Space optimization using bit manipulations

30 Jun, 2022
There are many situations where we use integer values as index in array to see presence or absence, we can use bit manipulations to optimize space in such problems.Let us consider below problem as an example.Given two numbers say a and b, mark the multiples of 2 and 5 between a and b using less than O(|b – a|) space and output each of the multiples. 
Note : We have to mark the multiples i.e save (key, value) pairs in memory such that each key either have value as 1 or 0 representing as multiple of 2 or 5 or not respectively. 
Examples :  
Input : 2 10
Output : 2 4 5 6 8 10

Input: 60 95
Output: 60 62 64 65 66 68 70 72 74 75 76 78 
        80 82 84 85 86 88 90 92 94 95
Approach 1 (Simple): Hash the indices in an array from a to b and mark each of the indices as 1 or 0. Space complexity : O(max(a, b)) 
Approach 2 (Better than simple): Save memory, by translating a to 0th index and b to (b-a)th index. Space complexity : O(|b-a|). 
Simply hash |b – a| positions of an array as 0 and 1.
C++
Java
Python3
C#
PHP
Javascript
// C++ program to mark numbers as multiple of 2 or 5#include <bits/stdc++.h>using namespace std; // Driver codeint main(){    int a = 2, b = 10;    int size = abs(b - a) + 1;    int* array = new int[size];     // Iterate through a to b, If it is a multiple    // of 2 or 5 Mark index in array as 1    for (int i = a; i <= b; i++)        if (i % 2 == 0 || i % 5 == 0)            array[i - a] = 1;     cout << "MULTIPLES of 2 and 5:\n";    for (int i = a; i <= b; i++)        if (array[i - a] == 1)            cout << i << " ";     return 0;}
// Java program to mark numbers as// multiple of 2 or 5import java.lang.*; class GFG {         // Driver code    public static void main(String[] args)    {        int a = 2, b = 10;        int size = Math.abs(b - a) + 1;        int array[] = new int[size];             // Iterate through a to b, If        // it is a multiple of 2 or 5        // Mark index in array as 1        for (int i = a; i <= b; i++)            if (i % 2 == 0 || i % 5 == 0)                array[i - a] = 1;             System.out.println("MULTIPLES of 2"                              + " and 5:");        for (int i = a; i <= b; i++)            if (array[i - a] == 1)                System.out.printf(i + " ");    }} // This code is contributed by// Smitha Dinesh Semwal
# Python3 program to mark numbers# as multiple of 2 or 5import math # Driver codea = 2b = 10size = abs(b - a) + 1array = [0] * size # Iterate through a to b,# If it is a multiple of 2# or 5 Mark index in array as 1for i in range(a, b + 1):    if (i % 2 == 0 or i % 5 == 0):            array[i - a] = 1 print("MULTIPLES of 2 and 5:")for i in range(a, b + 1):    if (array[i - a] == 1):            print(i, end=" ") # This code is contributed by# Smitha Dinesh Semwal
// C# program to mark numbers as// multiple of 2 or 5using System; class GFG {         // Driver code    static public void Main ()    {        int a = 2, b = 10;        int size = Math.Abs(b - a) + 1;        int[] array = new int[size];             // Iterate through a to b, If        // it is a multiple of 2 or 5        // Mark index in array as 1        for (int i = a; i <= b; i++)            if (i % 2 == 0 || i % 5 == 0)                array[i - a] = 1;             Console.WriteLine("MULTIPLES of 2" +                          " and 5:");        for (int i = a; i <= b; i++)            if (array[i - a] == 1)                Console.Write(i + " ");    }} // This code is contributed by Ajit.
