Do not be intimidated by the number of exercises in this homework. Most of them require fairly short answers and can be performed quite quickly.

1. Suppose I want to create a searching application which supports the following three operations:
   • insert a record into the search structure
   • find a record in the search structure
   • delete a record from the search structure

   Further suppose that:

   • the data is inserted almost in ascending order (i.e., the data is not completely ordered, but is close to being ordered).
   • the number of insertions and deletions roughly equals the number of finds.
   • ordered traversals and find min/find max operations are never required.

   Which data structure, a linked list, unbalanced tree, balanced tree, or hash table, would you choose to implement this search application? Justify your answer.

2. Suppose I have a data set with an estimated size of 5000 elements and I want to insert the elements into a hash table. What tablesize should I choose if I want the tablesize to be:
   1. a prime number
   2. one less than the first greater power of 2

3. Show the hash table that results if the integers 22, 28, 17, 21, 32 are inserted into the following hash table using:
   • Linear probing, and
   • the hash function h(k) = k % 11

   0:
   1:
   2:
   3:
   4:
   5:
   6:
   7:
   8:
   9:
   10:
   

4. Show the hash table that results if the integers 22, 28, 17, 21, 32 are inserted into the following hash table using:
   • separate chaining, and
   • the hash function h(k) = k % 11
   You should append integers to the end of your lists and you may show your lists as comma separated lists of integers.

   0:
   1:
   2:
   3:
   4:
   5:
   6:
   7:
   8:
   9:
   10:
   

5. Exercise 5.1 in the Weiss text. Use the above representation for your aways and represent linked lists as comma-separated lists of integers. Only do separate chaining, linear probing, and quadratic probing.

6. Weiss 6.2a

7. Weiss 6.3

Bit Manipulation

The rest of this homework assignment asks you to write a number of functions involving bit operations. You do not have to compile or execute the functions and you may assume that the parameters are error-free. All of the function bodies should requires only a few lines of code. Three of them can be done with a single line of code.

1. Write a function that takes a non-negative integer, n, as a parameter and that returns 2ⁿ as the result. Your function should consist of a single line of code and hence will need to make use of the bit operations you learned about in class.

2. In class I indicated that one common way to choose a table size for a hash table is to find an integer which is the smallest power of two greater than the estimated number of elements in the data set and then to subtract one from it. In other words, if n is the estimated number of elements in the data set, determine the smallest x such that n < 2^x and then use 2^x-1 as your table size. For example, if n is 11, then x is 4 and the table size should be 2⁴-1 = 15.

   Write a function that takes a non-negative integer, n, as a parameter and returns 2^x-1 as the result. if n is 11, your function should return 15. If it is 32 your function should return 63. Although you can solve this problem by writing a for loop that repeatedly divides n by 2, I want you to solve the problem using bit shifting. It is legitimate to divide a number by 2 by bit-shifting it. Hint: Think of n as a string of bits. For example, 13 can be represented as 1101 using bits. If bit positions are numbered from right to left with the first bit position being numbered 1, then x is equal to the bit position of the last 1 in the string. In the above example the last 1 will be in bit position 4.

3. Write a function that takes an unsigned integer, n, and two non-negative integers, left, and right as parameters. Your function should pretend that n is a bit string and print the bits between and including left and right. You should assume that bit positions are numbered from 31 to 0. For example, if n can be represented as 10110111011000011101011011010010 and left is 29 and right is 23, your function should print "1101110":

   	         10110111011000011101011011010010
   	position:  2     2
   		   9     3
           

   Hint: Use bit-shifting. In addition to bit-shifting, you may need to use bitwise-and (see the lecture notes on bits), a stack, or recursion in order to print the bits in the correct order. Your solution might use none of these three techniques but I thought of three separate solutions, each of which uses one of these techniques.