From 18dbed16ae2585603340cdee580e86742078e021 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Izan=20Beltr=C3=A1n?= <izanbf1803@gmail.com>
Date: Thu, 25 May 2017 23:01:32 +0200
Subject: [PATCH 1/2] Sieve of Eratosthenes optimization.

Complexity and space reduced to a half.
---
 math/primes_sieve_of_eratosthenes.py | 38 +++++++++++++++++++++++-----
 1 file changed, 31 insertions(+), 7 deletions(-)

diff --git a/math/primes_sieve_of_eratosthenes.py b/math/primes_sieve_of_eratosthenes.py
index 6b8977614..00799d7cb 100644
--- a/math/primes_sieve_of_eratosthenes.py
+++ b/math/primes_sieve_of_eratosthenes.py
@@ -1,14 +1,38 @@
 '''
 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):
+    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 = [2] # List of Primes
+    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

From f6103ff8732ecc95fd424a831e0ce71708cbdcfe Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Izan=20Beltr=C3=A1n?= <izanbf1803@gmail.com>
Date: Fri, 26 May 2017 10:47:01 +0200
Subject: [PATCH 2/2] Fixed error for input 0, 1

Now return []
---
 math/primes_sieve_of_eratosthenes.py | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/math/primes_sieve_of_eratosthenes.py b/math/primes_sieve_of_eratosthenes.py
index 00799d7cb..34289da64 100644
--- a/math/primes_sieve_of_eratosthenes.py
+++ b/math/primes_sieve_of_eratosthenes.py
@@ -27,12 +27,14 @@ def primes(x):
     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 = [2] # List of Primes
+    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
+                sieve[j] = 0
     return primes