Reconstructing the Array with Maximum Possible Sum by Given Operations
Last Updated :
29 Jan, 2024
Consider an array A[] of length N. Suppose, you can create obtain B[] from A[] using below steps:
- Iterate N/2 times on A and follow below the step below:
- Insert (Ai + Ai+(N/2)) and |Ai - Ai+(N/2)| into B[]
- After that rearrange B[] in any random order.
You are given B[], the task is to reconstructing any A[] if exists such that B[] can be obtained using above operations. If it is, then sum of A[] must be maximum possible else output -1.
Examples:
Input: N = 6, B[] = {4, 2, 8, 2, 10, 4}
Output: A[] = {7, 5, 3, 3, 3, 1}
Explanation: Considered that A[] is {7, 5, 3, 3, 3, 1} and B[] empty. Then (N=6)/2 = 3 iterations will be as follows:
- First Iteration (i = 1): (A1 + A1+(6/2)) and |A1 - A1+(6/2)| and (A1 + A1+(6/2)) and |A1 - A1+(6/2)| are 7+3 = 10 and |7-3| = 4 respectively. Updated B[] = {10, 4}
- Second Iteration (i = 2): (A2 + A2+(6/2)) and |A2 - A2+(6/2)| and (A2 + A2+(6/2)) and |A2 - A2+(6/2)| are 5+3 = 8 and ∣5−3∣= 2 respectively. Updated B[] = {10, 4, 8, 2}
- Third Iteration (i = 3): (A3 + A3+(6/2)) and |A3 - A3+(6/2)| and (A3 + A3+(6/2)) and |A3 - A3+(6/2)| are 3+1 = 4 and ∣3−1∣ = 2 respectively. Updated B[] = {10, 4, 8, 2, 4, 2}
It can be verified that obtained updated B[] = {10, 4, 8, 2, 4, 2} is an arrangement of input B[] = {4, 2, 8, 2, 10, 4}. Thus, B[] can be achieved by A[] using given operation. It can also verified that the sum of A[] among all possible such arrays is maximum possible, which is 22 (7+5+3+3+3+1).
Input: N = 2, B[] = {2, 3}
Output: -1
Explanation: It can be verified that no such A[] exists, by which we can obtain B[] using given operation.
Approach:
Firstly, we will check if the sum of B[] is even or not, the reason for checking is this we can confirm that whether such an array A[] really exist or not.
After this break the B[] in to arrays one with all even number and one with odd number. Next the two elements of A[] can be computed by basic math and we need max sum of A[]. Therefore, what we would do is this:
Greatest(EVEN) + Smallest(EVEN) and Greatest(EVEN) - Greatest(EVEN)
Same we will do with the subarray containing odd elements.
Step-by-step approach:
- Initialize two
arrays EV[] for even numbers and OD[] for odd numbers. - Iterate over the
B[] and add even numbers to EV[] and odd numbers to OD[]. - Check if the size of both
EV[] and OD[] is even. If not, output -1. - Sort both
EV[] and OD[]. - Initialize an array let say
A[] of size N to store the result. - Use two pointers
p and q to iterate over EV[] and OD[] from the smallest to the largest and from the largest to the smallest, respectively. - Calculate the sum
X and the difference Y for each pair of elements from EV[] and OD[]. - Store the average of the sum in the first half of array
A[] and the average of the difference in the second half of array A[]. - Continue this process until all elements are processed.
- Print the elements of array
A[].
