ANALYSIS OF ALGORITHMS. ANALYSIS OF ALGORITHMS IB DP Computer science Standard Level ICS3U

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Ethel Barber
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1 C A N A D I A N I N T E R N A T I O N A L S C H O O L O F H O N G K O N G When Things Get Out of Hand data cleanup: shuffle-left proceed through the list from left to right pointing with a finger on the left hand to keep the current place, and passing over non-zero values every 0 value, it gets squeezed out of the list by copying each remaining data item in the list one cell to the left data cleanup: shuffle-left must first examine each of the n elements in the list to see whether they are 0 a base of at least Θ(n) work units then copy the 0 values with n number, a base of at least Θ(n 2 ) work units in general, it is an Θ(n 2 ) algorithm 1

2 data cleanup: copy-over proceed through the list from left to right scans the list from left to right copying every legitimate value into a new list that it creates time/space tradeoff: fewer copies than the shuffleleft algorithm requires, but a lot of extra memory space is required data cleanup: copy-over must first examine each of the n elements in the list to see whether they are 0 a base of at least Θ(n) work units then copy the legitimate values with n number, a base of at least Θ(n) work units in general, it is an Θ(n) algorithm data cleanup: converging-pointers proceed in two directions simultaneously scans the list from left to right, reduce the legitimate value by 1, if one encounters a 0 value on the left AND copy the right hand value to the position on the left, the legitimate value is also reduced by 1 on the right 2

3 data cleanup: converging-pointers must first examine each of the n elements in the list to see whether they are 0 a base of at least Θ(n) work units then copy the legitimate values with n number, a base of at least Θ(n) work units in general, it is an Θ(n) algorithm binary search the binary search algorithm, is more efficient but it works only when the search list is already sorted the number of times a number n can be cut in half is the logarithm of n to the base 2 if a list is to be searched only a few times for a few particular names, then it is more efficient to do sequential search on the unsorted list, but if the list is to be searched repeatedly, it is more efficient to sort it and then use binary search pattern matching pattern matching usually involves a pattern length that is short compared to the text length, that is, when m is much less than n the pattern-matching algorithm is therefore Θ(n) in the best case and Θ(mxn) in the worst case. 3

4 other algorithms an Θ(2 n ) algorithm is called an exponential algorithm if no polynomially bounded algorithm exists, such problems are called intractable ; they are solvable, but the solution algorithms all require so much work as to be virtually useless problems for which no known polynomial solution algorithm exists are sometimes approached via approximation algorithms Pg. 129 # 1 Suppose that, using the list of seven numbers in this section, we try binary search to decide whether is in the list. What numbers would be compared with ? Pg. 129 # 2 Suppose that, using the list of seven numbers in this section, we try binary search to decide whether is in the list. What numbers would be compared with ? 4

5 Pg. 132 # 1 Use the first example pattern and text given in Section for the worst case of the patternmatching algorithm. What is m? What is n? What is mxn? This algorithm is Θ(mxn) in the worst case, but what is the exact number of comparisons done? Pg. 140 # 4 We have said that the average number of comparisons needed to find a name in an n-element list using sequential search is slightly higher than n/2. In this problem we find an exact expression for this average. part a part b part c Pg. 140 # 5 Here is a list of seven names: Sherman, Jane, Ted, Elise, Raul, Maki, John Search this list for each name in turn, using sequential search and counting the number of comparisons for each name. Now take the seven comparison counts and find their average. Did you get a number that you expected? Why? 5

6 Pg. 140 # 20 This exercise refers to short sequential search. 1. What is the worst-case number of comparisons of short sequential search on a sorted n-element list? 2. What is the approximate average number of comparisons to find an element that is in a sorted list using short sequential search? 3. Is short sequential search ever more efficient than regular sequential search? Explain. 6

Chapter 3: The Efficiency of Algorithms Invitation to Computer Science,

Chapter 3: The Efficiency of Algorithms Invitation to Computer Science, Objectives In this chapter, you will learn about Attributes of algorithms Measuring efficiency Analysis of algorithms When things