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Count Primes

Certification: Intermediate Level Accuracy: 25% Submissions: 4 Points: 10

Write a C# program to count the number of prime numbers less than a non-negative number n. Implement an optimized solution to efficiently calculate this count for large values of n.

Example 1
    
  
  • Input: n = 10
    • 
  
  • Output: 4
    • 
  
  • Explanation:
    
      
      
    • There are 4 prime numbers less than 10, they are 2, 3, 5, 7.
      • 
    
    
  
    •
Example 2
    
  
  • Input: n = 0
    • 
  
  • Output: 0
    • 
  
  • Explanation:
    
      
      
    • There are no prime numbers less than 0.
      • 
    
    
  
    •
Constraints
    
  
  • 0 ≤ n ≤ 5 * 10^6
    • 
  
  • The solution should handle large values of n efficiently
    • 
  
  • Time Complexity: O(n log log n)
    • 
  
  • Space Complexity: O(n)
    •
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Hint
Solution

Solution Hints

    
  
  • Use the Sieve of Eratosthenes algorithm for an efficient solution
    • 
  
  • Initialize an array to track which numbers are prime
    • 
  
  • Start with marking all numbers as potential primes
    • 
  
  • Iterate through numbers starting from 2, and mark all multiples as non-prime
    • 
  
  • Only check numbers up to the square root of n for optimization
    •

Steps to solve by this approach:

 Step 1: Handle edge cases - if n is less than or equal to 2, return 0 (there are no primes less than 2).

 Step 2: Create a boolean array of size n to mark which numbers are prime.

 Step 3: Initially, mark all numbers from 2 to n-1 as potential primes (true in the array).

 Step 4: Implement the Sieve of Eratosthenes algorithm - start from the smallest prime (2).

 Step 5: For each prime number, mark all its multiples as non-prime (false in the array).

 Step 6: Optimize by only checking numbers up to the square root of n, as larger factors would have been marked by smaller factors.

 Step 7: Count all remaining numbers marked as prime (true in the array) and return the count.

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