Nth Subset of the Sequence consisting of powers of K in increasing order of their Sum
Last Updated :
20 Oct, 2022
Given two integers N and K, the task is to find the Nth Subset from the sequence of subsets generated from the powers of K i.e. {1, K1, K2, K3, .....} such that the subsets are arranged in increasing order of their sum, the task is to find the Nth subset from the sequence.
Examples:
Input: N = 5, K = 3
Output: 1 9
Explanation:
The sequence of subsets along with their sum are:
- Subset = {1}, Sum = 1
- Subset = {3}, Sum = 3
- Subset = {1, 3}, Sum = 4
- Subset = {9}, Sum = 9
- Subset = {1, 9}, Sum = 10
Therefore, the subset at position 5 is {1, 9}.
Input: N = 4, K = 4
Output: 16
Approach:
Let's refer to the required sequence for K = 3 given below:
From the above sequence, it can be observed that the subset {3} has position 2, the subset {9} has position 4, and the subset {27} has position 8, and so on. The subset {1, 3}, {1, 9}, {1, 27} occupies positions 3, 5, and 9 respectively. Hence, all the elements of the required Nth subset can be obtained by finding the nearest power of 2 which is smaller than or equal to N.
Illustration:
N = 6, K = 3
1st iteration:
- p = log2(6) = 2
- 32 = 9, Subset = {9}
- N = 6 % 4 = 2
2nd iteration:
- p = log2(2) = 1
- 31 = 3, Subset = {3, 9}
- N = 2 % 2 = 0
Therefore, the required subset is {3, 9}
Follow the steps below to solve the problem:
- Calculate the nearest power of 2 which is smaller than or equal to N, say p. Therefore, p = log2N.
- Now, the element of the subset will be Kp. Insert it into the front of the subset.
- Update N to N % 2p.
- Repeat the above steps until N becomes 0, and consequently print the obtained subset.
Below is the implementation of the above approach:
C++
// C++ program for the above approach
#include <bits/stdc++.h>
#include <stdio.h>
using namespace std;
#define lli long long int
// Function to print the
// required N-th subset
void printSubset(lli n, int k)
{
vector<lli> answer;
while(n > 0)
{
// Nearest power of 2<=N
lli p = log2(n);
// Now insert k^p in the answer
answer.push_back(pow(k, p));
// update n
n %= (int)pow(2, p);
}
// Print the ans in sorted order
reverse(answer.begin(), answer.end());
for(auto x: answer)
{
cout << x << " ";
}
}
// Driver Code
int main()
{
lli n = 5;
int k = 4;
printSubset(n, k);
}
// This code is contributed by winter_soldier
// Java program for above approach
import java.util.*;
import java.lang.*;
import java.io.*;
class GFG
{
// Function to print the
// required N-th subset
static void printSubset(long n, int k)
{
ArrayList<Long> answer = new ArrayList<>();
while(n > 0)
{
// Nearest power of 2<=N
long p = (long)(Math.log(n) / Math.log(2));;
// Now insert k^p in the answer
answer.add((long)(Math.pow(k, p)));
// update n
n %= (int)Math.pow(2, p);
}
// Print the ans in sorted order
Collections.sort(answer);
for(Long x: answer)
{
System.out.print(x + " ");
}
}
// Driver function
public static void main (String[] args)
{
long n = 5;
int k = 4;
printSubset(n, k);
}
}
// This code is contributed by offbeat
# Python3 program for
# the above approach
import math
# Function to print the
# required N-th subset
def printSubset(N, K):
# Stores the subset
answer = ""
while(N > 0):
# Nearest power of 2 <= N
p = int(math.log(N, 2))
# Insert K ^ p in the subset
answer = str(K**p)+" "+answer
# Update N
N = N % (2**p)
# Print the subset
print(answer)
# Driver Code
N = 5
K = 4
printSubset(N, K)
// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {
// Function to print the
// required N-th subset
static void printSubset(int n, int k)
{
List<int> answer = new List<int>();
while(n > 0)
{
// Nearest power of 2<=N
int p = (int)Math.Log(n,2);
// Now insert k^p in the answer
answer.Add((int)Math.Pow(k, p));
// update n
n %= (int)Math.Pow(2, p);
}
// Print the ans in sorted order
answer.Reverse();
foreach(int x in answer)
{
Console.Write(x + " ");
}
}
// Driver code
static void Main() {
int n = 5;
int k = 4;
printSubset(n, k);
}
}
// This code is contributed by divyeshrabadiya07.
<script>
// Javascript program for the above approach
// Function to print the
// required N-th subset
function printSubset(n, k)
{
var answer = [];
while(n > 0)
{
// Nearest power of 2<=N
var p = parseInt(Math.log2(n));
// Now insert k^p in the answer
answer.push(Math.pow(k, p));
// update n
n %= parseInt(Math.pow(2, p));
}
// Print the ans in sorted order
answer.sort();
//reverse(answer.begin(), answer.end());
for(var i=0;i<answer.length;i++)
{
document.write(answer[i] + " ");
}
}
var n = 5;
var k = 4;
printSubset(n, k);
//This code is contributed by SoumikMondal
</script>
Time Complexity: O(logN)
Auxiliary Space: O(logN)
Approach:
- Initialize the count and x by 0. Also, a vector to store the elements of the subsets.
