Minimize maximum array element by splitting array elements into powers of two at most K times
Last Updated :
22 Feb, 2023
Given an array arr[] consisting of N positive integers and an integer K, the task is to minimize the maximum value of the array by splitting the array element into powers of 2 at most K times.
Examples:
Input: arr[] = {2, 4, 11, 2}, K = 2
Output: 2
Explanation:
Below are the operations performed on array elements at most K(= 2) times:
Operation 1: Remove the element at index 2, i.e., arr[2] = 11 and replace it with 11 numbers of 1s in it. Now the array arr[] modifies to {2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2}.
Operation 2: Remove the element at index 1, i.e., arr[1] = 4 and replace it with 4 numbers of 1s in it. Now the array arr[] modifies to {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2}.
After performing the above operations, the maximum value of the array is 2, which is minimum possible value.
Input: arr[]= {9}, K = 2
Output: 1
Approach: The given problem can be solved by using the fact that every number can be expressed the sum of 1 which is a power of 2. Follow the steps below to solve the problem:
Below is the implementation of the above approach:
C++
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to find the minimum value
// of the maximum element of the array
// by splitting at most K array element
// into perfect powers of 2
void minimumSize(int arr[], int N, int K)
{
// Sort the array element in
// the ascending order
sort(arr, arr + N);
// Reverse the array
reverse(arr, arr + N);
// If count of 0 is equal to N
if (count(arr, arr + N, 0) == N)
cout << 0;
// Otherwise, if K is greater
// than or equal to N
else if (K >= N)
cout << 1 << endl;
// Otherwise
else
cout << arr[K] << endl;
}
// Driver Code
int main()
{
int arr[] = { 2, 4, 8, 2 };
int K = 2;
int N = sizeof(arr) / sizeof(arr[0]);
minimumSize(arr, N, K);
return 0;
}
// Java program for the above approach
import java.lang.*;
import java.util.*;
class GFG{
// Function to find the minimum value
// of the maximum element of the array
// by splitting at most K array element
// into perfect powers of 2
static void minimumSize(int arr[], int N, int K)
{
// Sort the array element in
// the ascending order
Arrays.sort(arr);
// Reverse the array
reverse(arr);
// If count of 0 is equal to N
if (count(arr, 0) == N)
System.out.println(0);
// Otherwise, if K is greater
// than or equal to N
else if (K >= N)
System.out.println(1);
// Otherwise
else
System.out.println(arr[K]);
}
static void reverse(int[] a)
{
int i, k, t;
int n = a.length;
for(i = 0; i < n / 2; i++)
{
t = a[i];
a[i] = a[n - i - 1];
a[n - i - 1] = t;
}
}
static int count(int[] a, int n)
{
int freq = 0;
for(int i = 0; i < a.length; i++)
{
if (a[i] == n)
freq++;
}
return freq;
}
// Driver code
public static void main(String[] args)
{
int arr[] = { 2, 4, 8, 2 };
int K = 2;
int N = arr.length;
minimumSize(arr, N, K);
}
}
// This code is contributed by offbeat
# Python program for the above approach
# Function to find the minimum value
# of the maximum element of the array
# by splitting at most K array element
# into perfect powers of 2
def minimumSize(arr, N, K):
# Sort the array element in
# the ascending order
arr.sort()
# Reverse the array
arr.reverse()
# If count of 0 is equal to N
zero = arr.count(0)
if zero == N:
print(0)
# Otherwise, if K is greater
# than or equal to N
elif K >= N:
print(1)
# Otherwise
else:
print(arr[K])
# Driver Code
arr = [2, 4, 8, 2]
K = 2
N = len(arr)
minimumSize(arr, N, K)
# This code is contributed by sudhanshugupta2019a.
// C#program for the above approach
using System;
class GFG
{
// Function to find the minimum value
// of the maximum element of the array
// by splitting at most K array element
// into perfect powers of 2
static void minimumSize(int[] arr, int N, int K)
{
// Sort the array element in
// the ascending order
Array.Sort(arr);
// Reverse the array
Array.Reverse(arr);
// If count of 0 is equal to N
if (count(arr, 0) == N)
Console.WriteLine(0);
// Otherwise, if K is greater
// than or equal to N
else if (K >= N)
Console.WriteLine(1);
// Otherwise
else
Console.WriteLine(arr[K]);
}
static void reverse(int[] a)
{
int i, t;
int n = a.Length;
for (i = 0; i < n / 2; i++) {
t = a[i];
a[i] = a[n - i - 1];
a[n - i - 1] = t;
}
}
static int count(int[] a, int n)
{
int freq = 0;
for (int i = 0; i < a.Length; i++) {
if (a[i] == n)
freq++;
}
return freq;
}
// Driver code
public static void Main(string[] args)
{
int[] arr = { 2, 4, 8, 2 };
int K = 2;
int N = arr.Length;
minimumSize(arr, N, K);
}
}
// This code is contributed by ukasp.
<script>
// JavaScript program for the above approach
// Function to find the minimum value
// of the maximum element of the array
// by splitting at most K array element
// into perfect powers of 2
function minimumSize(arr,N,K)
{
// Sort the array element in
// the ascending order
(arr).sort(function(a,b){return a-b;});
// Reverse the array
reverse(arr);
// If count of 0 is equal to N
if (count(arr, 0) == N)
document.write(0);
// Otherwise, if K is greater
// than or equal to N
else if (K >= N)
document.write(1);
// Otherwise
else
document.write(arr[K]);
}
function reverse(a)
{
let i, k, t;
let n = a.length;
for(i = 0; i < n / 2; i++)
{
t = a[i];
a[i] = a[n - i - 1];
a[n - i - 1] = t;
}
}
function count(a,n)
{
let freq = 0;
for(let i = 0; i < a.length; i++)
{
if (a[i] == n)
freq++;
}
return freq;
}
// Driver code
let arr=[2, 4, 8, 2];
let K = 2;
let N = arr.length;
minimumSize(arr, N, K);
// This code is contributed by avanitrachhadiya2155
</script>
Time Complexity: O(N * log N)
Auxiliary Space: O(1)
Another efficient approach: In the previous approach, we are sorting the array and found the (K+1)th maximum element for the K<N. Instead of sorting the array, we can use a priority queue to find the (K+1)th maximum element.
The time complexity for this approach in the worst case is O(N*log(K)) for k<N otherwise, the time complexity is O(1). Hence the given approach is much better than the previous approach for a smaller value of k.
Below is the implementation of the above approach:
C++
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to find the minimum value
// of the maximum element of the array
// by splitting at most K array element
// into perfect powers of 2
void minimumSize(int arr[], int N, int K)
{
// If count of 0 is equal to N
if (count(arr, arr + N, 0) == N)
cout << 0;
// Otherwise, if K is greater
// than or equal to N
else if (K >= N)
cout << 1 << endl;
// Otherwise
else
{
// Finding (K+1)th maximum element
// using a priority_queue
priority_queue<int, vector<int>, greater<int> >pq;
for (int i = 0; i < N; ++i) {
// Insert elements into
// the priority queue
pq.push(arr[i]);
// If size of the priority
// queue exceeds k+1
if (pq.size() > (K+1)) {
pq.pop();
}
}
// Print the (K+1)th maximum element
cout<<pq.top()<<endl;
}
}
// Driver Code
int main()
{
int arr[] = { 2, 4, 8, 2 };
int K = 2;
int N = sizeof(arr) / sizeof(arr[0]);
minimumSize(arr, N, K);
return 0;
}
// This code is contributed by Pushpesh raj
# Python program for the above approach
import bisect
# Function to find the minimum value
# of the maximum element of the array
# by splitting at most K array element
# into perfect powers of 2
def minimumSize(arr,N,K):
# If count of 0 is equal to N
if(arr.count(0)==N):
print(0)
# Otherwise, if K is greater
# than or equal to N
elif(K>=N):
print(1)
# Otherwise
else:
# Finding (K+1)th maximum element
# using a priority_queue
pq=[]
for i in range(N):
# Insert elements into
# the priority queue
bisect.insort(pq,arr[i])
# If size of the priority
# queue exceeds k+1
if(len(pq)>(K+1)):
pq.pop()
# Print the (K+1)th maximum element
print(pq[0])
# Driver Code
arr=[2,4,8,2]
K=2
N=len(arr)
minimumSize(arr,N,K)
