Chapter 13 Review Exercise Solutions
R13.1
Recursion
Recursion solves a problem by using the solution of the same problem with simpler values.
Iteration
Iteration uses some form of a loop (for loop, while loop) to solve a problem.
Infinite recursion
Infinite recursion occurs when the recursion never stops. A method calls itself over and over again and never reaches and end.
Recursive helper method
A recursive helper method is a recursive method that is called by another, usually non-recursive, method.
R13.2
If the array only has one element, return that element.
Otherwise, recursively find the smallest element of the array that is obtained by removing the last element. Then return the smaller of the last element and the value returned by that recursive call.
R13.3
If the array has no elements or only one element, you are done.
Otherwise, first find the smallest element in the array. After obtaining the smallest number, swap that number with the first element in the array and recursively sort the remaining numbers in the array.
R13.4
If n is 1, return a collection containing the empty set and the set { 1 }.
Otherwise, first find all subsets of the set { 1 ... n - 1 }. For each subset S in that collection, add two sets to the answer, namely S and S ∪ {n}.
R13.5
We can prove that the algorithm generates all permutations of a sequence by induction.
We assume that the algorithm works correctly for n = k
Now, we prove it works correctly for the case n = k + 1: 
sequence: (0, 1, . . ., k)
The algorithm produces: 
( 0 ) | (the permutations of (1, 2, . . ., k)) 
( 1 ) | (the permutations of (0, 2, . . ., k)) 
. . . 
( k ) | (the permutations of (0, 1, . . ., k - 1))
which are all the permutations of the sequence, and is correct because the permutations of the reduced sequences are assumed to be right since n = k - 1 is assumed to work fine.
Now, we prove that the base case works correctly: n = 1 
sequence: ( 0 ) 
The algorithm produces: 
( 0 ) | () which is equal to ( 0 ), which are all the permutations of the sequence. 
Q.E.D.
R13.6
x0 = 1
xn = x * xn-1
R13.7
The approach is significantly faster because it reduces the calls to "pow" to as little as 2 + log2 n, when n is a power of 2. 
For example:
Using the first method: 
x1023 took 1024 calls
x1024 took 1025 calls
Using the second method: 
x1023 took 20 calls
x1024 took 12 calls
R13.8
0! = 1
n! = n * (n-1)!
R13.9
When computing fib(n), the function is called 2 * fib(n) - 1 times.

import java.util.Scanner;

/**
   This program computes Fibonacci numbers using a recursive
   method.
*/
public class FibTester
{
   private static int fibCount;

   public static void main(String[] args)
   {
      Scanner in = new Scanner(System.in);
      System.out.print("Enter n: ");
      int n = in.nextInt();

      for (int i = 1; i <= n; i++)
      {
         fibCount = 0;
         long f = fib(i);
         System.out.println("fib(" + i + ") = " + f);
         System.out.println("number of times fib(n) is called = " + fibCount);
      }
   }

   /**
      Computes a Fibonacci number.
      @param n an integer
      @return the nth Fibonacci number
   */
   public static long fib(int n)
   {
      fibCount++;
      if (n <= 2) return 1;
      else return fib(n - 1) + fib(n - 2);
   }
}
R13.10

To guess this formula, compute

The formula can then be verified by induction.