Algorithms by R. Sedgewick

Algorithms by R. Sedgewick

By R. Sedgewick

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A simple method which avoids recomputation and uses no extra space is known as Homer’s rule: by alternat:ing the multiplication and addition operations appropriately, a degree-N polynomial can be evaluated using only 45 CHAPTER 4 46 N - 1 multiplications and N additions. The parenthesization P(X) = x(x(x(x + 3) - 6) + 2) + 1 makes the order of computation obvious: Y:=PN; for i:=N-I downto 0 do y:=x*y+p[i]; This program (and the others in this section) assume the array representation for polynomials that we discussed in Chapter 2.

Also, we will see examples of algorithms which gain efficiency by using random numbers to do sampling or to aid in decision making. Linear Congruential Method The most well-known method for generating random numbers, which has been used almost exclusively since it was introduced by D. Lehmer in 1951, is the so-called linear congruential method. If a [I] contains some arbitrary number, then the following statement fills up an array with N random numbers using this method: for i:=2 to N do a[i]:=(a[i-l]*b $1) mod m That is, to get a new random number, take the previous one, multiply it by a constant b, add 1 and take the remainder when divided by a second constant m.

A few of them are outlined here. One obvious application is in cryptography, where the major goal is to encode a message so that it can’t be read by anyone but the intended recipient. As we will see in Chapter 23, one way to do this is to make the message look random using a pseudo-random sequence to encode the message, in such a way that the recipient can use the same pseudorandom sequence to decode it. Another area in which random numbers have been widely used is in simulation. A typical simulation involves a large program which models some aspect of the real world: random numbers are natural for the input to such programs.

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