Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

This definition leads immediately to a recursive algorithm that uses O(n) method
calls to compute p(x,n). We can compute the power function much faster than this,
however, by using the following alternative definition, also based on linear
recursion, which employs a squaring technique:
To illustrate how this definition works, consider the following examples:
2 4 = 2 (4/2)2 = (2 4/2 ) 2 = (2 2 ) 2 = 4 2 = 16
2 5 = 2 1 + (4/2)2 = 2(2 4/2 ) 2 = 2(2 2 ) 2 = 2(4 2 ) = 32
2 6 = 2 (6/2)2 = (2 6/2 ) 2 = (2 3 ) 2 = 8 2 = 64
2 7 = 2 1 + (6/2)2 = 2(2 6/2 ) 2 = 2(2 3 ) 2 = 2(8 2 ) = 128.
This definition suggests the algorithm of Code Fragment 4.3.
Code Fragment 4.3: Computing the power function
using linear recursion.
To analyze the running time of the algorithm, we observe that each recursive call of
method Power(x, n) divides the exponent, n, by two. Thus, there are O(logn)
recursive calls, not O(n). That is, by using linear recursion and the squaring
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