and current values by the help of  variables which will reduce the space complexity from O(N) to O(1).  

Implementation Steps:

• Create 2 variables dpLeft and dpRight to keep track of previous and next values of DP.
• Initialize base case dpLeft = dpRight = 1.
• Create a variable totalPrizes to store current value.
• Iterate over subproblem using loop and update totalPrizes.
• After every iteration update dpLeft and dpRight for further iterations.
• At last return totalPrizes.

Implementation 

C++

#include <bits/stdc++.h>

using namespace std;

// Function to find the minimum
// number of required such that
// adjacent smaller elements gets
// less number of prizes
int minPrizes(int arr[], int n)
{
    int dpLeft = 1, dpRight = 1, totalPrizes = 1;

    // Loop to compute the smaller
    // elements at the left and right
    for (int i = 1; i < n; i++) {
        if (arr[i] > arr[i - 1]) {
            dpLeft++;
        } else {
            dpLeft = 1;
        }

        if (arr[n - i - 1] > arr[n - i]) {
            dpRight++;
        } else {
            dpRight = 1;
        }

        totalPrizes += max(dpLeft, dpRight);
    }

    cout << totalPrizes << endl;

    return 0;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 2, 3 };
    int n = sizeof(arr) / sizeof(arr[0]);

    minPrizes(arr, n);

    return 0;
}

Java

import java.util.*;

public class Main {

    // Function to find the minimum number
    // of required such that adjacent smaller
    // elements gets less number of prizes
    static int minPrizes(int arr[], int n)
    {
        int dpLeft = 1, dpRight = 1, totalPrizes = 1;

        // Loop to compute the smaller
        // elements at the left and right
        for (int i = 1; i < n; i++) {
            if (arr[i] > arr[i - 1]) {
                dpLeft++;
            }
            else {
                dpLeft = 1;
            }

            if (arr[n - i - 1] > arr[n - i]) {
                dpRight++;
            }
            else {
                dpRight = 1;
            }

            totalPrizes += Math.max(dpLeft, dpRight);
        }

        System.out.println(totalPrizes);

        return 0;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 1, 2, 2, 3 };
        int n = arr.length;

        minPrizes(arr, n);
    }
}

Python3

# Python program for above approach

# Function to find the minimum
# number of required such that
# adjacent smaller elements gets
# less number of prizes
def minPrizes(arr, n):
    dpLeft = 1
    dpRight = 1
    totalPrizes = 1

    # Loop to compute the smaller
    # elements at the left and right
    for i in range(1, n):
        if arr[i] > arr[i-1]:
            dpLeft += 1
        else:
            dpLeft = 1

        if arr[n-i-1] > arr[n-i]:
            dpRight += 1
        else:
            dpRight = 1

        totalPrizes += max(dpLeft, dpRight)

    print(totalPrizes)

    return 0

# Driver Code
if __name__ == '__main__':
    arr = [1, 2, 2, 3]
    n = len(arr)

    minPrizes(arr, n)

C#

using System;

class Program
{
    // Function to find the minimum number of required prizes
    // such that adjacent smaller elements get fewer prizes
    static void MinPrizes(int[] arr, int n)
    {
        int dpLeft = 1;
        int dpRight = 1;
        int totalPrizes = 1;

        // Loop to compute the smaller elements at the left and right
        for (int i = 1; i < n; i++)
        {
            if (arr[i] > arr[i - 1])
                dpLeft += 1;
            else
                dpLeft = 1;

            if (arr[n - i - 1] > arr[n - i])
                dpRight += 1;
            else
                dpRight = 1;

            totalPrizes += Math.Max(dpLeft, dpRight);
        }

        Console.WriteLine(totalPrizes);
    }
// Driver code
    static void Main()
    {
        int[] arr = { 1, 2, 2, 3 };
        int n = arr.Length;

        MinPrizes(arr, n);
    }
}

Javascript

// Function to find the minimum number of required prizes
function minPrizes(arr) {
    let dpLeft = 1;
    let dpRight = 1;
    let totalPrizes = 1;
    const n = arr.length;

    // Loop to compute the smaller elements at the left and right
    for (let i = 1; i < n; i++) {
        if (arr[i] > arr[i - 1]) {
            dpLeft++;
        } else {
            dpLeft = 1;
        }

        if (arr[n - i - 1] > arr[n - i]) {
            dpRight++;
        } else {
            dpRight = 1;
        }

        totalPrizes += Math.max(dpLeft, dpRight);
    }

    console.log(totalPrizes);
}

// Driver code
const arr = [1, 2, 2, 3];
minPrizes(arr);

Output

6

Performance Analysis:

• Time Complexity: O(N)
• Auxiliary Space: O(1)