From cb6f7e89e263cd846777046ec17cc4006adb0f70 Mon Sep 17 00:00:00 2001
From: piyushkumar0707 <121piyush466mits@gmail.com>
Date: Sat, 4 Oct 2025 09:48:17 +0530
Subject: [PATCH] Add Sieve of Eratosthenes algorithm implementation

- Implement both basic and optimized versions of the sieve
- Include segmented sieve for finding primes in ranges
- Add performance comparison with trial division
- Comprehensive examples and edge case handling
- Update DIRECTORY.md to include the new algorithm
---
 DIRECTORY.md                        |   1 +
 mathematics/sieve_of_eratosthenes.r | 212 ++++++++++++++++++++++++++++
 2 files changed, 213 insertions(+)
 create mode 100644 mathematics/sieve_of_eratosthenes.r

diff --git a/DIRECTORY.md b/DIRECTORY.md
index 8f377748..d3e69a3e 100644
--- a/DIRECTORY.md
+++ b/DIRECTORY.md
@@ -47,6 +47,7 @@
   * [Perfect Square](https://github.com/TheAlgorithms/R/blob/HEAD/mathematics/perfect_square.r)
   * [Pi Monte Carlo](https://github.com/TheAlgorithms/R/blob/HEAD/mathematics/pi_monte_carlo.r)
   * [Prime](https://github.com/TheAlgorithms/R/blob/HEAD/mathematics/prime.r)
+  * [Sieve Of Eratosthenes](https://github.com/TheAlgorithms/R/blob/HEAD/mathematics/sieve_of_eratosthenes.r)
 
