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# Segmented Sieve

Given a number n, print all primes smaller than n. For example, if the given number is 10, output 2, 3, 5, 7.

Recommended Practice

A Naive approach is to run a loop from 0 to n-1 and check each number for primeness. A Better Approach is to use Simple Sieve of Eratosthenes.

## C++

 `// C++ code to implement the approach``#include ``using` `namespace` `std;` `void` `simpleSieve(``int` `limit)``{``    ``// Create a boolean array "mark[0..limit-1]" and``    ``// initialize all entries of it as true. A value``    ``// in mark[p] will finally be false if 'p' is Not``    ``// a prime, else true.``    ``bool` `mark[limit];``    ``for``(``int` `i = 0; i

## Java

 `// This functions finds all primes smaller than 'limit'``// using simple sieve of eratosthenes.``static` `void` `simpleSieve(``int` `limit)``{``  ` `    ``// Create a boolean array "mark[0..limit-1]" and``    ``// initialize all entries of it as true. A value``    ``// in mark[p] will finally be false if 'p' is Not``    ``// a prime, else true.``    ``boolean` `[]mark = ``new` `boolean``[limit];``    ``Arrays.fill(mark, ``true``);`` ` `    ``// One by one traverse all numbers so that their``    ``// multiples can be marked as composite.``    ``for` `(``int` `p = ``2``; p * p < limit; p++)``    ``{``        ``// If p is not changed, then it is a prime``        ``if` `(mark[p] == ``true``)``        ``{``            ``// Update all multiples of p``            ``for` `(``int` `i = p * p; i < limit; i += p)``                ``mark[i] = ``false``;``        ``}``    ``}`` ` `    ``// Print all prime numbers and store them in prime``    ``for` `(``int` `p = ``2``; p < limit; p++)``        ``if` `(mark[p] == ``true``)``            ``System.out.print(p + ``" "``);``}` `// This code is contributed by rutvik_56.`

## Python3

 `# This functions finds all primes smaller than 'limit'``# using simple sieve of eratosthenes.``def` `simpleSieve(limit):` `    ``# Create a boolean array "mark[0..limit-1]" and``    ``# initialize all entries of it as true. A value``    ``# in mark[p] will finally be false if 'p' is Not``    ``# a prime, else true.``    ``mark ``=` `[``True` `for` `i ``in` `range``(limit)]` `    ``# One by one traverse all numbers so that their``    ``# multiples can be marked as composite.``    ``for` `p ``in` `range``(p ``*` `p, limit ``-` `1``, ``1``):` `        ``# If p is not changed, then it is a prime``        ``if` `(mark[p] ``=``=` `True``):` `            ``# Update all multiples of p``            ``for` `i ``in` `range``(p ``*` `p, limit ``-` `1``, p):``                ``mark[i] ``=` `False` `    ``# Print all prime numbers and store them in prime``    ``for` `p ``in` `range``(``2``, limit ``-` `1``, ``1``):``        ``if` `(mark[p] ``=``=` `True``):``            ``print``(p, end``=``" "``)` `# This code is contributed by Dharanendra L V.`

## C#

 `// This functions finds all primes smaller than 'limit'``// using simple sieve of eratosthenes.``static` `void` `simpleSieve(``int` `limit)``{``  ` `    ``// Create a boolean array "mark[0..limit-1]" and``    ``// initialize all entries of it as true. A value``    ``// in mark[p] will finally be false if 'p' is Not``    ``// a prime, else true.``    ``bool` `[]mark = ``new` `bool``[limit];``    ``Array.Fill(mark, ``true``);`` ` `    ``// One by one traverse all numbers so that their``    ``// multiples can be marked as composite.``    ``for` `(``int` `p = 2; p * p < limit; p++)``    ``{``      ` `        ``// If p is not changed, then it is a prime``        ``if` `(mark[p] == ``true``)``        ``{``          ` `            ``// Update all multiples of p``            ``for` `(``int` `i = p * p; i < limit; i += p)``                ``mark[i] = ``false``;``        ``}``    ``}`` ` `    ``// Print all prime numbers and store them in prime``    ``for` `(``int` `p = 2; p < limit; p++)``        ``if` `(mark[p] == ``true``)``            ``Console.Write(p + ``" "``);``}` `// This code is contributed by pratham76.`

## Javascript

 ``

## C

 `// This functions finds all primes smaller than 'limit'``// using simple sieve of eratosthenes.``void` `simpleSieve(``int` `limit)``{``    ``// Create a boolean array "mark[0..limit-1]" and``    ``// initialize all entries of it as true. A value``    ``// in mark[p] will finally be false if 'p' is Not``    ``// a prime, else true.``    ``bool` `mark[limit];``    ``for``(``int` `i = 0; i

Problems with Simple Sieve:
The Sieve of Eratosthenes looks good, but consider the situation when n is large, the Simple Sieve faces the following issues.