- Do the following while n is greater than 0.
- Set x = n & 1, for finding if the last bit of the number is set or not.
- Now Push element 3count into the subset if n is not 0.
- Reduce the number n by two with the help of right shifting by 1 unit.
- Increase the count value by 1.
- Finally, the elements in the array are the elements of the Nth subset.
Below is the implementation of the above approach:
C++
// C++ program to print subset
// at the nth position ordered
// by the sum of the elements
#include <bits/stdc++.h>
using namespace std;
// Function to print the elements of
// the subset at pos n
void printsubset(int n,int k)
{
// Initialize count=0 and x=0
int count = 0, x = 0;
// create a vector for
// storing the elements
// of subsets
vector<int> vec;
// doing until all the
// set bits of n are used
while (n) {
x = n & 1;
// this part is executed only
// when the last bit is
// set
if (x) {
vec.push_back(pow(k, count));
}
// right shift the bit by one position
n = n >> 1;
// increasing the count each time by one
count++;
}
// printing the values os elements
for (int i = 0; i < vec.size(); i++)
cout << vec[i] << " ";
}
// Driver Code
int main()
{
int n = 7,k=4;
printsubset(n,k);
return 0;
}
// This code is contributed by shivkant
// Java program to print subset
// at the nth position ordered
// by the sum of the elements
import java.util.*;
import java.lang.*;
class GFG{
// Function to print the
// elements of the subset
// at pos n
static void printsubset(int n,
int k)
{
// Initialize count=0 and x=0
int count = 0, x = 0;
// Create a vector for
// storing the elements
// of subsets
ArrayList<Integer> vec =
new ArrayList<>();
// Doing until all the
// set bits of n are used
while (n != 0)
{
x = n & 1;
// This part is executed only
// when the last bit is
// set
if (x != 0)
{
vec.add((int)Math.pow(k,
count));
}
// Right shift the bit
// by one position
n = n >> 1;
// Increasing the count
// each time by one
count++;
}
// Printing the values os elements
for (int i = 0; i < vec.size(); i++)
System.out.print(vec.get(i) + " ");
}
// Driver function
public static void main (String[] args)
{
int n = 7, k = 4;
printsubset(n, k);
}
}
// This code is contributed by offbeat
# Python3 program to print subset
# at the nth position ordered
# by the sum of the elements
import math
# Function to print the elements of
# the subset at pos n
def printsubset(n, k):
# Initialize count=0 and x=0
count = 0
x = 0
# Create a vector for
# storing the elements
# of subsets
vec = []
# Doing until all the
# set bits of n are used
while (n > 0):
x = n & 1
# This part is executed only
# when the last bit is
# set
if (x):
vec.append(pow(k, count))
# Right shift the bit by one position
n = n >> 1
# Increasing the count each time by one
count += 1
# Printing the values os elements
for item in vec:
print(item, end = " ")
# Driver Code
n = 7
k = 4
printsubset(n, k)
# This code is contributed by Stream_Cipher
// C# program to print subset
// at the nth position ordered
// by the sum of the elements
using System.Collections.Generic;
using System;
class GFG{
// Function to print the
// elements of the subset
// at pos n
static void printsubset(int n, int k)
{
// Initialize count=0 and x=0
int count = 0, x = 0;
// Create a vector for
// storing the elements
// of subsets
List<int> vec = new List<int>();
// Doing until all the
// set bits of n are used
while (n != 0)
{
x = n & 1;
// This part is executed only
// when the last bit is
// set
if (x != 0)
{
vec.Add((int)Math.Pow(k, count));
}
// Right shift the bit
// by one position
n = n >> 1;
// Increasing the count
// each time by one
count++;
}
// Printing the values os elements
for(int i = 0; i < vec.Count; i++)
Console.Write(vec[i] + " ");
}
// Driver code
public static void Main ()
{
int n = 7, k = 4;
printsubset(n, k);
}
}
// This code is contributed by Stream_Cipher
<script>
// Javascript program to print subset
// at the nth position ordered
// by the sum of the elements
// Function to print the
// elements of the subset
// at pos n
function printsubset(n, k)
{
// Initialize count=0 and x=0
let count = 0, x = 0;
// Create a vector for
// storing the elements
// of subsets
let vec = [];
// Doing until all the
// set bits of n are used
while (n != 0)
{
x = n & 1;
// This part is executed only
// when the last bit is
// set
if (x != 0)
{
vec.push(Math.pow(k, count));
}
// Right shift the bit
// by one position
n = n >> 1;
// Increasing the count
// each time by one
count++;
}
// Printing the values os elements
for(let i = 0; i < vec.length; i++)
document.write(vec[i] + " ");
}
let n = 7, k = 4;
printsubset(n, k);
</script>
Time Complexity: O(log2N)
Auxiliary Space: O(log2N)
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem