![]() By utilizing the divide-and-conquer approach, the algorithm reduces the number of recursive multiplications required and achieves a faster runtime compared to traditional long multiplication. The Karatsuba algorithm provides an efficient solution to multiply large numbers without relying on the BigInteger class. Now, let’s see how we can implement the Karatsuba algorithm in Python without using the built-in BigInteger class:ĭef karatsuba ( x, y ): # Base case: if either x or y is a single-digit number if x < 10 or y < 10 : return x * y # Calculate the number of digits in x and y n = max ( len ( str ( x )), len ( str ( y ))) n_half = n // 2 # Split x and y into halves a, b = divmod ( x, 10 ** n_half ) c, d = divmod ( y, 10 ** n_half ) # Recursive steps ac = karatsuba ( a, c ) bd = karatsuba ( b, d ) ad_plus_bc = karatsuba ( a + b, c + d ) - ac - bd # Compute the final product return ac * 10 ** ( 2 * n_half ) + ad_plus_bc * 10 ** n_half + bd Conclusion Implementing the Karatsuba Algorithm in Python The key observation is that the number of recursive multiplications required is reduced from four to three, which results in a significant improvement in performance when dealing with large numbers. Compute the final product using the formula: result = z2 * 10^(n/2) + z1 * 10^(n/2) + z0.Compute three intermediate multiplications:.Split the two numbers x and y into two halves, x1 and x0, and y1 and y0, respectively.The Karatsuba algorithm can be summarized in the following steps: Let’s say we want to multiply two large numbers, x and y, with n digits each. This algorithm has a time complexity of O(n^log2(3)) which is faster than the traditional long multiplication algorithm, which has a time complexity of O(n^2). The key idea behind the Karatsuba algorithm is to split the given numbers into smaller parts, perform intermediate multiplications, and combine the results to obtain the final product. It is a divide-and-conquer algorithm that reduces the number of recursive multiplications required to compute the product of two large numbers. ![]() The Karatsuba algorithm is a fast multiplication algorithm that was discovered by Anatolii Alexeevitch Karatsuba in 1960. This is where the Karatsuba algorithm comes to the rescue. Traditional multiplication algorithms like long multiplication can be time-consuming and inefficient when dealing with numbers that have a large number of digits. ![]() One such problem is multiplying large numbers. | Miscellaneous Karatsuba Algorithm without BigInteger UsageĪs a data scientist or software engineer, you are constantly faced with challenges that require efficient algorithms to solve complex mathematical problems.
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