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Therefore, the loop will print (n)(n 1)! distinct strings all having the same left prefix string This proves the conjecture by mathematical induction Now it follows from that conjecture that the call print(str) will print n! distinct permutations of the characters in the string str, where n is its length Since that is precisely the total number of permutations that the string has, it follows that the method is correct 925 For the factorial function implemented in Example 92 on page 165, integer overflow occurs on the return type long with n = 13 on the author s computer So the computable domain for this function is 0 n 12 For the sum(b,n) function implemented in Problem 92 on page 177 with b = 2, floating point overflow occurs on the return type double with n = 1,023 on the author s computer.

So the computable domain for this function is 0 n 1,022 For the Fibonacci function implemented in Example 96 on page 167, the overhead from the recursive calls degrades the run-time performance noticeably after n = 36 on the author s computer So the computable domain for this function is about 0 n 40 For the binomial coefficient function implemented in Example 97 on page 167, the overhead from the recursive calls degrades the run-time performance noticeably after n = 25 on the author s computer So the computable domain for this function is about 0 n 30 The Towers of Hanoi program performs: a 31 moves for 5 disks b 63 moves for 6 disks c 2n 1 moves for n disks To prove that the Towers of Hanoi program performs 2n 1 disk moves for n disks, use mathematical induction.

6.5 6.6 6.7

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The basis is established in Example 915 on page 174 To move n + 1 disks, it takes 2n 1 moves to move all but the last disk to peg B (by the inductive hypothesis) Then it takes 1 move to move the last disk to peg C, and 2n 1 more moves to move the rest of the disks from peg B to peg C on top of that last disk The total is (2n 1) + 1 + (2n 1) = 2n+1 1 For the Ackermann function implemented in Problem 914 on page 178, exceptions are thrown for m = 17 when n = 2, for m = 5 when n = 3, for m = 4 when n = 4, and for m = 3 when n = 5.

So the computable domain for this function is restricted to 0 m 16 when n = 2, to 0 m 4 when n = 3, to 0 m 3 when n = 4, and to 0 m 2 when n = 5 The call tree for Example 915 on page 174 is:.

Why is it that threads are faster to create than processes What advantage do kernel threads provide over user threads Give an example of a user thread package. When a process is multithreaded, which of these resources are shared among the threads, and which are private to each thread Memory Program counter Global variables Open files Register contents Stack/local/automatic variables

These are more accurate recursive implementation of the sine and cosine functions:

public static double s(double x) { if (Math.abs(x) < 0.00005) { return x - x*x*x/6; // basis }

6.8 6.9

CHAP. 9]

return 2*s(x/2)*c(x/2); } public static double c(double x) { if (Math.abs(x) < 0.00005) { return 1.0 - x*x/2; // basis } return 1 - 2*s(x/2)*s(x/2); // recursion } // recursion

List four events that transfer control to the operating system in privileged mode. How can a semaphore be used to insure that Process First will complete before Process Second executes

These are faster converging implementation of the sine and cosine functions:

public static double s(double x) { if (-0.005 < x && x < 0.005) { return x - x*x*x/6 + x*x*x*x*x/120; } return 2*s(x/2)*c(x/2); // recursion } public static double c(double x) { if (-0.005 < x && x < 0.005) { return 1.0 - x*x/2 + x*x*x*x*x/24; } return 1 - 2*s(x/2)*s(x/2); // recursion } // basis

6.10 Describe how transaction logging might apply to a file system, and give an example of a file system that employs transaction logging.

These are mutually recursive implementations of the hyperbolic sine and cosine functions:

public static double s(double x) { if (-0.005 < x && x < 0.005) { return x + x*x*x/6; // basis } return 2*s(x/2)*c(x/2); // recursion } public static double c(double x) { if (-0.005 < x && x < 0.005) { return 1.0 + x*x/2; // basis } return 1 + 2*s(x/2)*s(x/2); // recursion }

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