# This code is contributed by Aman Kumar.
import java.util.PriorityQueue;
public class Main {
// Function to find the minimum value of the maximum
// element of the array by splitting at most K array
// element into perfect powers of 2
public static void minimumSize(int[] arr, int N, int K)
{
// If count of 0 is equal to N
if (isZeroArray(arr, N)) {
System.out.println(0);
// Otherwise, if K is greater than or equal to N
}
else if (K >= N) {
System.out.println(1);
// Otherwise
}
else {
// Finding (K+1)th maximum element using a
// priority_queue
PriorityQueue<Integer> pq
= new PriorityQueue<>();
for (int i = 0; i < N; ++i) {
// Insert elements into the priority queue
pq.add(arr[i]);
// If size of the priority queue exceeds k+1
if (pq.size() > (K + 1)) {
pq.poll();
}
}
// Print the (K+1)th maximum element
System.out.println(pq.peek());
}
}
// Function to check if the given array only contains
// zeros
private static boolean isZeroArray(int[] arr, int N)
{
for (int i = 0; i < N; i++) {
if (arr[i] != 0) {
return false;
}
}
return true;
}
// Driver code
public static void main(String[] args)
{
int[] arr = { 2, 4, 8, 2 };
int K = 2;
int N = arr.length;
minimumSize(arr, N, K);
}
} // this code is contributed by devendra1
// Define a function to find the minimum value of the maximum
// element of the array by splitting at most K array element into perfect powers of 2
function minimumSize(arr, N, K)
{
// If count of 0 is equal to N
if (isZeroArray(arr, N)) {
console.log(0);
// Otherwise, if K is greater than or equal to N
}
else if (K >= N) {
console.log(1);
// Otherwise
}
else
{
// Finding (K+1)th maximum element using a priority_queue
// Create an empty array to serve as the priority queue
const pq = [];
for (let i = 0; i < N; ++i)
{
// Insert elements into the priority queue
pq.push(arr[i]);
// Sort the array in descending order
pq.sort((a, b) => b - a);
// If size of the priority queue exceeds k+1
if (pq.length > (K + 1))
{
// Remove the smallest element from the queue
pq.pop();
}
}
// Print the (K+1)th maximum element
console.log(pq[K]);
}
}
// Function to check if the given array only contains zeros
function isZeroArray(arr, N) {
for (let i = 0; i < N; i++) {
if (arr[i] !== 0) {
return false;
}
}
return true;
}
// Driver code
const arr = [2, 4, 8, 2];
const K = 2;
const N = arr.length;
minimumSize(arr, N, K);
using System;
using System.Collections.Generic;
using System.Linq;
public class Program
{
// Function to find the minimum value
// of the maximum element of the array
// by splitting at most K array element
// into perfect powers of 2
public static void minimumSize(int[] arr, int N, int K)
{
// If count of 0 is equal to N
if (arr.Count(x => x == 0) == N)
{
Console.WriteLine(0);
}
// Otherwise, if K is greater
// than or equal to N
else if (K >= N)
{
Console.WriteLine(1);
}
// Otherwise
else
{
// Finding (K+1)th maximum element
// using a priority_queue
var pq = new SortedSet<int>();
foreach (var x in arr)
{
// Insert elements into
// the priority queue
pq.Add(x);
// If size of the priority
// queue exceeds k+1
if (pq.Count > (K + 1))
{
pq.Remove(pq.Max);
}
}
// Print the (K+1)th maximum element
Console.WriteLine(pq.Min);
}
}
// Driver Code
public static void Main()
{
int[] arr = { 2, 4, 8, 2 };
int K = 2;
int N = arr.Length;
minimumSize(arr, N, K);
}
}
Time Complexity: O(N*log(K))
Auxiliary Space: O(K)
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