How does Karatsuba algorithm work?

The point of the Karatsuba algorithm is to break large numbers down into smaller numbers so that any multiplications that occur happen on smaller numbers. Karatsuba can be used to multiply numbers in all base systems (base-10, base-2, etc.).

What is the time complexity of Karatsuba?

Time Complexity: Time complexity of the above solution is O(nlog23) = O(n1.59). Time complexity of multiplication can be further improved using another Divide and Conquer algorithm, fast Fourier transform.

What is Karatsuba integer multiplication?

The Karatsuba algorithm is a fast multiplication algorithm that uses a divide and conquer approach to multiply two numbers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. Using this algorithm, multiplication of two n-digit numbers is reduced from O(N^2) to O(N^(log 3) that is O(N^1.585).

What kind of algorithm is the Karatsuba algorithm?

The Karatsuba algorithm is a fast multiplication algorithm that uses a divide and conquer approach to multiply two numbers.

What can Karatsuba be used for in math?

Karatsuba can be used to multiply numbers in all base systems (base-10, base-2, etc.). The naive way to multiple numbers is commonly taught in elementary school.

Which is the most efficient step of Karatsuba?

Karatsuba’s basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n /2, rounded up. In particular, if n is 2 k, for some integer k, and the recursion stops only when n is 1, then the number of single-digit multiplications is 3 k, which is nc where c = log 2 3.

How many multiplications can Karatsuba be computed in?

Karatsuba observed that can be computed in only three multiplications, at the cost of a few extra additions. With and as before we can calculate A more efficient implementation of Karatsuba multiplication can be set as , where .