<?php// PHP program to mark// numbers as multiple// of 2 or 5 // Driver Code$a = 2;$b = 10;$size = abs($b - $a) + 1;$array = array_fill(0, $size, 0); // Iterate through a to b,// If it is a multiple of// 2 or 5 Mark index in// array as 1for ($i = $a; $i <= $b; $i++)    if ($i % 2 == 0 || $i % 5 == 0)        $array[$i - $a] = 1; echo "MULTIPLES of 2 and 5:\n";for ($i = $a; $i <= $b; $i++)    if ($array[$i - $a] == 1)        echo $i . " "; // This code is contributed by mits.?>
<script> // JavaScript program to mark numbers as// multiple of 2 or 5 // Driver codelet a = 2, b = 10;let size = Math.abs(b - a) + 1;let array = []; // Iterate through a to b, If// it is a multiple of 2 or 5// Mark index in array as 1for(let i = a; i <= b; i++)    if (i % 2 == 0 || i % 5 == 0)        array[i - a] = 1; document.write("MULTIPLES of 2" +               " and 5:" + "<br/>");for(let i = a; i <= b; i++)    if (array[i - a] == 1)        document.write(i + " ");         // This code is contributed by code_hunt </script>
MULTIPLES of 2 and 5:
2 4 5 6 8 10 
Approach 3 (Using Bit Manipulations):Here is a space optimized which uses bit manipulation technique that can be applied to problems mapping binary values in arrays. Size of int variable in 64-bit compiler is 4 bytes. 1 byte is represented by 8 bit positions in memory. So, an integer in memory is represented by 32 bit positions(4 Bytes) these 32 bit positions can be used instead of just one index to hash binary values.
We can Implement the above Approach for 32-bit integers by following these steps
 Find the actual index in int[] that needs to be bit manipulated it will be bitwise index/ 32. Find the index of bit in those 32 bits that needs to be turned on it will be bitwise index % 32.  Let’s Call it X Turn on the bit by doing | (bitwise OR) with (1 << X) (here we turn on the Xth bit by bit manipulation) To get the value of a bit at a bitwise index we calculate the same indices and do a bitwise & so that if Xth bit is on it will return an integer not equal to 0 which is true in C++.Now instead of using arithmetic operators we can use bitwise operations for efficiency
 Find the actual index in int[] that needs to be bit manipulated it will be bitwise index/ 32.
 Find the index of bit in those 32 bits that needs to be turned on it will be bitwise index % 32.  Let’s Call it X
 Turn on the bit by doing | (bitwise OR) with (1 << X) (here we turn on the Xth bit by bit manipulation)
 To get the value of a bit at a bitwise index we calculate the same indices and do a bitwise & so that if Xth bit is on it will return an integer not equal to 0 which is true in C++.
Now instead of using arithmetic operators we can use bitwise operations for efficiency
C++
Java
Python3
C#
Javascript
// C++ code to for marking multiples#include <bits/stdc++.h>using namespace std; // index >> 5 corresponds to dividing index by 32// index & 31 corresponds to modulo operation of// index by 32 // Function to check value of bit position whether// it is zero or onebool checkbit(int array[], int index){    return array[index >> 5] & (1 << (index & 31));} // Sets value of bit for corresponding indexvoid setbit(int array[], int index){    array[index >> 5] |= (1 << (index & 31));} /* Driver program to test above functions*/int main(){    int a = 2, b = 10;    int size = abs(b - a)+1;     // Size that will be used is actual_size/32    // ceil is used to initialize the array with    // positive number    size = ceil((double)size/32);         // Array is dynamically initialized as    // we are calculating size at run time    int* array = new int[size];     // Iterate through every index from a to b and    // call setbit() if it is a multiple of 2 or 5    for (int i = a; i <= b; i++)        if (i % 2 == 0 || i % 5 == 0)            setbit(array, i - a);     cout << "MULTIPLES of 2 and 5:\n";    for (int i = a; i <= b; i++)        if (checkbit(array, i - a))            cout << i << " ";     return 0;}
// Java code to for marking multiplesimport java.io.*;import java.util.*; class GFG{    // index >> 5 corresponds to dividing index by 32    // index & 31 corresponds to modulo operation of    // index by 32     // Function to check value of bit position whether    // it is zero or one    static boolean checkbit(int array[], int index)    {            int val = array[index >> 5] & (1 << (index & 31));            if (val == 0)                return false;            return true;    }     // Sets value of bit for corresponding index    static void setbit(int array[], int index)    {            array[index >> 5] |= (1 << (index & 31));    }     // Driver code    public static void main(String args[])    {            int a = 2, b = 10;            int size = Math.abs(b-a);             // Size that will be used is actual_size/32            // ceil is used to initialize the array with            // positive number            size = (int)Math.ceil((double)size / 32);             // Array is dynamically initialized as            // we are calculating size at run time            int[] array = new int[size];             // Iterate through every index from a to b and            // call setbit() if it is a multiple of 2 or 5            for (int i = a; i <= b; i++)                if (i % 2 == 0 || i % 5 == 0)                    setbit(array, i - a);             System.out.println("MULTIPLES of 2 and 5:");            for (int i = a; i <= b; i++)                if (checkbit(array, i - a))                    System.out.print(i + " ");    }} // This code is contributed by rachana soma
# Python3 code to for marking multiplesimport math # index >> 5 corresponds to dividing index by 32# index & 31 corresponds to modulo operation of# index by 32 # Function to check value of bit position whether# it is zero or onedef checkbit( array, index):    return array[index >> 5] & (1 << (index & 31)) # Sets value of bit for corresponding indexdef setbit( array, index):    array[index >> 5] |= (1 << (index & 31)) # Driver codea = 2b = 10size = abs(b - a) # Size that will be used is actual_size/32# ceil is used to initialize the array with# positive numbersize = math.ceil(size / 32)     # Array is dynamically initialized as# we are calculating size at run timearray = [0 for i in range(size)] # Iterate through every index from a to b and# call setbit() if it is a multiple of 2 or 5for i in range(a, b + 1):    if (i % 2 == 0 or i % 5 == 0):        setbit(array, i - a) print("MULTIPLES of 2 and 5:")for i in range(a, b + 1):    if (checkbit(array, i - a)):        print(i, end = " ") # This code is contributed by rohitsingh07052
// C# code to for marking multiplesusing System; class GFG{    // index >> 5 corresponds to dividing index by 32    // index & 31 corresponds to modulo operation of    // index by 32     // Function to check value of bit position     // whether it is zero or one    static bool checkbit(int []array, int index)    {        int val = array[index >> 5] &                 (1 << (index & 31));        if (val == 0)            return false;        return true;    }     // Sets value of bit for corresponding index    static void setbit(int []array, int index)    {            array[index >> 5] |= (1 << (index & 31));    }     // Driver code    public static void Main(String []args)    {        int a = 2, b = 10;        int size = Math.Abs(b-a);         // Size that will be used is actual_size/32        // ceil is used to initialize the array with        // positive number        size = (int)Math.Ceiling((double)size / 32);         // Array is dynamically initialized as        // we are calculating size at run time        int[] array = new int[size];         // Iterate through every index from a to b and        // call setbit() if it is a multiple of 2 or 5        for (int i = a; i <= b; i++)            if (i % 2 == 0 || i % 5 == 0)                setbit(array, i - a);         Console.WriteLine("MULTIPLES of 2 and 5:");        for (int i = a; i <= b; i++)            if (checkbit(array, i - a))                Console.Write(i + " ");    }} // This code is contributed by 29AjayKumar
<script> // JavaScript code to for marking multiples // index >> 5 corresponds to dividing index// by 32 index & 31 corresponds to modulo// operation of index by 32 // Function to check value of bit// position whether it is zero or onefunction checkbit(array, index){    return array[index >> 5] &                    (1 << (index & 31));} // Sets value of bit for corresponding indexfunction setbit(array, index){    array[index >> 5] |=             (1 << (index & 31));} // Driver codelet a = 2, b = 10;let size = Math.abs(b - a); // Size that will be used is actual_size/32// ceil is used to initialize the array with// positive numbersize = Math.ceil(size / 32); // Array is dynamically initialized as// we are calculating size at run timelet array = new Array(size); // Iterate through every index from a to b and// call setbit() if it is a multiple of 2 or 5for(let i = a; i <= b; i++)    if (i % 2 == 0 || i % 5 == 0)        setbit(array, i - a); document.write("MULTIPLES of 2 and 5:<br>");for(let i = a; i <= b; i++)    if (checkbit(array, i - a))        document.write(i + " "); // This code is contributed by Surbhi Tyagi. </script>
MULTIPLES of 2 and 5:
2 4 5 6 8 10 
Smitha Dinesh Semwal
jit_t
Mithun Kumar
rachana soma
29AjayKumar
code_hunt
rohitsingh07052
surbhityagi15
bajrangi07
himanshusaini1100
Arrays
Bit Magic
Arrays
Bit Magic
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