Add description for Sieve of Eratosthenes algorithm (Fixes: TheAlgorithms#2724) (TheAlgorithms#2725)

Author: Boiarshinov

Others/SieveOfEratosthenes.java

@@ -1,44 +1,76 @@
 package Others;
 
-/** @author Varun Upadhyay (https://github.com/varunu28) */
+import java.util.Arrays;
+
+/**
+ * Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
+ * It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime,
+ * starting with the first prime number, 2.
+ * The multiples of a given prime are generated as a sequence of numbers starting from that prime,
+ * with constant difference between them that is equal to that prime.
+ * This is the sieve's key distinction from using trial division to sequentially test each
+ * candidate number for divisibility by each prime.
+ * Once all the multiples of each discovered prime have been marked as composites, the remaining
+ * unmarked numbers are primes.
+ * <p>
+ * Poetry about Sieve of Eratosthenes:
+ *    <p><i>Sift the Two's and Sift the Three's:</i></p>
+ *    <p><i>The Sieve of Eratosthenes.</i></p>
+ *    <p><i>When the multiples sublime,</i></p>
+ *    <p><i>The numbers that remain are Prime.</i></p>
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes">Wiki</a>
+ */
 public class SieveOfEratosthenes {
 
   /**
-   * This method implements the Sieve of Eratosthenes Algorithm
-   *
-   * @param n The number till which we have to check for prime Prints all the prime numbers till n
+   * @param n The number till which we have to check for prime Prints all the prime numbers till n.
+   *          Should be more than 1.
+   * @return array of all prime numbers between 0 to n
    */
-  public static void findPrimesTillN(int n) {
-    int[] arr = new int[n + 1];
-
-    for (int i = 0; i <= n; i++) {
-      arr[i] = 1;
-    }
+  public static int[] findPrimesTill(int n) {
+    // Create array where index is number and value is flag - is that number a prime or not.
+    // size of array is n + 1 cause in Java array indexes starts with 0
+    Type[] numbers = new Type[n + 1];
 
-    arr[0] = arr[1] = 0;
+    // Start with assumption that all numbers except 0 and 1 are primes.
+    Arrays.fill(numbers, Type.PRIME);
+    numbers[0] = numbers[1] = Type.NOT_PRIME;
 
-    for (int i = 2; i <= Math.sqrt(n); i++) {
-      if (arr[i] == 1) {
+    double cap = Math.sqrt(n);
+    // Main algorithm: mark all numbers which are multiples of some other values as not prime
+    for (int i = 2; i <= cap; i++) {
+      if (numbers[i] == Type.PRIME) {
         for (int j = 2; i * j <= n; j++) {
-          arr[i * j] = 0;
+          numbers[i * j] = Type.NOT_PRIME;
         }
       }
     }
 
+    //Write all primes to result array
+    int primesCount = (int) Arrays.stream(numbers)
+        .filter(element -> element == Type.PRIME)
+        .count();
+    int[] primes = new int[primesCount];
+
+    int primeIndex = 0;
     for (int i = 0; i < n + 1; i++) {
-      if (arr[i] == 1) {
-        System.out.print(i + " ");
+      if(numbers[i] == Type.PRIME) {
+        primes[primeIndex++] = i;
       }
     }
 
-    System.out.println();
+    return primes;
+  }
+
+  private enum Type {
+    PRIME, NOT_PRIME
   }
 
-  // Driver Program
   public static void main(String[] args) {
     int n = 100;
-
-    // Prints 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
-    findPrimesTillN(n);
+    System.out.println("Searching for all primes from zero to " + n);
+    int[] primes = findPrimesTill(n);
+    System.out.println("Found: " + Arrays.toString(primes));
   }
 }