A Practical Introduction to Data Structures and Algorithm Analysis

564 Chap. 17 Limits to ComputationProblem A:ITransform 1I’Problem BSLN’Transform 2SLNFigure 17.2 The general process for reduction shown as a “blackbox” diagram.Asymptotically faster (but more complicated) algorithms are known, but none is sofast as to be in O(n).Next we ask the question: Is squaring an n-digit number as difficult as multiplyingtwo n-digit numbers? We might hope that something about this special casewill allow for a faster algorithm than is required by the more general multiplicationproblem. However, a simple reduction proof serves to show that squaring is “ashard” as multiplying.The key to the reduction is the following formula:X × Y = (X + Y )2 − (X − Y ) 2.4The significance of this formula is that it allows us to convert an arbitrary instanceof multiplication to a series of operations involving three addition/subtractions(each of which can be done in linear time), two squarings, and a division by 4.Note that the division by 4 can be done in linear time (simply convert to binary,shift right by two digits, and convert back).This reduction shows that if a linear time algorithm for squaring can be found,it can be used to construct a linear time algorithm for multiplication.