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The general solution to the Towers of Hanoi game is naturally recursive: Part I: Move the smaller n 1 disks from peg A to peg B Part II: Move the remaining disk from peg A to peg C Part III: Move the smaller n 1 disks from peg B to peg C The first and third steps are recursive: Apply the complete solution to n 1 disks The basis to this recursive solution is the case where n = 0 In that case, do nothing The solution for the case of n = 1 disk is: 1 Move the disk from peg A to peg C The solution for the case of n = 2 disks is: 1 Move the top disk from peg A to peg B 2 Move the second disk from peg A to peg C 3 Move the top disk from peg B to peg C.

Q(z) =

+ H ( z ) . Kz-2

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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

The solution for the case of n = 3 disks is: 1 Move the top disk from peg A to peg C 2 Move the second disk from peg A to peg B 3 Move the top disk from peg C to peg B 4 Move the remaining disk from peg A to peg C 5 Move the top disk from peg B to peg A 6 Move the second disk from peg B to peg C 7 Move the top disk from peg A to peg C Here, steps 1 3 constitute Part I of the general solution, step 4 constitutes Part II, and steps 5 7 constitute Part III Since the general recursive solution requires the substitution of different peg labels, it is better to use variables.

H(z)

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 input to a causal linear shift-invariant system is x(n) = u-17 - I ) The z-transform of the output of this system is

Then, naming this three-step algorithm hanoi(n, x, y, z), it becomes: Part I: Move the smaller n 1 disks from peg x to peg z Part II: Move the remaining disk from peg x to peg y Part III: Move the smaller n 1 disks from peg z to peg y The general solution is implemented in Example 915 EXAMPLE 915 The Towers of Hanoi.

+( ~ U ( I , )

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