Below is the implementation of the above approach:
C++
#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
// Function to output A[], if it exists
void FindA(int N, vector<int>& B) {
vector<int> A(N);
vector<int> EV; // Vector to store even numbers
vector<int> OD; // Vector to store odd numbers
for (int i = 0; i < N; i++) {
if (B[i] % 2 == 0) {
EV.push_back(B[i]); // Add even numbers to EV
} else {
OD.push_back(B[i]); // Add odd numbers to OD
}
}
// Check if the number of even and odd numbers are even
if (EV.size() % 2 != 0 || OD.size() % 2 != 0) {
cout << "-1" << endl;
} else {
sort(EV.begin(), EV.end()); // Sort the array of even numbers
sort(OD.begin(), OD.end()); // Sort the array of odd numbers
int n = EV.size();
int p = 0;
int q = n - 1;
int j = 0;
while (p < q) {
int a = EV[q]; // Get the greatest even number
int b = EV[p]; // Get the smallest even number
int X = a + b; // Sum of the greatest and smallest even numbers
int Y = a - b; // Difference of the greatest and smallest even numbers
A[j] = X / 2; // Store the average of the sum in the first half of array A
A[j + (N / 2)] = Y / 2; // Store the average of the difference in the
// second half of array
j++;
q--;
p++;
}
n = OD.size();
p = 0;
q = n - 1;
while (p < q) {
int a = OD[q]; // Get the greatest odd number
int b = OD[p]; // Get the smallest odd number
int X = a + b; // Sum of the greatest and smallest odd numbers
int Y = a - b; // Difference of the greatest and smallest odd numbers
A[j] = X / 2; // Store the average of the sum in the first half of array A
A[j + (N / 2)] = Y / 2; // Store the average of the difference
// in the second half of array
j++;
q--;
p++;
}
for (int i : A) {
cout << i << " ";
}
cout << endl;
}
}
// Driver Code
int main() {
// Input
int N = 6;
vector<int> B = {4, 2, 8, 2, 10, 4};
// Function call
FindA(N, B);
return 0;
}
// Java code to implement the approach
import java.io.*;
import java.lang.*;
import java.util.*;
// Driver Class
class Main {
public static void main(String[] args)
throws java.lang.Exception
{
// Input
int N = 6;
int[] B = { 4, 2, 8, 2, 10, 4 };
// Function_call
FindA(N, B);
}
// method to output A[], If it exists
public static void FindA(int N, int[] B)
{
int[] A = new int[N];
ArrayList<Integer> EV
= new ArrayList<Integer>(); // Array to store
// even numbers
ArrayList<Integer> OD
= new ArrayList<Integer>(); // Array to store
// odd numbers
for (int i = 0; i < N; i++) {
if (B[i] % 2 == 0) {
EV.add(B[i]); // Add even numbers to EV
}
else {
OD.add(B[i]); // Add odd numbers to OD
}
}
// Check if the number of even and odd numbers are
// even
if ((EV.size()) % 2 != 0 || (OD.size()) % 2 != 0) {
System.out.println("-1");
}
else {
Collections.sort(
EV); // Sort the array of even numbers
Collections.sort(
OD); // Sort the array of odd numbers
int n = EV.size();
int p = 0;
int q = n - 1;
int j = 0;
while (p < q) {
int a = EV.get(
q); // Get the greatest even number
int b = EV.get(
p); // Get the smallest even number
int X = (a + b); // Sum of the greatest and
// smallest even numbers
int Y
= (a - b); // Difference of the greatest
// and smallest even numbers
A[j]
= X / 2; // Store the average of the sum
// in the first half of array A
A[j + (N / 2)]
= Y / 2; // Store the average of the
// difference in the second
// half of array A
j++;
q--;
p++;
}
n = OD.size();
p = 0;
q = n - 1;
while (p < q) {
int a = OD.get(
q); // Get the greatest odd number
int b = OD.get(
p); // Get the smallest odd number
int X = (a + b); // Sum of the greatest and
// smallest odd numbers
int Y
= (a - b); // Difference of the greatest
// and smallest odd numbers
A[j]
= X / 2; // Store the average of the sum
// in the first half of array A
A[j + (N / 2)]
= Y / 2; // Store the average of the
// difference in the second
// half of array A
j++;
q--;
p++;
}
for (int i : A) {
System.out.print(i + " ");
}
System.out.println();
}
}
}
def GFG(N, B):
A = [0] * N
EV = []
OD = []
# Separate even and odd numbers
for i in range(N):
if B[i] % 2 == 0:
EV.append(B[i])
else:
OD.append(B[i])