• An array of size Θ(n) may not fit in memory
• The simple Sieve is not cached friendly even for slightly bigger n. The algorithm traverses the array without locality of reference

Segmented Sieve
The idea of a segmented sieve is to divide the range [0..n-1] in different segments and compute primes in all segments one by one. This algorithm first uses Simple Sieve to find primes smaller than or equal to √(n). Below are steps used in Segmented Sieve.

1. Use Simple Sieve to find all primes up to the square root of ‘n’ and store these primes in an array “prime[]”. Store the found primes in an array ‘prime[]’.
2. We need all primes in the range [0..n-1]. We divide this range into different segments such that the size of every segment is at-most √n
3. Do following for every segment [low..high]
• Create an array mark[high-low+1]. Here we need only O(x) space where x is a number of elements in a given range.
• Iterate through all primes found in step 1. For every prime, mark its multiples in the given range [low..high].

In Simple Sieve, we needed O(n) space which may not be feasible for large n. Here we need O(√n) space and we process smaller ranges at a time (locality of reference)

Below is the implementation of the above idea.

## C++

 `// C++ program to print  all primes smaller than``// n using segmented sieve``#include ``using` `namespace` `std;` `// This functions finds all primes smaller than 'limit'``// using simple sieve of eratosthenes. It also stores``// found primes in vector prime[]``void` `simpleSieve(``int` `limit, vector<``int``> &prime)``{``    ``// Create a boolean array "mark[0..n-1]" and initialize``    ``// all entries of it as true. A value in mark[p] will``    ``// finally be false if 'p' is Not a prime, else true.``    ``vector<``bool``> mark(limit + 1, ``true``);` `    ``for` `(``int` `p=2; p*p prime;``    ``prime.reserve(limit);``    ``simpleSieve(limit, prime);` `    ``// Divide the range [0..n-1] in different segments``    ``// We have chosen segment size as sqrt(n).``    ``int` `low = limit;``    ``int` `high = 2*limit;` `    ``// While all segments of range [0..n-1] are not processed,``    ``// process one segment at a time``    ``while` `(low < n)``    ``{``        ``if` `(high >= n)``           ``high = n;``        ` `        ``// To mark primes in current range. A value in mark[i]``        ``// will finally be false if 'i-low' is Not a prime,``        ``// else true.``        ``bool` `mark[limit+1];``        ``memset``(mark, ``true``, ``sizeof``(mark));` `        ``// Use the found primes by simpleSieve() to find``        ``// primes in current range``        ``for` `(``int` `i = 0; i < prime.size(); i++)``        ``{``            ``// Find the minimum number in [low..high] that is``            ``// a multiple of prime[i] (divisible by prime[i])``            ``// For example, if low is 31 and prime[i] is 3,``            ``// we start with 33.``            ``int` `loLim = ``floor``(low/prime[i]) * prime[i];``            ``if` `(loLim < low)``                ``loLim += prime[i];` `            ``/* Mark multiples of prime[i] in [low..high]:``                ``We are marking j - low for j, i.e. each number``                ``in range [low, high] is mapped to [0, high-low]``                ``so if range is [50, 100] marking 50 corresponds``                ``to marking 0, marking 51 corresponds to 1 and``                ``so on. In this way we need to allocate space only``                ``for range */``            ``for` `(``int` `j=loLim; j

