CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Discussion Notes #4

from Wednesday 1 Oct 2003

Multiplying Large Integers (Levitin 4.5)

Please answer the questions during the discussion period.

Java provides a BigInteger class for storing integers too long for an int or long. Here we will look at the complexity of addition and multiplication of BigInteger objects, and determine whether the unusual multiplication algorithm of Levitin section 4.5 is useful in this context.

We'll assume that a (non-negative) BigInteger is written in the form "the sum for i from 0 to n of b_irⁱ, where r is a base such as MAXINT+1. Each b_i is kept in an int variable, and these int variables are kept in some sort of list. We can get b_i from b, for example, in O(1) time by the call b.getTerm(i), and change b_i to x in O(1) time by b.setTerm(i,x).

• Question 1: Here are my sketched versions of two methods from the BigInteger class:

  
          BigInteger add (BigInteger c) 
          {// returns sum of this and c
             int n = max(this.size, c.size);
             BigInteger result = new BigInteger(n+1);
             int carryBit = 0;
             for (int i=0; i < n; i++) {
                result.setTerm(i, this.getTerm(i) + c.getTerm(i) + carryBit);
                carryBit = addCarry(this.getTerm(i), c.getTerm(i), carryBit);}
             result.setTerm(n, carryBit);
             return result;}
  
          BigInteger multiply (int x)
          {// returns this times x
             int n = this.size;
             BigInteger result = new BigInteger(n+1);
             int carryInt = 0;
             for (int i=0; i < n; i++) {
                result.setTerm(i, this.getTerm(i) * x + carryInt);
                carryInt = multCarry(this.getTerm(i), x, carryInt);}
             result.setTerm(n,carryInt);
             return result;}
  

  Argue that each of these methods takes $O(n)$ time.

• Question 2: Write Java pseudocode using the add and multiply methods above to implement BigInteger multiply (BigInteger c) using the ordinary algorithm for multiplication. Analyze the running time.

• Question 3: Write Java pseudocode to reimplement multiply utilizing the trick from section 4.5, where it is observed that if a = a₁r^n/2 + a₀ and b = b₁r^n/2 + b₀, then ab = (a₁b₁)rⁿ + ((a₁ + a₀)(b₁ + b₀) - a₁b₁ - a₀b₀)r^n/2 + (a₀b₀).

  Assume you also have a method shift in the BigInteger class, taking an int argument, such that b.shift(i) is b times rⁱ if i is positive and b divided by rⁱ (with no remainder) if i is negative. There is also a method tail such that b.tail(i) gives the remainder when b is divided by rⁱ. Each of these methods takes Θ(i) time.

  Analyze the running time of your method.

Last modified 2 October 2003