Sieb des Eratosthenes in 0(n) Zeitkomplexität
Der klassische Sieve-Algorithmus von Eratosthenes benötigt O(N log (log N)) Zeit, um alle Primzahlen kleiner als N zu finden. In diesem Artikel wird ein modifiziertes Sieve diskutiert, das in O(N)-Zeit arbeitet.
Beispiel :
Given a number N, print all prime numbers smaller than N Input : int N = 15 Output : 2 3 5 7 11 13 Input : int N = 20 Output : 2 3 5 7 11 13 17 19
Der Algorithmus des manipulierten Siebs von Eratosthenes funktioniert wie folgt:
For every number i where i varies from 2 to N-1: Check if the number is prime. If the number is prime, store it in prime array. For every prime numbers j less than or equal to the smallest prime factor p of i: Mark all numbers i*p as non_prime. Mark smallest prime factor of i*p as j
Unten ist die Implementierung der obigen Idee.
C++
// C++ program to generate all prime numbers // less than N in O(N) #include<bits/stdc++.h> using namespace std; const long long MAX_SIZE = 1000001; // isPrime[] : isPrime[i] is true if number is prime // prime[] : stores all prime number less than N // SPF[] that store smallest prime factor of number // [for Exp : smallest prime factor of '8' and '16' // is '2' so we put SPF[8] = 2 , SPF[16] = 2 ] vector<long long >isprime(MAX_SIZE , true); vector<long long >prime; vector<long long >SPF(MAX_SIZE); // function generate all prime number less then N in O(n) void manipulated_seive(int N) { // 0 and 1 are not prime isprime[0] = isprime[1] = false ; // Fill rest of the entries for (long long int i=2; i<N ; i++) { // If isPrime[i] == True then i is // prime number if (isprime[i]) { // put i into prime[] vector prime.push_back(i); // A prime number is its own smallest // prime factor SPF[i] = i; } // Remove all multiples of i*prime[j] which are // not prime by making isPrime[i*prime[j]] = false // and put smallest prime factor of i*Prime[j] as prime[j] // [ for exp :let i = 5 , j = 0 , prime[j] = 2 [ i*prime[j] = 10 ] // so smallest prime factor of '10' is '2' that is prime[j] ] // this loop run only one time for number which are not prime for (long long int j=0; j < (int)prime.size() && i*prime[j] < N && prime[j] <= SPF[i]; j++) { isprime[i*prime[j]]=false; // put smallest prime factor of i*prime[j] SPF[i*prime[j]] = prime[j] ; } } } // driver program to test above function int main() { int N = 13 ; // Must be less than MAX_SIZE manipulated_seive(N); // pint all prime number less then N for (int i=0; i<prime.size() && prime[i] <= N ; i++) cout << prime[i] << " "; return 0; }
Java
// Java program to generate all prime numbers // less than N in O(N) import java.util.Vector; class Test { static final int MAX_SIZE = 1000001; // isPrime[] : isPrime[i] is true if number is prime // prime[] : stores all prime number less than N // SPF[] that store smallest prime factor of number // [for Exp : smallest prime factor of '8' and '16' // is '2' so we put SPF[8] = 2 , SPF[16] = 2 ] static Vector<Boolean>isprime = new Vector<>(MAX_SIZE); static Vector<Integer>prime = new Vector<>(); static Vector<Integer>SPF = new Vector<>(MAX_SIZE); // method generate all prime number less then N in O(n) static void manipulated_seive(int N) { // 0 and 1 are not prime isprime.set(0, false); isprime.set(1, false); // Fill rest of the entries for (int i=2; i<N ; i++) { // If isPrime[i] == True then i is // prime number if (isprime.get(i)) { // put i into prime[] vector prime.add(i); // A prime number is its own smallest // prime factor SPF.set(i,i); } // Remove all multiples of i*prime[j] which are // not prime by making isPrime[i*prime[j]] = false // and put smallest prime factor of i*Prime[j] as prime[j] // [for exp :let i = 5, j = 0, prime[j] = 2 [ i*prime[j] = 10] // so smallest prime factor of '10' is '2' that is prime[j] ] // this loop run only one time for number which are not prime for (int j=0; j < prime.size() && i*prime.get(j) < N && prime.get(j) <= SPF.get(i); j++) { isprime.set(i*prime.get(j),false); // put smallest prime factor of i*prime[j] SPF.set(i*prime.get(j),prime.get(j)) ; } } } // Driver method public static void main(String args[]) { int N = 13 ; // Must be less than MAX_SIZE // initializing isprime and spf for (int i = 0; i < MAX_SIZE; i++){ isprime.add(true); SPF.add(2); } manipulated_seive(N); // pint all prime number less then N for (int i=0; i<prime.size() && prime.get(i) <= N ; i++) System.out.print(prime.get(i) + " "); } }
