Sieve of Eratosthenes Find All Primes in DSA C# - Time & Space Complexity

Time Complexity: Sieve of Eratosthenes Find All Primes

O(n log log n)

Understanding Time Complexity

We want to understand how the time needed to find all prime numbers grows as the input number gets bigger.

Specifically, how does the Sieve of Eratosthenes perform when finding primes up to a number n?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

bool[] isPrime = new bool[n + 1];
for (int i = 2; i <= n; i++)
    isPrime[i] = true;
for (int i = 2; i * i <= n; i++)
{
    if (isPrime[i])
    {
        for (int j = i * i; j <= n; j += i)
            isPrime[j] = false;
    }
}
// isPrime array now marks primes as true
    

This code marks all prime numbers up to n by crossing out multiples of each prime.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Marking multiples of each prime number as not prime.
• How many times: For each prime i, we mark multiples starting from i*i up to n, stepping by i.

How Execution Grows With Input

As n grows, the number of operations grows a bit faster than n but much slower than n squared.

Input Size (n)	Approx. Operations
10	~20
100	~300
1000	~5000

Pattern observation: The work grows roughly proportional to n times the logarithm of the logarithm of n, which is slower than n log n but faster than n.

Final Time Complexity

Time Complexity: O(n log log n)

This means the time grows a bit faster than n but much slower than n squared, making it efficient for large n.

Common Mistake

[X] Wrong: "The inner loop runs n times for each i, so total time is O(n²)."

[OK] Correct: The inner loop starts at i*i and skips by i, so it does fewer steps as i grows, reducing total work significantly.

Interview Connect

Understanding this algorithm's time complexity shows you can analyze loops that don't run the same number of times each iteration, a valuable skill in interviews.

Self-Check

"What if we started marking multiples from 2*i instead of i*i? How would the time complexity change?"

Practice

1. What is the main purpose of the Sieve of Eratosthenes algorithm?

easy

A. To sort an array of numbers

B. To find the greatest common divisor of two numbers

C. To find all prime numbers up to a given limit

D. To reverse a string

Solution

1. Step 1: Understand the algorithm's goal

   The Sieve of Eratosthenes is designed to identify prime numbers efficiently.

2. Step 2: Recognize the output

   It marks multiples of primes as non-prime to find all primes up to a limit.

3. Final Answer:

   To find all prime numbers up to a given limit -> Option C

4. Quick Check:

   Sieve of Eratosthenes = Find primes [OK]

Hint: Remember: Sieve filters out non-primes to list primes [OK]

Common Mistakes:

• Confusing it with sorting algorithms
• Thinking it finds factors instead of primes
• Assuming it works on strings

2. Which of the following C# code snippets correctly initializes a boolean array for the sieve to find primes up to n?

easy

A. bool[] isPrime = new bool[n + 1]; for(int i = 0; i <= n; i++) isPrime[i] = false;

B. bool[] isPrime = new bool[n + 1]; for(int i = 2; i <= n; i++) isPrime[i] = true;

C. bool[] isPrime = new bool[n]; for(int i = 0; i < n; i++) isPrime[i] = false;

D. bool[] isPrime = new bool[n]; for(int i = 2; i < n; i++) isPrime[i] = true;

Solution

1. Step 1: Check array size and initialization

   The array must be of size n+1 to include index n, and primes start from 2.

2. Step 2: Verify initialization values

   All entries from 2 to n should be set to true initially to mark potential primes.

3. Final Answer:

   bool[] isPrime = new bool[n + 1]; for(int i = 2; i <= n; i++) isPrime[i] = true; -> Option B

4. Quick Check:

   Array size n+1 and true from 2 to n [OK]

Hint: Array size n+1 and true from 2 to n for primes [OK]

Common Mistakes:

• Using array size n instead of n+1
• Initializing all values to false
• Starting initialization from 0 or 1

3. What will be the output list of primes when running the Sieve of Eratosthenes for n = 10?

medium

A. [2, 3, 5, 7, 9]

B. [1, 2, 3, 5, 7]

C. [2, 3, 4, 5, 7]

D. [2, 3, 5, 7]

Solution

1. Step 1: Identify primes up to 10

   Primes are numbers greater than 1 with no divisors other than 1 and themselves: 2, 3, 5, 7.

2. Step 2: Exclude non-primes

   Numbers 1, 4, 6, 8, 9, 10 are not prime.

3. Final Answer:

   [2, 3, 5, 7] -> Option D

4. Quick Check:

   Primes up to 10 = [2, 3, 5, 7] [OK]

Hint: Remember primes > 1 and exclude multiples [OK]

Common Mistakes:

• Including 1 as prime
• Including even numbers > 2
• Including non-prime odd numbers like 9

4. Consider this code snippet for the sieve:

for (int i = 2; i * i <= n; i++) {
  if (isPrime[i]) {
    for (int j = i * 2; j <= n; j += i) {
      isPrime[j] = false;
    }
  }
}

Which of the following improves the efficiency of this code?

medium

A. The inner loop should start from i * i, not i * 2

B. The outer loop condition should be i <= n, not i * i <= n

C. The inner loop increments should be j += 1, not j += i

D. The isPrime array should be initialized with false, not true

Solution

1. Step 1: Understand sieve optimization

   Marking multiples starts from i squared because smaller multiples were marked by smaller primes.

2. Step 2: Identify suboptimal start of inner loop

   Starting from i*2 repeats marking multiples already handled; correct start is i*i.

3. Final Answer:

   The inner loop should start from i * i, not i * 2 -> Option A

4. Quick Check:

   Start inner loop at i*i for efficiency [OK]

Hint: Start marking multiples from i squared, not double i [OK]

Common Mistakes:

• Starting inner loop at i*2 causing redundant work
• Changing outer loop condition incorrectly
• Incorrect increment steps in inner loop

5. You want to find all prime numbers up to n = 30 using the Sieve of Eratosthenes. After running the sieve, which of the following lists correctly shows the primes and their count?

hard

A. Primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29], Count: 10

B. Primes: [2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29], Count: 11

C. Primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29], Count: 11

D. Primes: [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29], Count: 11

Solution

1. Step 1: List primes up to 30

   Primes are numbers >1 with no divisors except 1 and themselves: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

2. Step 2: Count and verify correctness

   Count is 10; numbers like 9, 25, and 1 are not prime and should be excluded.

3. Final Answer:

   Primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29], Count: 10 -> Option A

4. Quick Check:

   Correct primes and count up to 30 [OK]

Hint: Exclude 1 and non-primes like 9, 25; count primes carefully [OK]

Common Mistakes:

• Including 1 as prime
• Including composite numbers like 9 or 25
• Miscounting the total primes