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# Construct a distinct elements array with given size, sum and element upper bound

• Difficulty Level : Hard
• Last Updated : 05 Apr, 2021

Given N size of the original array, SUM total sum of all elements present in the array and K such that no element in array is greater than K, construct the original array where all elements in the array are unique. If there is no solution, print “Not Possible”.
Note : All elements should be positive.
Examples:

```Input : N = 3, SUM = 15, K = 8
Output: array[] = {1, 6, 8}
The constructed array has size 3, sum
15 and all elements are smaller than
or equal to 8.

Input : N = 2,  SUM = 9, K = 10
Output: array[]={1, 8}

Input  : N = 3, SUM = 23, K = 8
Output : Not Possible```

We must choose N elements and all elements must be positive and no element should be greater than K. Since elements are positive and distinct, smallest possible sum is equal to sum of first N natural numbers which is N * (N + 1)/2.
Since all elements should be smaller than or equal to K, largest possible sum is K + (K-1) + (K-2) + ….. + (K-N+1) = (N*K)- (N*(N-1))/2.
So if the given sum is in between smallest and largest possible sum then only the array can be formed, or else we have to print -1.
Following is the complete algorithm if it is feasible to construct the array.
1. Create an array of size N and fill it with first N numbers. So total sum of the array will be smallest possible sum.
2. Find the largest element of the array, but as the array is sorted so array[N] will be largest.

• If the largest element is less than K, we replace it with K and check the new sum of the array.
• If it is less than given SUM, we move to N-1 position in the array because array[N] can’t be further incremented and to maintain uniqueness we decrease K by 1.
• If it is greater than given SUM, we replace an element such that sum will be given SUM, and will come out of loop.
• If the largest element is equal to K, we move to N-1 position in the array because array[N] can’t be further incremented and to maintain uniqueness we will decrease K by 1.

3. Print the array.

## C++

 `// CPP program to construct a distinct element``// array with given size, sum, element upper``// bound and all elements positive``#include ``using` `namespace` `std;` `void` `printArray(``int` `N, ``int` `SUM, ``int` `K)``{``    ``// smallest possible sum``    ``int` `minSum = (N * (N + 1)) / 2;` `    ``// largest possible sum``    ``int` `maxSum = (N * K) - (N * (N - 1)) / 2;` `    ``if` `(minSum > SUM || maxSum < SUM) {``        ``printf``(``"Not Possible"``);``        ``return``;``    ``}` `    ``// Creating array with minimum possible``    ``// sum.``    ``int` `arr[N + 1];``    ``for` `(``int` `i = 1; i <= N; i++)``        ``arr[i] = i;``    ``int` `sum = minSum;` `    ``// running the loop from last because the``    ``// array is sorted and running from last``    ``// will give largest numbers``    ``for` `(``int` `i = N; i >= 1; i--) {` `        ``// replacing i with K, Note arr[i] = i``        ``int` `x = sum + (K - i);``        ``if` `(x < SUM) {``            ``sum = sum + (K - i);``            ``arr[i] = K; ``// can't be incremented further``            ``K--; ``// to maintain uniqueness``        ``}` `        ``else` `{` `            ``// directly replacing with a suitable``            ``// element to make sum as given sum``            ``arr[i] += (SUM - sum);``            ``sum = SUM;``            ``break``;``        ``}``    ``}` `    ``for` `(``int` `i = 1; i <= N; i++)``        ``cout << arr[i] << ``" "``;``}` `// Driver code``int` `main()``{``    ``int` `N = 3, SUM = 15, K = 8;``    ``printArray(N, SUM, K);``    ``return` `0;``}`

## Java

 `// Java program to construct a distinct element``// array with given size, sum, element upper``// bound and all elements positive` `import` `java.io.*;` `class` `GFG {``    ``static` `void` `printArray(``int` `N, ``int` `SUM, ``int` `K)``    ``{``        ``// smallest possible sum``        ``int` `minSum = (N * (N + ``1``)) / ``2``;` `        ``// largest possible sum``        ``int` `maxSum = (N * K) - (N * (N - ``1``)) / ``2``;` `        ``if` `(minSum > SUM || maxSum < SUM) {``            ``System.out.println(``"Not Possible"``);``            ``return``;``        ``}` `        ``// Creating array with``        ``// minimum possible sum.``        ``int` `arr[] = ``new` `int``[N + ``1``];``        ``for` `(``int` `i = ``1``; i <= N; i++)``            ``arr[i] = i;` `        ``int` `sum = minSum;` `        ``// running the loop from last because the``        ``// array is sorted and running from last``        ``// will give largest numbers``        ``for` `(``int` `i = N; i >= ``1``; i--) {` `            ``// replacing i with K, Note arr[i] = i``            ``int` `x = sum + (K - i);``            ``if` `(x < SUM) {``                ``sum = sum + (K - i);` `                ``// can't be incremented further``                ``arr[i] = K;` `                ``// to maintain uniqueness``                ``K--;``            ``}` `            ``else` `{` `                ``// directly replacing with a suitable``                ``// element to make sum as given sum``                ``arr[i] += (SUM - sum);``                ``sum = SUM;``                ``break``;``            ``}``        ``}` `        ``for` `(``int` `i = ``1``; i <= N; i++)``            ``System.out.print(arr[i] + ``" "``);``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `N = ``3``, SUM = ``15``, K = ``8``;``        ``printArray(N, SUM, K);``    ``}``}` `// This code is contributed by vt_m`