 ## Regression Algorithms
   * [Ann](https://github.com/TheAlgorithms/R/blob/HEAD/regression_algorithms/ann.r)
diff --git a/mathematics/sieve_of_eratosthenes.r b/mathematics/sieve_of_eratosthenes.r
new file mode 100644
index 00000000..14c50d4c
--- /dev/null
+++ b/mathematics/sieve_of_eratosthenes.r
@@ -0,0 +1,212 @@
+# Sieve of Eratosthenes Algorithm
+#
+# The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers
+# up to a given limit. It works by iteratively marking the multiples of each prime
+# starting from 2, and the unmarked numbers that remain are primes.
+#
+# Time Complexity: O(n log log n)
+# Space Complexity: O(n)
+#
+# Input: A positive integer n (the upper limit)
+# Output: A vector of all prime numbers from 2 to n
+
+sieve_of_eratosthenes <- function(n) {
+  # Handle edge cases
+  if (n < 2) {
+    return(integer(0))  # No primes less than 2
+  }
+  
+  # Create a boolean array "prime[0..n]" and initialize all entries as TRUE
+  prime <- rep(TRUE, n + 1)
+  prime[1] <- FALSE  # 1 is not a prime number
+  
+  p <- 2
+  while (p * p <= n) {
+    # If prime[p] is not changed, then it is a prime
+    if (prime[p]) {
+      # Update all multiples of p starting from p^2
+      for (i in seq(p * p, n, by = p)) {
+        prime[i] <- FALSE
+      }
+    }
+    p <- p + 1
+  }
+  
+  # Collect all prime numbers
+  primes <- which(prime)[-1]  # Remove index 1 (since 1 is not prime)
+  return(primes)
+}
+
+# Optimized version that only checks odd numbers after 2
+sieve_of_eratosthenes_optimized <- function(n) {
+  # Handle edge cases
+  if (n < 2) {
+    return(integer(0))
+  }
+  if (n == 2) {
+    return(2)
+  }
+  
+  # Start with 2 (the only even prime)
+  primes <- c(2)
+  
+  # Create boolean array for odd numbers only (3, 5, 7, ...)
+  # Index i represents number (2*i + 3)
+  size <- (n - 1) %/% 2
+  is_prime <- rep(TRUE, size)
+  
+  # Sieve process for odd numbers
+  for (i in 1:size) {
+    if (is_prime[i]) {
+      num <- 2 * i + 1  # Convert index to actual odd number
+      
+      # Mark multiples of num starting from num^2
+      if (num * num <= n) {
+        start_idx <- (num * num - 1) %/% 2  # Convert num^2 to index
+        for (j in seq(start_idx, size, by = num)) {
+          if (j <= size) {
+            is_prime[j] <- FALSE
+          }
+        }
+      }
+    }
+  }
+  
+  # Collect odd primes
+  odd_primes <- 2 * which(is_prime) + 1
+  primes <- c(primes, odd_primes)
+  
+  return(primes)
+}
+
+# Function to count primes up to n (useful for large n)
+count_primes_sieve <- function(n) {
+  if (n < 2) {
+    return(0)
+  }
+  
+  prime <- rep(TRUE, n + 1)
+  prime[1] <- FALSE
+  
+  p <- 2
+  while (p * p <= n) {
+    if (prime[p]) {
+      for (i in seq(p * p, n, by = p)) {
+        prime[i] <- FALSE
+      }
+    }
+    p <- p + 1
+  }
+  
+  return(sum(prime))
+}
+
+# Function to check if a number is prime using trial division (for comparison)
+is_prime_trial_division <- function(n) {
+  if (n <= 1) return(FALSE)
+  if (n <= 3) return(TRUE)
+  if (n %% 2 == 0 || n %% 3 == 0) return(FALSE)
+  
+  i <- 5
+  while (i * i <= n) {
+    if (n %% i == 0 || n %% (i + 2) == 0) {
+      return(FALSE)
+    }
+    i <- i + 6
+  }
+  return(TRUE)
+}
+
+# Segmented sieve for finding primes in a range [low, high]
+segmented_sieve <- function(low, high) {
+  # First, find all primes up to sqrt(high)
+  limit <- floor(sqrt(high))
+  primes <- sieve_of_eratosthenes(limit)
+  
+  # Create a boolean array for range [low, high]
+  size <- high - low + 1
+  is_prime <- rep(TRUE, size)
+  
+  # Mark multiples of each prime in the range
+  for (prime in primes) {
+    # Find the minimum number in [low, high] that is a multiple of prime
+    start <- max(prime * prime, low + (prime - low %% prime) %% prime)
+    
+    # Mark multiples of prime in the range
+    for (j in seq(start, high, by = prime)) {
+      is_prime[j - low + 1] <- FALSE
+    }
+  }
+  
+  # Handle the case where low = 1 (1 is not prime)
+  if (low == 1) {
+    is_prime[1] <- FALSE
+  }
+  
+  # Collect primes in the range
+  range_primes <- (low:high)[is_prime]
+  return(range_primes)
+}
+
+# Example usage and testing
+cat("=== Sieve of Eratosthenes Algorithm ===\n")
+
+# Test with small number
+cat("Primes up to 30:\n")
+primes_30 <- sieve_of_eratosthenes(30)
+cat(paste(primes_30, collapse = ", "), "\n")
+cat("Count:", length(primes_30), "\n\n")
+
+# Test optimized version
+cat("Optimized sieve - Primes up to 30:\n")
+primes_30_opt <- sieve_of_eratosthenes_optimized(30)
+cat(paste(primes_30_opt, collapse = ", "), "\n")
+cat("Count:", length(primes_30_opt), "\n\n")
+
+# Test with larger number
+cat("Primes up to 100:\n")
+primes_100 <- sieve_of_eratosthenes(100)
+cat("Count:", length(primes_100), "\n")
+cat("First 10 primes:", paste(primes_100[1:10], collapse = ", "), "\n")
+cat("Last 10 primes:", paste(tail(primes_100, 10), collapse = ", "), "\n\n")
+
+# Performance comparison for counting primes
+cat("=== Performance Comparison ===\n")
+n <- 1000
+
+# Count using sieve
+start_time <- Sys.time()
+sieve_count <- count_primes_sieve(n)
+sieve_time <- as.numeric(Sys.time() - start_time, units = "secs")
+
+# Count using trial division
+start_time <- Sys.time()
+trial_count <- sum(sapply(2:n, is_prime_trial_division))
+trial_time <- as.numeric(Sys.time() - start_time, units = "secs")
+
+cat("Primes up to", n, ":\n")
+cat("Sieve method:", sieve_count, "primes (", sprintf("%.4f", sieve_time), "seconds )\n")
+cat("Trial division:", trial_count, "primes (", sprintf("%.4f", trial_time), "seconds )\n")
+cat("Speedup:", sprintf("%.2f", trial_time / sieve_time), "x\n\n")
+
+# Test segmented sieve
+cat("=== Segmented Sieve Example ===\n")
+cat("Primes between 50 and 100:\n")
+range_primes <- segmented_sieve(50, 100)
+cat(paste(range_primes, collapse = ", "), "\n")
+cat("Count:", length(range_primes), "\n\n")
+
+# Edge cases
+cat("=== Edge Cases ===\n")
+cat("Primes up to 1:", paste(sieve_of_eratosthenes(1), collapse = ", "), "\n")
+cat("Primes up to 2:", paste(sieve_of_eratosthenes(2), collapse = ", "), "\n")
+cat("Primes up to 3:", paste(sieve_of_eratosthenes(3), collapse = ", "), "\n")
+
+# Large example (uncomment for larger tests)
+# cat("\n=== Large Scale Test ===\n")
+# large_n <- 10000
+# start_time <- Sys.time()
+# large_primes <- sieve_of_eratosthenes(large_n)
+# end_time <- Sys.time()
+# cat("Found", length(large_primes), "primes up to", large_n, "\n")
+# cat("Computation time:", as.numeric(end_time - start_time, units = "secs"), "seconds\n")
\ No newline at end of file