Added sieve

Updated this folder

Author: kedarj0shi

Number_Theory/sieve.cpp

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+// Prime Seive aka Sieve of Eratosthenes
+// Time complexity : O(n*log(log(n)))
+// Space complexity: O(n)
+// DESC : here we will discuss how to find prime numbers in a given range 
+#include <bits/stdc++.h>
+
+using namespace std;
+
+void sieve(vector<bool>& isprime, int n){
+    // since 0 and 1 are non-prime numbers we mark them as false;
+    isprime[0] = isprime[1] = false;
+    // 1st method
+    for(int i = 2; i * i <= n; i++){
+        if(isprime[i] == true){ //if we find i'th number to be a prime number, we will mark every multiple of that number as non-prime
+            for(int j = i*i ; j <= n; j += i){
+                isprime[j] = false;
+            }
+        }
+    }
+    // here we are iterating every element from 2 to sqrt(n), but since we know that 2 is the only even prime number
+    // this can be further optimized to
+    // first we will mark every multiple of 2, and then in the next loop we will iterating only for odd numbers 
+    // 2. more optimized
+    for(int i = 2; i < 3; i++){
+        if(isprime[i] == true){ //if we find i'th number to be a prime number, we will mark every multiple of that number as non-prime
+            for(int j = i*i ; j <= n; j += i){
+                isprime[j] = false;
+            }
+        }
+    }
+    for(int i = 3; i * i <= n; i += 2){ // will only check for odd number
+        if(isprime[i] == true){ //if we find i'th number to be a prime number, we will mark every multiple of that number as non-prime
+            for(int j = i*i ; j <= n; j += i){
+                isprime[j] = false;
+            }
+        }
+    }
+    // now in the isprime vector we have prime numbers from 0 to 1000000
+    // note that we are using isprime vector of boolean data type and not integer as it will reduce space (integer takes 4bytes, whereas bool takes 1byte)
+}
+
+signed main(){
+    int n=1e6;
+    //initialize a vector isprime such that if value at an index is true then it is a prime number, else a composite
+    //initially we consider every number as prime
+    vector<bool> isprime(n+1,true);  
+    sieve(v,n);
+
+    return 0;
+}