Merge pull request keon#76 from izanbf1803/master

Sieve of Eratosthenes optimization.

Author: ankit167

math/primes_sieve_of_eratosthenes.py

@@ -1,14 +1,40 @@
 '''
 Using sieve of Eratosthenes, primes(x) returns list of all primes less than x
+
+Modification:
+We don't need to check all even numbers, we can make the sieve excluding even
+numbers and adding 2 to the primes list by default.
+
+We are going to make an array of: x / 2 - 1 if number is even, else x / 2 
+(The -1 with even number it's to exclude the number itself)
+Because we just need numbers [from 3..x if x is odd]
+
+# We can get value represented at index i with (i*2 + 3)
+
+For example, for x = 10, we start with an array of x / 2 - 1 = 4
+[1, 1, 1, 1]
+ 3  5  7  9
+
+For x = 11:
+[1, 1, 1, 1, 1]
+ 3  5  7  9  11  # 11 is odd, it's included in the list
+
+With this, we have reduced the array size to a half,
+and complexity it's also a half now.
 '''
 
 def primes(x):
-    assert(x>=0)
-    sieve = [1 for v in range(x+1)]     # Sieve
-    primes = []                         # List of Primes
-    for i in range(2,x+1):
-        if sieve[i]==1:
-            primes.append(i)
-            for j in range(i,x+1,i):
-                sieve[j]=0
+    assert(x >= 0)
+    # If x is even, exclude x from list (-1):
+    sieve_size = (x//2 - 1) if x % 2 == 0 else (x//2)
+    sieve = [1 for v in range(sieve_size)]   # Sieve
+    primes = []                              # List of Primes
+    if x >= 2:
+        primes.append(2)                     # Add 2 by default
+    for i in range(0, sieve_size):
+        if sieve[i] == 1:
+            value_at_i = i*2 + 3
+            primes.append(value_at_i)
+            for j in range(i, sieve_size, value_at_i):
+                sieve[j] = 0
     return primes