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Most of the chapter's code is provided, including a Tic-Tac-Toe program An improved version of the Tic-Tac-Toe algorithm that uses fancier data structures is discussed in 11 The following are the filenames

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RecSumcpp PrintIntcpp

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The routine shown in Figure 81 with a simple

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The routine given in Figure 84 for printing a number in any base, plus a main BinarySearchReccpp Virtually the same as BinarySearchcpp (in 7 ) , but with the binarysearch shown in Figure 81 1 Ruler-java The routine shown in Figure 813, ready to run It contains code that forces the drawing to be slow FractalStarjava The routine given in Figure 815, ready to run It contains code that allows the drawing to be slow The math routines presented in Section 84, the primality testing routine, and a main that illustrates the RSA computations The four maximum contiguous subsequence sum routines The routine shown in Figure 825, with a simple main The Tic-Tac-Toe algorithm, with a primitive

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In Short

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81 What are the four fundamental rules of recursion

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82 Modify the program given in Figure 81 so that zero is returned for negative n Make the minimum number of changes

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Exercises

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Following are four alternatives for line 12 of the routine power (in Figure 816) Why is each alternative wrong

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HugeInt HugeInt HugeInt HugeInt tmp tmp tmp tn rp

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power( x * x, n/2, p

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= power( power( x, 2 , p ) , n/2, p ) ; = power( power( x , n/2, p ) , 2 , p ) ; = power( x, n/2, p ) * power( x, n/2, p )

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Show how the recursive calls are processed in the calculation 263m0d 37 Compute gcd( 1995, 1492) Bob chooses p and q equal to 37 and 41, respectively Determine acceptable values for the remaining parameters in the RSA algorithm Show that the greedy change-making algorithm fails if 5-cent pieces are not part of United States currency

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Prove by induction the formula

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Prove the following identities relating to the Fibonacci numbers a F I + F 2 + + F N= F , + , - 1 b Fl + F, + + F2,- = F2, C Fo + F 2 + - + F,, = F,,, -1 d F N - , F N += ( - l ) N + F i , e F l F z + F 2 F 3 + + F 2 N - l F 2 N= F;, f F , F , + F , F , + + F 2 N F 2 N + 1 = FiN+I1 -

g F i + F i + , =

F2,+,

Show that if A = B(mod N), then for any C, D, and P, the following are true a A + C = B + C ( m o d N ) b AD r BD(mod N) c A P = BP(mod N ) Prove that if A L B, then A mod B < A / 2 (Hint: Consider the cases A B I / 2 and B > A / 2 separately) How does this result show that the running time of gcd is logarithmic

Prove by induction the formula for the number of calls to the recursive function f i b in Section 834 Prove by induction that if A > B 2 0 and the invocation gcd ( a ,b) performs k 2 1 recursive calls, then A 2 Fk and B 2 Fk

Prove by induction that in the extended gcd algorithm, lY <A

11 < B x

Write an alternative gcd algorithm, based on the following observations (arrange that A > B ) a gcd(A, B) = 2 gcd(A / 2, B / 2) if A and B are both even b gcd(A, B) = gcd(A I 2, B)if A is even and B is odd c gcd(A, B) = gcd(A, B / 2) if A is odd and B is even d gcd(A, B) = gcd((A + B) 12, (A - B) / 2) if A and B are both odd Solve the following equation Assume that A 2 I , B > I , and P 2 0

Strassen's algorithm for matrix multiplication multiplies two N x N matrices by performing seven recursive calls to multiply two N / 2 x N / 2 matrices The additional overhead is quadratic What is the running time of Strassen's algorithm