Python3
# Python3 program to generate all # prime numbers less than N in O(N) MAX_SIZE = 1000001 # isPrime[] : isPrime[i] is true if # number is prime # prime[] : stores all prime number # less than N # SPF[] that store smallest prime # factor of number [for ex : smallest # prime factor of '8' and '16' # is '2' so we put SPF[8] = 2 , # SPF[16] = 2 ] isprime = [True] * MAX_SIZE prime = [] SPF = [None] * (MAX_SIZE) # function generate all prime number # less then N in O(n) def manipulated_seive(N): # 0 and 1 are not prime isprime[0] = isprime[1] = False # Fill rest of the entries for i in range(2, N): # If isPrime[i] == True then i is # prime number if isprime[i] == True: # put i into prime[] vector prime.append(i) # A prime number is its own smallest # prime factor SPF[i] = i # Remove all multiples of i*prime[j] # which are not prime by making is # Prime[i * prime[j]] = false and put # smallest prime factor of i*Prime[j] # as prime[j] [ for exp :let i = 5 , j = 0 , # prime[j] = 2 [ i*prime[j] = 10 ] # so smallest prime factor of '10' is '2' # that is prime[j] ] this loop run only one # time for number which are not prime j = 0 while (j < len(prime) and i * prime[j] < N and prime[j] <= SPF[i]): isprime[i * prime[j]] = False # put smallest prime factor of i*prime[j] SPF[i * prime[j]] = prime[j] j += 1 # Driver Code if __name__ == "__main__": N = 13 # Must be less than MAX_SIZE manipulated_seive(N) # print all prime number less then N i = 0 while i < len(prime) and prime[i] <= N: print(prime[i], end = " ") i += 1 # This code is contributed by Rituraj Jain
PHP
<?php // PHP program to generate all // prime numbers less than N in O(N) $MAX_SIZE = 10001; // isPrime[] : isPrime[i] is true if // number is prime // prime[] : stores all prime number // less than N // SPF[] that store smallest prime // factor of number [for ex : smallest // prime factor of '8' and '16' // is '2' so we put SPF[8] = 2 , // SPF[16] = 2 ] $isprime = array_fill(0, $MAX_SIZE, true); $prime = array(); $SPF = array_fill(0, $MAX_SIZE, 0); // function generate all prime number // less then N in O(n) function manipulated_seive($N) { global $isprime, $MAX_SIZE, $SPF, $prime; // 0 and 1 are not prime $isprime[0] = $isprime[1] = false; // Fill rest of the entries for ($i = 2; $i < $N; $i++) { // If isPrime[i] == True then // i is prime number if ($isprime[$i]) { // put i into prime[] vector array_push($prime, $i); // A prime number is its own // smallest prime factor $SPF[$i] = $i; } // Remove all multiples of i*prime[j] // which are not prime by making is // Prime[i * prime[j]] = false and put // smallest prime factor of i*Prime[j] // as prime[j] [ for exp :let i = 5 , j = 0 , // prime[j] = 2 [ i*prime[j] = 10 ] // so smallest prime factor of '10' is '2' // that is prime[j] ] this loop run only // one time for number which are not prime $j = 0; while ($j < count($prime) && $i * $prime[$j] < $N && $prime[$j] <= $SPF[$i]) { $isprime[$i * $prime[$j]] = false; // put smallest prime factor of i*prime[j] $SPF[$i * $prime[$j]] = $prime[$j]; $j += 1; } } } // Driver Code $N = 13; // Must be less than MAX_SIZE manipulated_seive($N); // print all prime number less then N $i = 0; while ($i < count($prime) && $prime[$i] <= $N) { print($prime[$i] . " "); $i += 1; } // This code is contributed by mits ?>
Ausgabe :
2 3 5 7 11
Illustration:
isPrime[0] = isPrime[1] = 0 After i = 2 iteration : isPrime[] [F, F, T, T, F, T, T, T] SPF[] [0, 0, 2, 0, 2, 0, 0, 0] index 0 1 2 3 4 5 6 7 After i = 3 iteration : isPrime[] [F, F, T, T, F, T, F, T, T, F ] SPF[] [0, 0, 2, 3, 2, 0, 2, 0, 0, 3 ] index 0 1 2 3 4 5 6 7 8 9 After i = 4 iteration : isPrime[] [F, F, T, T, F, T, F, T, F, F] SPF[] [0, 0, 2, 3, 2, 0, 2, 0, 2, 3] index 0 1 2 3 4 5 6 7 8 9
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