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Now to prove that T(n) = n 1 for all n > 0, let f(n) = n 1 and apply the (weak) principle of mathematical induction. The basis case is where n = 1. In that case, T(n) = T(1) = 0 and f(n) = f(1) = (1) 1 = 0. For the inductive step, we assume that T(n) = f(n) for some n > 0 and then deduce from that assumption that T(n +1) = f(n +1): T(n +1) = 1 + T(n) = 1 + f(n) = 1 + (n 1) = n f(n +1) = (n +1) 1 = n That completes the proof. Now that we have determined that the complexity function for this recursive implementation of the factorial function T(n) = n 1, we can conclude that this implementation will run in O(n) time. This means that its execution time will be proportional to the size of its argument n. If it takes 3 milliseconds to compute 8!, then it should take about 6 milliseconds to compute 16!. DYNAMIC PROGRAMMING In most cases, recursion is very inefficient because of its frequent function calls. So an iterative implementation may be better if it is not too complex. Another alternative is to implement the recurrence relation by storing previously computed values in an array instead of recomputing them with recursive function calls. This method is called dynamic programming. EXAMPLE 9.14 Dynamic Programming Implementation of the Fibonacci Function

are numerous reasons why the private sector holds money balances. These reasons can be categorized into types of demand, as follows: (1) a transaction demand, since money is needed to purchase goods and services, to pay employees, etc.; (2) a precautionary demand, since money may be held to meet emergency and unforeseen needs that may arise; (3) a portfolio (asset) demand, since some money balances are held in the expectation of opportunities in the nancial markets. When there is a xed demand for money, an increase in the M1 money supply lowers the shortterm nominal rate of interest, ceteris paribus. Example 7.2 L in Figure 7-1 depicts the demand for money. The amount of money demanded is inversely related to the rate of interest since the holder of money forgoes a higher interest return from an alternative nancial asset. When the Fed purchases government securities in the open market, bank reserves increase as does the M1 money supply. Thus, money supply curve S in Figure 7-1 shifts rightward to S as the M1 money supply increases and the short-term rate of interest falls from i0 to i1.

public static int fib(int n) { if (n < 2) { 3 return n; 4 } 5 int[] f = new int[n]; 6 f[0] = 0; 7 f[1] = 1; 8 for (int i=2; i<n; i++) { // store the Fibonacci numbers 9 f[i] = f[i-1] + f[i-2]; 10 } 11 return f[n-1] + f[n-2]; 12 } This implementation uses a dynamic array f[] of n integers to store the first n Fibonacci numbers.

THE TOWERS OF HANOI We have seen important examples of functions that are more naturally defined and more easily understood by using recursion. For some problems, recursion is the only reasonable method of solution. The Towers of Hanoi puzzle is a classic example of a problem whose solution demands recursion. The game consists of a board with three vertical pegs labeled A, B, and C, and a Figure 9.5 The Towers of Hanoi puzzle sequence of n disks with holes in their centers. (See Figure 9.5.) The radii of the disks are in an arithmetic progression (e.g., 5cm, 6cm, 7cm, 8cm, . . .) and are mounted on peg A. The rule is that no disk may be above a smaller disk on the same peg. The objective of the game is to move all the disks from peg A to peg C, one disk at a time, without violating the rule.