A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 7.12 Projects 2757.16 (a) Devise an algorithm to sort three numbers. It should make as few comparisonsas possible. How many comparisons and swaps are requiredin the best, worst, and average cases?(b) Devise an algorithm to sort five numbers. It should make as few comparisonsas possible. How many comparisons and swaps are requiredin the best, worst, and average cases?(c) Devise an algorithm to sort eight numbers. It should make as few comparisonsas possible. How many comparisons and swaps are requiredin the best, worst, and average cases?7.17 Devise an efficient algorithm to sort a set of numbers with values in the range0 to 30,000. There are no duplicates. Keep memory requirements to a minimum.7.18 Which of the following operations are best implemented by first sorting thelist of numbers? For each operation, briefly describe an algorithm to implementit, and state the algorithm’s asymptotic complexity.(a) Find the minimum value.(b) Find the maximum value.(c) Compute the arithmetic mean.(d) Find the median (i.e., the middle value).(e) Find the mode (i.e., the value that appears the most times).7.19 Consider a recursive Mergesort implementation that calls Insertion Sort onsublists smaller than some threshold. If there are n calls to Mergesort, howmany calls will there be to Insertion Sort? Why?7.20 Implement Mergesort for the case where the input is a linked list.7.21 Counting sort (assuming the input key values are integers in the range 0 tom − 1) works by counting the number of records with each key value in thefirst pass, and then uses this information to place the records in order in asecond pass. Write an implementation of counting sort (see the implementationof radix sort for some ideas). What can we say about the relative valuesof m and n for this to be effective? If m < n, what is the running time ofthis algorithm?7.22 Use an argument similar to that given in Section 7.9 to prove that log n is aworst-case lower bound for the problem of searching for a given value in asorted array containing n elements.7.12 Projects7.1 One possible improvement for Bubble Sort would be to add a flag variableand a test that determines if an exchange was made during the current iteration.If no exchange was made, then the list is sorted and so the algorithm