Lecture 2 - Problems

Goals

• Be able to analyze the runtime of algorithms that contain nested loops and method calls that don’t run in constant time.

For each of the following algorithms, give its worst-case and best-case asymptotic runtime class.

1. Let \(n\) = side_length and give your answer in terms of \(n\).

   /** Print a multiplication table listing the pairwise products 
     * of all integers in 0..side_length. */ 
   public static void multTable(int side_length) {
     for (int i = 0; i < side_length; i++) {
       for (int j = 0; j < side_length; j++) {
        System.out.print(i*j);
         System.out.print(" ");
       }
       System.out.println();
     }
   }

2. Give your answer in terms of n.

   /** Print n sets of nested parentheses. Example: nest(4) prints (((()))) 
     * Pre: n >= 0. */ 
   public static void nest(int n) {
     for (int i = 0; i < n; i++) {
       System.out.print("(");
     }
     for (int i = 0; i < n; i++) {
       System.out.print(")");
     }
     System.out.println();
   }

3. Let \(n\) = A.length and give your answer in terms of \(n\).

   /** Print the first min(10, A.length) elements of A. */ 
   public static void firstTen(int[] A) {
     int i = 0;
     while (i < 10 && i < A.length) {
       System.out.print(A[i]);
       i += 1;
     }
   }

4. Give your answer in terms of n.

   /** Return the number of times n is evenly divisible by 7  */ 
   public static void sevens(int n) {
     int k = n;
     int count = 0;
     while (k > 1 && k % 7 == 0) {
       k /= 7;
       count++;
     }
     return count;
   }

5. Let \(n\) = height and give your answer in terms of \(n\).

   /** Print an ASCII art picture of a width-6 ladder with the given number of rungs.
     * For example, printLadder(4) prints:
     * |----|
     * |----|
     * |----|
     * |----|
     */ 
   public static void printLadder(int height) {
     for (int i = 0; i < height; i++) {
       System.out.print("|");
       for (int j = 0; j < 4; j++) {
         System.out.print("-");
       }
       System.out.println("|");
     }
   }

6. Assume that this method has access to the linearSearch method we’ve discussed. Let \(m\) = A.length and give your answer in terms of n and \(m\).

   /** Return the number of integers in 0..n that appear in A. */
   public static int numFirstN(int[] A, int n) {
     int count = 0;
     for (int i = 0; i < n; i++) {
       if (linearSearch(A, i) >= 0) {
         count++;
       }
     }
     return count;
   }

7. Let \(n\) = side_length and give your answer in terms of \(n\).

    /** Print a lower-triangular multiplication table listing the products i*j
       * for all pairs i <= j in 0..side_length. */ 
     public static void multTable(int side_length) {
       for (int i = 0; i < side_length; i++) {
         for (int j = 0; j < side_length; j++) {
           if (i <= j) {
             System.out.print(i*j);
             System.out.print(" ");
           }
         }
         System.out.println();
       }
     }

8. Give your answer in terms of n.

   public static void nonsense(int n) {
     int i = n/2;
     int j = i;
     while (i > 0) {
       i /= 2;
     }
     while (j > 0) {
        j -= 2;
     }
   }