Added segmented sieve for generating prime number in a given range, providing more memory efficiency

Author: CaptainDaVinci

Prime/code.py

@@ -1,9 +1,55 @@
-def SieveOfEratosthenes(range_to):
+import time
+from math import sqrt, ceil
 
-    # creating a boolean array first
+
+def segmented_sieve(range_from, range_to):
+    """
+    Segmented sieve helps in providing a efficient way in terms of
+    memory to find the prime numbers in a given range.
+
+    Space complexity: O(l)
+    Time complexity: O(l log(logl))
+
+    l =  range_to - range_from
+    """
+
+    # if the range starts from 1, incerement it.
+    if range_from == 1:
+        range_from += 1
+
+    # initialize a boolean list of size l, with True.
+    primes_range = [True for _ in range(range_to - range_from + 1)]
+
+    # generate primes upto sqrt(range_to) using SieveOfEratosthenes.
+    primes = sieve_eratosthenes(ceil(sqrt(range_to)))
+
+    # for every prime, all its multiples will not be prime
+    for prime in primes:
+        prime_multiple = max(range_from // prime, 2) * prime
+
+        # cross off all the multiple of the prime number.
+        for i in range(prime_multiple, range_to + 1, prime):
+            if i >= range_from:
+                primes_range[i - range_from] = False
+
+    # return a list of prime numbers from range_from to range_to
+    return [range_from + i for i, prime in enumerate(primes_range) if prime]
+
+
+def sieve_eratosthenes(range_to):
+    """
+    A Very efficient way to generate all prime numbers upto a number.
+
+    Space complexity: O(n)
+    Time complexity: O(n * log(logn))
+
+    n = the number upto which prime numbers are to be generated.
+    """
+
+    # creating a boolean list first
     prime = [True for i in range(range_to + 1)]
     p = 2
-    while (p * p <= range_to):    
+    while (p * p <= range_to):
 
         # If prime[p] is not changed, then it is a prime
         if (prime[p] == True):
@@ -14,27 +60,49 @@ def SieveOfEratosthenes(range_to):
 
         p += 1
 
-    print([p for p in range(2, range_to) if prime[p]])
+    # return the list of primes
+    return [p for p in range(2, range_to + 1) if prime[p]]
 
 
 def is_prime(num):
-        for i in range(2, int(num / 2) + 1):
-            if num % i == 0:
-                return False
-        return True
+    """
+        Returns true is the number is prime.
+
+        To check if a number is prime, we need to check
+        all its divisors upto only sqrt(number).
+
+        Reference: https://stackoverflow.com/a/5811176/7213370
+    """
+
+    # corner case. 1 is not a prime
+    if num == 1:
+        return False
+
+    # check if for a divisor upto sqrt(number)
+    i = 2
+    while i * i <= num:
+        if num % i == 0:
+            return False
+        i += 1
+
+    return True
 
 
 def prime(range_from, range_to):
-    allList = [num for num in range(range_from, range_to + 1) if is_prime(num)]
+    """
+    Brute Force. O(l * sqrt(num)) [l = range_to - range_from]
+
+    Returns list of primes in the given range, by checking
+    if the number is a prime or not.
+    """
+    return [num for num in range(range_from, range_to + 1) if is_prime(num)]
+
 
-    print("Prime List Between", range_from, "and", range_to, "is")
-    print(allList)
-  
-  
 if __name__ == "__main__":
     print("Enter the range of number")
     range_from = int(input("From: "))
     range_to = int(input("To: "))
 
-    prime(range_from, range_to)
-    SieveOfEratosthenes(range_to)
+    print("Brute force:", prime(range_from, range_to), end="\n\n")
+    print("Sieve of Eratosthenes:", sieve_eratosthenes(range_to), end="\n\n")
+    print("Segmented Sieve:", segmented_sieve(range_from, range_to))