## Python3

 `# Python 3 program to construct a distinct``# element array with given size, sum,``# element upper bound and all elements``# positive``def` `printArray(N, ``SUM``, K):``    ` `    ``# smallest possible sum``    ``minSum ``=` `(N ``*` `(N ``+` `1``)) ``/` `2` `    ``# largest possible sum``    ``maxSum ``=` `(N ``*` `K) ``-` `(N ``*` `(N ``-` `1``)) ``/` `2` `    ``if` `(minSum > ``SUM` `or` `maxSum < ``SUM``):``        ``print``(``"Not Possible"``)``        ``return``    ` `    ``# Creating array with minimum``    ``# possible sum.``    ``arr ``=` `[``0` `for` `i ``in` `range``(N ``+` `1``)]``    ``for` `i ``in` `range``(``1``, N ``+` `1``, ``1``):``        ``arr[i] ``=` `i``    ``sum` `=` `minSum` `    ``# running the loop from last because``    ``# the array is sorted and running``    ``# from last will give largest numbers``    ``i ``=` `N``    ``while``(i >``=` `1``):``        ` `        ``# replacing i with K, Note arr[i] = i``        ``x ``=` `sum` `+` `(K ``-` `i)``        ``if` `(x < ``SUM``):``            ``sum` `=` `sum` `+` `(K ``-` `i)``            ``arr[i] ``=` `K``            ` `            ``# can't be incremented further``            ``K ``-``=` `1``            ` `            ``# to maintain uniqueness``        ``else``:``            ` `            ``# directly replacing with a suitable``            ``# element to make sum as given sum``            ``arr[i] ``+``=` `(``SUM` `-` `sum``)``            ``sum` `=` `SUM``            ``break``        ``i ``-``=` `1` `    ``for` `i ``in` `range``(``1``, N ``+` `1``, ``1``):``        ``print``(``int``(arr[i]), end ``=` `" "``)` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``N ``=` `3``    ``SUM` `=` `15``    ``K ``=` `8``    ``printArray(N, ``SUM``, K)` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# program to construct a distinct element``// array with given size, sum, element upper``// bound and all elements positive``using` `System;` `class` `GFG {``    ` `    ``static` `void` `printArray(``int` `N, ``int` `SUM, ``int` `K)``    ``{``        ` `        ``// smallest possible sum``        ``int` `minSum = (N * (N + 1)) / 2;` `        ``// largest possible sum``        ``int` `maxSum = (N * K) - (N * (N - 1)) / 2;` `        ``if` `(minSum > SUM || maxSum < SUM) {``            ``Console.WriteLine(``"Not Possible"``);``            ``return``;``        ``}` `        ``// Creating array with``        ``// minimum possible sum.``        ``int``[] arr = ``new` `int``[N + 1];``        ``for` `(``int` `i = 1; i <= N; i++)``            ``arr[i] = i;` `        ``int` `sum = minSum;` `        ``// running the loop from last because the``        ``// array is sorted and running from last``        ``// will give largest numbers``        ``for` `(``int` `i = N; i >= 1; i--) {` `            ``// replacing i with K, Note arr[i] = i``            ``int` `x = sum + (K - i);``        ` `            ` `            ``if` `(x < SUM) {``                ``sum = sum + (K - i);` `                ``// can't be incremented further``                ``arr[i] = K;` `                ``// to maintain uniqueness``                ``K--;``            ``}` `            ``else` `{` `                ``// directly replacing with a suitable``                ``// element to make sum as given sum``                ``arr[i] += (SUM - sum);``                ``sum = SUM;``                ``break``;``            ``}``        ``}` `        ``for` `(``int` `i = 1; i <= N; i++)``            ``Console.Write(arr[i] + ``" "``);``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ` `        ``int` `N = 3, SUM = 15, K = 8;``        ` `        ``printArray(N, SUM, K);``    ``}``}` `// This code is contributed by vt_m.`

## PHP

 ` ``\$SUM` `|| ``\$maxSum` `< ``\$SUM``) {``        ``echo``"Not Possible"``;``        ``return``;``    ``}` `    ``// Creating array with minimum possible``    ``// sum.``     ``\$arr` `= ``array``();``    ``for` `(``\$i` `= 1; ``\$i` `<= ``\$N``; ``\$i``++)``        ``\$arr``[``\$i``] = ``\$i``;``      ``\$sum` `= ``\$minSum``;` `    ``// running the loop from last because the``    ``// array is sorted and running from last``    ``// will give largest numbers``    ``for` `(``\$i` `= ``\$N``; ``\$i` `>= 1; ``\$i``--) {` `        ``// replacing i with K, Note arr[i] = i``        ``\$x` `= ``\$sum` `+ (``\$K` `- ``\$i``);``        ``if` `(``\$x` `<``\$SUM``) {``            ``\$sum` `= ``\$sum` `+ (``\$K` `- ``\$i``);``            ``\$arr``[``\$i``] =``\$K``; ``// can't be incremented further``            ``\$K``--; ``// to maintain uniqueness``        ``}` `        ``else` `{` `            ``// directly replacing with a suitable``            ``// element to make sum as given sum``            ``\$arr``[``\$i``] += (``\$SUM` `- ``\$sum``);``            ``\$sum` `= ``\$SUM``;``            ``break``;``        ``}``    ``}` `    ``for` `(``\$i` `= 1; ``\$i` `<= ``\$N``; ``\$i``++)``        ``echo` `\$arr``[``\$i``] , ``" "``;``}` `// Driver code``    ``\$N` `= 3; ``\$SUM` `= 15;``\$K` `= 8;``    ``printArray(``\$N``, ``\$SUM``, ``\$K``);``// This code is contributed by inder_verma..` `?>`

## Javascript

 ``

Output:

`1 6 8`

Time Complexity : O(N)

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