# Space optimization using bit manipulations

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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++

 `// CPP program to mark numbers as multiple of 2 or 5 ` `#include ` `using` `namespace` `std; ` ` `  `// Driver code ` `int` `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

 `// Java program to mark numbers as  ` `// multiple of 2 or 5 ` `import` `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 `

## Python 3

 `# Python 3 program to mark numbers ` `# as multiple of 2 or 5 ` `import` `math ` ` `  `# Driver code ` `a ``=` `2` `b ``=` `10` `size ``=` `abs``(b ``-` `a) ``+` `1` `array ``=` `[``0``] ``*` `size ` ` `  `# Iterate through a to b,  ` `# If it is a multiple of 2  ` `# or 5 Mark index in array as 1 ` `for` `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#

 `// C# program to mark numbers as  ` `// multiple of 2 or 5 ` `using` `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

 ` `

Output :

```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. ## C++

 `// CPP code to for marking multiples ` `#include ` `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 one ` `bool` `checkbit(``int` `array[], ``int` `index) ` `{ ` `    ``return` `array[index >> 5] & (1 << (index & 31)); ` `} ` ` `  `// Sets value of bit for corresponding index ` `void` `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); ` ` `  `    ``// Size that will be used is actual_size/32 ` `    ``// ceil is used to initialize the array with ` `    ``// positive number ` `    ``size = ``ceil``(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

 `// Java code to for marking multiples ` `import` `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 `

## C#

 `// C# code to for marking multiples ` `using` `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 `

Output:

```MULTIPLES of 2 and 5:
2 4 5 6 8 10
```

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