fix, test: Refactor of sieve_of_eratosthenes (#969)

* fix, test: Refactor of sieve_of_eratosthenes

* Add missing include.

* Modernize the vector initialization.

* Add @details for the documentation.

Author: fhlasek

math/sieve_of_eratosthenes.cpp

@@ -1,58 +1,72 @@
 /**
  * @file
  * @brief Get list of prime numbers using Sieve of Eratosthenes
- * Sieve of Eratosthenes is an algorithm to find the primes
- * that is between 2 to N (as defined in main).
+ * @details
+ * Sieve of Eratosthenes is an algorithm that finds all the primes
+ * between 2 and N.
  *
- * Time Complexity  : \f$O(N \cdot\log N)\f$
+ * Time Complexity  : \f$O(N \cdot\log \log N)\f$
  * <br/>Space Complexity : \f$O(N)\f$
  *
  * @see primes_up_to_billion.cpp prime_numbers.cpp
  */
 
-#include <iostream> // for io operations
+#include <cassert>
+#include <iostream>
+#include <vector>
 
 /**
- * This is the function that finds the primes and eliminates
- * the multiples.
+ * This is the function that finds the primes and eliminates the multiples.
+ * Contains a common optimization to start eliminating multiples of
+ * a prime p starting from p * p since all of the lower multiples
+ * have been already eliminated.
  * @param N number of primes to check
- * @param [out] isprime a boolean array of size `N` identifying if `i`^th number is prime or not
+ * @return is_prime a vector of `N + 1` booleans identifying if `i`^th number is a prime or not
  */
-void sieve(uint32_t N, bool *isprime) {
-    isprime[0] = true;
-    isprime[1] = true;
+std::vector<bool> sieve(uint32_t N) {
+    std::vector<bool> is_prime(N + 1, true);
+    is_prime[0] = is_prime[1] = false;
     for (uint32_t i = 2; i * i <= N; i++) {
-        if (!isprime[i]) {
-            for (uint32_t j = (i << 1); j <= N; j = j + i) {
-                isprime[j] = true;
+        if (is_prime[i]) {
+            for (uint32_t j = i * i; j <= N; j += i) {
+                is_prime[j] = false;
             }
         }
     }
+    return is_prime;
 }
 
 /**
  * This function prints out the primes to STDOUT
  * @param N number of primes to check
- * @param [in] isprime a boolean array of size `N` identifying if `i`^th number is prime or not
+ * @param is_prime a vector of `N + 1` booleans identifying if `i`^th number is a prime or not
  */
-void print(uint32_t N, const bool *isprime) {
+void print(uint32_t N, const std::vector<bool> &is_prime) {
     for (uint32_t i = 2; i <= N; i++) {
-        if (!isprime[i]) {
+        if (is_prime[i]) {
             std::cout << i << ' ';
         }
     }
     std::cout << std::endl;
 }
 
+/**
+ * Test implementations
+ */
+void tests() {
+  //                    0      1      2     3     4      5     6      7     8      9      10
+  std::vector<bool> ans{false, false, true, true, false, true, false, true, false, false, false};
+  assert(sieve(10) == ans);
+}
+
 /**
  * Main function
  */
 int main() {
-    uint32_t N = 100;
-    bool *isprime = new bool[N];
-    sieve(N, isprime);
-    print(N, isprime);
-    delete[] isprime;
+    tests();
 
+    uint32_t N = 100;
+    std::vector<bool> is_prime = sieve(N);
+    print(N, is_prime);
     return 0;
 }