## Java

 `// Java program to print all primes smaller than``// n using segmented sieve`  `import` `java.util.Vector;``import` `static` `java.lang.Math.sqrt;``import` `static` `java.lang.Math.floor;` `class` `Test``{``    ``// This method finds all primes smaller than 'limit'``    ``// using simple sieve of eratosthenes. It also stores``    ``// found primes in vector prime[]``    ``static` `void` `simpleSieve(``int` `limit, Vector prime)``    ``{``        ``// Create a boolean array "mark[0..n-1]" and initialize``        ``// all entries of it as true. A value in mark[p] will``        ``// finally be false if 'p' is Not a prime, else true.``        ``boolean` `mark[] = ``new` `boolean``[limit+``1``];``        ` `        ``for` `(``int` `i = ``0``; i < mark.length; i++)``            ``mark[i] = ``true``;``     ` `        ``for` `(``int` `p=``2``; p*p prime = ``new` `Vector<>(); ``        ``simpleSieve(limit, prime);``     ` `        ``// Divide the range [0..n-1] in different segments``        ``// We have chosen segment size as sqrt(n).``        ``int` `low  = limit;``        ``int` `high = ``2``*limit;``     ` `        ``// While all segments of range [0..n-1] are not processed,``        ``// process one segment at a time``        ``while` `(low < n)``        ``{``            ``if` `(high >= n)``                ``high = n;` `            ``// To mark primes in current range. A value in mark[i]``            ``// will finally be false if 'i-low' is Not a prime,``            ``// else true.``            ``boolean` `mark[] = ``new` `boolean``[limit+``1``];``            ` `            ``for` `(``int` `i = ``0``; i < mark.length; i++)``                ``mark[i] = ``true``;``     ` `            ``// Use the found primes by simpleSieve() to find``            ``// primes in current range``            ``for` `(``int` `i = ``0``; i < prime.size(); i++)``            ``{``                ``// Find the minimum number in [low..high] that is``                ``// a multiple of prime.get(i) (divisible by prime.get(i))``                ``// For example, if low is 31 and prime.get(i) is 3,``                ``// we start with 33.``                ``int` `loLim = (``int``) (floor(low/prime.get(i)) * prime.get(i));``                ``if` `(loLim < low)``                    ``loLim += prime.get(i);``     ` `                ``/*  Mark multiples of prime.get(i) in [low..high]:``                    ``We are marking j - low for j, i.e. each number``                    ``in range [low, high] is mapped to [0, high-low]``                    ``so if range is [50, 100]  marking 50 corresponds``                    ``to marking 0, marking 51 corresponds to 1 and``                    ``so on. In this way we need to allocate space only``                    ``for range  */``                ``for` `(``int` `j=loLim; j