# Check if the number of even and odd numbers is even
if len(EV) % 2 != 0 or len(OD) % 2 != 0:
print("-1")
else:
EV.sort() # Sort the array of even numbers
OD.sort() # Sort the array of odd numbers
n = len(EV)
p = 0
q = n - 1
j = 0
# Process even numbers
while p < q:
a = EV[q]
b = EV[p]
X = a + b
Y = a - b
A[j] = X // 2 # Store the average of the sum in the first half of array A
A[j + N // 2] = Y // 2 # Store the average of the difference in the second half of array A
j += 1
q -= 1
p += 1
n = len(OD)
p = 0
q = n - 1
# Process odd numbers
while p < q:
a = OD[q]
b = OD[p]
X = a + b
Y = a - b
A[j] = X // 2
A[j + N // 2] = Y // 2
j += 1
q -= 1
p += 1
# Print the result
print(" ".join(map(str, A)))
# Driver Code
def main():
# Input
N = 6
B = [4, 2, 8, 2, 10, 4]
# Function call
GFG(N, B)
if __name__ == "__main__":
main()
using System;
using System.Collections.Generic;
using System.Linq;
class Program
{
// Function to output A[], if it exists
static void FindA(int N, List<int> B)
{
List<int> A = new List<int>(new int[N]);
List<int> EV = new List<int>(); // List to store even numbers
List<int> OD = new List<int>(); // List to store odd numbers
// Separate even and odd numbers
foreach (int num in B)
{
if (num % 2 == 0)
{
EV.Add(num); // Add even numbers to EV
}
else
{
OD.Add(num); // Add odd numbers to OD
}
}
// Check if the number of even and odd numbers are even
if (EV.Count % 2 != 0 || OD.Count % 2 != 0)
{
Console.WriteLine("-1");
}
else
{
EV.Sort(); // Sort the list of even numbers
OD.Sort(); // Sort the list of odd numbers
int n = EV.Count;
int p = 0;
int q = n - 1;
int j = 0;
// Merge even numbers
while (p < q)
{
int a = EV[q]; // Get the greatest even number
int b = EV[p]; // Get the smallest even number
int X = a + b; // Sum of the greatest and smallest even numbers
int Y = a - b; // Difference of the greatest and smallest even numbers
A[j] = X / 2; // Store the average of the sum in the first half of array A
A[j + (N / 2)] = Y / 2; // Store the average of the difference in the second half of array
j++;
q--;
p++;
}
n = OD.Count;
p = 0;
q = n - 1;
// Merge odd numbers
while (p < q)
{
int a = OD[q]; // Get the greatest odd number
int b = OD[p]; // Get the smallest odd number
int X = a + b; // Sum of the greatest and smallest odd numbers
int Y = a - b; // Difference of the greatest and smallest odd numbers
A[j] = X / 2; // Store the average of the sum in the first half of array A
A[j + (N / 2)] = Y / 2; // Store the average of the difference in the second half of array
j++;
q--;
p++;
}
// Output the resulting array A
foreach (int i in A)
{
Console.Write(i + " ");
}
Console.WriteLine();
}
}
// Driver Code
static void Main()
{
// Input
int N = 6;
List<int> B = new List<int> { 4, 2, 8, 2, 10, 4 };
// Function call
FindA(N, B);
}
}
function GFG(N, B) {
let A = new Array(N);
let EV = [];
let OD = [];
// Separate even and odd numbers
for (let i = 0; i < N; i++) {
if (B[i] % 2 === 0) {
EV.push(B[i]);
} else {
OD.push(B[i]);
}
}
// Check if the number of the even and odd numbers are even
if (EV.length % 2 !== 0 || OD.length % 2 !== 0) {
console.log("-1");
} else {
EV.sort((a, b) => a - b); // Sort the array of the even numbers
OD.sort((a, b) => a - b); // Sort the array of odd numbers
let n = EV.length;
let p = 0;
let q = n - 1;
let j = 0;
// Process even numbers
while (p < q) {
let a = EV[q];
let b = EV[p];
let X = a + b;
let Y = a - b;
A[j] = X / 2; // Store the average of the sum in the first half of array A
A[j + N / 2] = Y / 2; // Store the average of the difference in the second half of array A
j++;
q--;
p++;
}
n = OD.length;
p = 0;
q = n - 1;
// Process odd numbers
while (p < q) {
let a = OD[q];
let b = OD[p];
let X = a + b;
let Y = a - b;
A[j] = X / 2;
A[j + N / 2] = Y / 2;
j++;
q--;
p++;
}
// Print the result
console.log(A.join(" "));
}
}
// Driver Code
function main() {
// Input
let N = 6;
let B = [4, 2, 8, 2, 10, 4];
// Function call
GFG(N, B);
}
main();
Time Complexity: O(NLogN), As sorting is performed.
Auxiliary Space: O(N), As ArrayLists EV and OD of length N are used.
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