feat: add count distinct primes from binary string algorithm

bit_manipulation/count_distinct_primes_from_binary_string.cpp

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+/**
+ * @file count_distinct_primes_from_binary_string.cpp
+ * @brief Count distinct primes formed from binary strings using allowed operations.
+ * 
+ * @author Rudraksh Tank
+ * @date July 2025
+ *
+ * @details
+ * Given a binary string, the task is to count how many distinct prime decimal numbers
+ * can be formed by:
+ * - Swapping any two characters (makes position irrelevant)
+ * - Changing any '1' to '0' (not the reverse)
+ *
+ * Efficient solution using bit manipulation and Sieve of Eratosthenes.
+ *
+ * Tags: Bit Manipulation, Prime Numbers, Combinatorics, Greedy, Bitmask
+ */
+
+#include <iostream>
+#include <vector>
+#include <unordered_set>
+#include <algorithm>
+
+const int MAX = 1e6;
+std::vector<bool> is_prime;
+
+/**
+ * @brief Precomputes prime numbers up to MAX using the Sieve of Eratosthenes.
+ */
+void precomputePrimes() {
+    is_prime.assign(MAX + 1, true);
+    is_prime[0] = is_prime[1] = false;
+    for (int i = 2; i * i <= MAX; i++) {
+        if (is_prime[i]) {
+            for (int j = i * i; j <= MAX; j += i) {
+                is_prime[j] = false;
+            }
+        }
+    }
+}
+
+/**
+ * @brief Counts distinct prime numbers that can be formed from the given binary string.
+ * @param s Binary string input
+ * @return Number of distinct primes possible after allowed transformations
+ */
+int countPrimeBinaryStrings(const std::string &s) {
+    int n = s.length();
+    int k = std::count(s.begin(), s.end(), '1');
+    int cnt = 0;
+    int limit = 1 << n;
+
+    std::unordered_set<int> seen;
+
+    for (int i = 2; i < limit; i++) {
+        if (__builtin_popcount(i) <= k && is_prime[i]) {
+            if (!seen.count(i)) {
+                cnt++;
+                seen.insert(i);
+            }
+        }
+    }
+
+    return cnt;
+}
+
+/**
+ * @brief Main function to test the algorithm.
+ */
+int main() {
+    precomputePrimes();
+    std::string s;
+    std::cin >> s;
+    std::cout << countPrimeBinaryStrings(s) << std::endl;
+    return 0;
+}
+