## Python3

 `# Python3 program to print all primes``# smaller than n, using segmented sieve``import` `math``prime ``=` `[]` `# This method finds all primes``# smaller than 'limit' using``# simple sieve of eratosthenes.``# It also stores found primes in list prime``def` `simpleSieve(limit):``    ` `    ``# Create a boolean list "mark[0..n-1]" and ``    ``# initialize all entries of it as True.``    ``# A value in mark[p] will finally be False``    ``# if 'p' is Not a prime, else True.``    ``mark ``=` `[``True` `for` `i ``in` `range``(limit ``+` `1``)]``    ``p ``=` `2``    ``while` `(p ``*` `p <``=` `limit):``        ` `        ``# If p is not changed, then it is a prime``        ``if` `(mark[p] ``=``=` `True``):``            ` `            ``# Update all multiples of p``            ``for` `i ``in` `range``(p ``*` `p, limit ``+` `1``, p):``                ``mark[i] ``=` `False` `        ``p ``+``=` `1``        ` `    ``# Print all prime numbers``    ``# and store them in prime``    ``for` `p ``in` `range``(``2``, limit):``        ``if` `mark[p]:``            ``prime.append(p)``            ``print``(p,end ``=` `" "``)``            ` `# Prints all prime numbers smaller than 'n'``def` `segmentedSieve(n):``    ` `    ``# Compute all primes smaller than or equal``    ``# to square root of n using simple sieve``    ``limit ``=` `int``(math.floor(math.sqrt(n)) ``+` `1``)``    ``simpleSieve(limit)``    ` `    ``# Divide the range [0..n-1] in different segments``    ``# We have chosen segment size as sqrt(n).``    ``low ``=` `limit``    ``high ``=` `limit ``*` `2``    ` `    ``# While all segments of range [0..n-1] are not processed,``    ``# process one segment at a time``    ``while` `low < n:``        ``if` `high >``=` `n:``            ``high ``=` `n``            ` `        ``# To mark primes in current range. A value in mark[i]``        ``# will finally be False if 'i-low' is Not a prime,``        ``# else True.``        ``mark ``=` `[``True` `for` `i ``in` `range``(limit ``+` `1``)]``        ` `        ``# Use the found primes by simpleSieve()``        ``# to find primes in current range``        ``for` `i ``in` `range``(``len``(prime)):``            ` `            ``# Find the minimum number in [low..high]``            ``# that is a multiple of prime[i]``            ``# (divisible by prime[i])``            ``# For example, if low is 31 and prime[i] is 3,``            ``# we start with 33.``            ``loLim ``=` `int``(math.floor(low ``/` `prime[i]) ``*``                                         ``prime[i])``            ``if` `loLim < low:``                ``loLim ``+``=` `prime[i]``                ` `            ``# Mark multiples of prime[i] in [low..high]:``            ``# We are marking j - low for j, i.e. each number``            ``# in range [low, high] is mapped to [0, high-low]``            ``# so if range is [50, 100] marking 50 corresponds``            ``# to marking 0, marking 51 corresponds to 1 and``            ``# so on. In this way we need to allocate space``            ``# only for range``            ``for` `j ``in` `range``(loLim, high, prime[i]):``                ``mark[j ``-` `low] ``=` `False``                ` `        ``# Numbers which are not marked as False are prime``        ``for` `i ``in` `range``(low, high):``            ``if` `mark[i ``-` `low]:``                ``print``(i, end ``=` `" "``)``                ` `        ``# Update low and high for next segment``        ``low ``=` `low ``+` `limit``        ``high ``=` `high ``+` `limit` `# Driver Code``n ``=` `100``print``(``"Primes smaller than"``, n, ``":"``)``segmentedSieve(``100``)` `# This code is contributed by bhavyadeep`

## C#

 `// C# program to print``// all primes smaller than``// n using segmented sieve``using` `System;``using` `System.Collections;` `class` `GFG``{``    ``// This method finds all primes``    ``// smaller than 'limit' using simple``    ``// sieve of eratosthenes. It also stores``    ``// found primes in vector prime[]``    ``static` `void` `simpleSieve(``int` `limit,``                            ``ArrayList prime)``    ``{``        ``// Create a boolean array "mark[0..n-1]"``        ``// and initialize all entries of it as``        ``// true. A value in mark[p] will finally be``        ``// false if 'p' is Not a prime, else true.``        ``bool``[] mark = ``new` `bool``[limit + 1];``        ` `        ``for` `(``int` `i = 0; i < mark.Length; i++)``            ``mark[i] = ``true``;``    ` `        ``for` `(``int` `p = 2; p * p < limit; p++)``        ``{``            ``// If p is not changed, then it is a prime``            ``if` `(mark[p] == ``true``)``            ``{``                ``// Update all multiples of p``                ``for` `(``int` `i = p * p; i < limit; i += p)``                    ``mark[i] = ``false``;``            ``}``        ``}``    ` `        ``// Print all prime numbers and store them in prime``        ``for` `(``int` `p = 2; p < limit; p++)``        ``{``            ``if` `(mark[p] == ``true``)``            ``{``                ``prime.Add(p);``                ``Console.Write(p + ``" "``);``            ``}``        ``}``    ``}``    ` `    ``// Prints all prime numbers smaller than 'n'``    ``static` `void` `segmentedSieve(``int` `n)``    ``{``        ``// Compute all primes smaller than or equal``        ``// to square root of n using simple sieve``        ``int` `limit = (``int``) (Math.Floor(Math.Sqrt(n)) + 1);``        ``ArrayList prime = ``new` `ArrayList();``        ``simpleSieve(limit, prime);``    ` `        ``// Divide the range [0..n-1] in``        ``// different segments We have chosen``        ``// segment size as sqrt(n).``        ``int` `low = limit;``        ``int` `high = 2*limit;``    ` `        ``// While all segments of range``        ``// [0..n-1] are not processed,``        ``// process one segment at a time``        ``while` `(low < n)``        ``{``            ``if` `(high >= n)``                ``high = n;` `            ``// To mark primes in current range.``            ``// A value in mark[i] will finally``            ``// be false if 'i-low' is Not a prime,``            ``// else true.``            ``bool``[] mark = ``new` `bool``[limit + 1];``            ` `            ``for` `(``int` `i = 0; i < mark.Length; i++)``                ``mark[i] = ``true``;``    ` `            ``// Use the found primes by``            ``// simpleSieve() to find``            ``// primes in current range``            ``for` `(``int` `i = 0; i < prime.Count; i++)``            ``{``                ``// Find the minimum number in``                ``// [low..high] that is a multiple``                ``// of prime.get(i) (divisible by``                ``// prime.get(i)) For example,``                ``// if low is 31 and prime.get(i)``                ``//  is 3, we start with 33.``                ``int` `loLim = ((``int``)Math.Floor((``double``)(low /``                            ``(``int``)prime[i])) * (``int``)prime[i]);``                ``if` `(loLim < low)``                    ``loLim += (``int``)prime[i];``    ` `                ``/* Mark multiples of prime.get(i) in [low..high]:``                    ``We are marking j - low for j, i.e. each number``                    ``in range [low, high] is mapped to [0, high-low]``                    ``so if range is [50, 100] marking 50 corresponds``                    ``to marking 0, marking 51 corresponds to 1 and``                    ``so on. In this way we need to allocate space only``                    ``for range */``                ``for` `(``int` `j = loLim; j < high; j += (``int``)prime[i])``                    ``mark[j-low] = ``false``;``            ``}``    ` `            ``// Numbers which are not marked as false are prime``            ``for` `(``int` `i = low; i < high; i++)``                ``if` `(mark[i - low] == ``true``)``                    ``Console.Write(i + ``" "``);``    ` `            ``// Update low and high for next segment``            ``low = low + limit;``            ``high = high + limit;``        ``}``    ``}``    ` `    ``// Driver code``    ``static` `void` `Main()``    ``{``        ``int` `n = 100;``        ``Console.WriteLine(``"Primes smaller than "` `+ n + ``":"``);``        ``segmentedSieve(n);``    ``}``}` `// This code is contributed by mits`

