Karatsuba Multiplication in R

mathematics/karatsuba.r

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+# Karatsuba Multiplication Algorithm
+# Multiply two integers using the Karatsuba multiplication algorithm
+# This is a fast multiplication algorithm that uses divide-and-conquer
+# Time Complexity: O(n^log2(3)) ≈ O(n^1.585)
+
+karatsuba <- function(x, y) {
+  # Base case for recursion
+  if (x < 10 || y < 10) {
+    return(x * y)
+  }
+
+  # Determine the number of digits in the largest number
+  n <- max(nchar(as.character(x)), nchar(as.character(y)))
+  half <- floor(n / 2)
+
+  # Split x and y into high and low parts
+  divisor <- 10^half
+  high_x <- floor(x / divisor)
+  low_x <- x %% divisor
+  high_y <- floor(y / divisor)
+  low_y <- y %% divisor
+
+  # Recursive calls
+  z0 <- karatsuba(low_x, low_y)
+  z1 <- karatsuba((low_x + high_x), (low_y + high_y))
+  z2 <- karatsuba(high_x, high_y)
+
+  # Karatsuba formula: (z2 * 10^(2*half)) + ((z1 - z2 - z0) * 10^half) + z0
+  result <- (z2 * 10^(2 * half)) + ((z1 - z2 - z0) * 10^half) + z0
+
+  return(result)
+}
+
+# Example usage
+a <- 12345678
+b <- 87654321
+result <- karatsuba(a, b)
+cat(sprintf("Karatsuba(%d, %d) = %.0f\n", a, b, result))
+
+# Verify with standard multiplication
+standard_result <- a * b
+cat(sprintf("Standard multiplication: %d * %d = %.0f\n", a, b, standard_result))
+cat(sprintf("Results match: %s\n", result == standard_result))
+
+# Additional test cases
+cat("\nAdditional test cases:\n")
+test_cases <- list(
+  c(1234, 5678),
+  c(999, 888),
+  c(12, 34),
+  c(123456, 789012)
+)
+
+for (test in test_cases) {
+  x <- test[1]
+  y <- test[2]
+  karatsuba_result <- karatsuba(x, y)
+  standard_result <- x * y
+  cat(sprintf("karatsuba(%d, %d) = %.0f (Expected: %.0f, Match: %s)\n", 
+              x, y, karatsuba_result, standard_result, 
+              karatsuba_result == standard_result))
+}