A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 3.14 Projects 95a summation for the running time of your algorithm on n pieces, and thenderive a closed-form solution for the summation.3.22 Can the average case cost for an algorithm be worse than the worst case cost?Can it be better than the best case cost? Explain why or why not.3.23 Prove that if an algorithm is Θ(f(n)) in the average case, then it is Ω(f(n))in the worst case.3.24 Prove that if an algorithm is Θ(f(n)) in the average case, then it is O(f(n))in the best case.3.14 Projects3.1 Imagine that you are trying to store 32 boolean values, and must access themfrequently. Compare the time required to access Boolean values stored alternativelyas a single bit field, a character, a short integer, or a long integer.There are two things to be careful of when writing your program. First, besure that your program does enough variable accesses to make meaningfulmeasurements. A single access is much smaller than the measurement ratefor all four methods. Second, be sure that your program spends as much timeas possible doing variable accesses rather than other things such as callingtiming functions or incrementing for loop counters.3.2 Implement sequential search and binary search algorithms on your computer.Run timings for each algorithm on arrays of size n = 10 i for i ranging from1 to as large a value as your computer’s memory and compiler will allow. Forboth algorithms, store the values 0 through n − 1 in order in the array, anduse a variety of random search values in the range 0 to n − 1 on each sizen. Graph the resulting times. When is sequential search faster than binarysearch for a sorted array?3.3 Implement a program that runs and gives timings for the two Fibonacci sequencefunctions provided in Exercise 2.11. Graph the resulting runningtimes for as many values of n as your computer can handle.