## Javascript

 `// JavaSCript program to print all primes smaller than``// n using segmented sieve` `// This functions finds all primes smaller than 'limit'``// using simple sieve of eratosthenes. It also stores``// found primes in vector prime[]` `let res = ``""``;` `function` `simpleSieve(limit, prime)``{``    ``// Create a boolean array "mark[0..n-1]" and initialize``    ``// all entries of it as true. A value in mark[p] will``    ``// finally be false if 'p' is Not a prime, else true.``    ``let mark = ``new` `Array(limit+1).fill(``true``);``    ` `    ``for` `(let p=2; p*p= n){``            ``high = n;``        ``}``            ` `        ``// To mark primes in current range. A value in mark[i]``        ``// will finally be false if 'i-low' is Not a prime,``        ``// else true.``        ``let mark = ``new` `Array(limit+1).fill(``true``);` `        ``// Use the found primes by simpleSieve() to find``        ``// primes in current range``        ``for` `(let i = 0; i < prime.length; i++)``        ``{``            ``// Find the minimum number in [low..high] that is``            ``// a multiple of prime[i] (divisible by prime[i])``            ``// For example, if low is 31 and prime[i] is 3,``            ``// we start with 33.``            ``let loLim = Math.floor(low/prime[i]) * prime[i];``            ``if` `(loLim < low){``                ``loLim += prime[i];``            ``}``                ` `            ``/* Mark multiples of prime[i] in [low..high]:``                ``We are marking j - low for j, i.e. each number``                ``in range [low, high] is mapped to [0, high-low]``                ``so if range is [50, 100] marking 50 corresponds``                ``to marking 0, marking 51 corresponds to 1 and``                ``so on. In this way we need to allocate space only``                ``for range */``            ``for` `(let j=loLim; j

Output

```Primes smaller than 100:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

Time Complexity : O(n * ln(sqrt(n)))

Auxiliary Space: O(sqrt(n))

Note that time complexity (or a number of operations) by Segmented Sieve is the same as Simple Sieve. It has advantages for large ‘n’ as it has better locality of reference thus allowing better caching by the CPU and also requires less memory space.