24 November 2015Birkbeck College, U. London1 Introduction to Computer Systems Lecturer: Steve Maybank Department of Computer Science and Information Systems Autumn 2015 Week 9a: Revision Exercises and Sorting

17 November 2015Birkbeck College2 Example 1: Qu. 11 from the 2003 Exam Write an algorithm for calculation of tax for a yearly salary. The tax is zero when the salary is £3000 or less. It is 10% for the next salary band from £3001 to £8000. It is 20% for the part from £8001 to £20000, and it is 40% for any part of the salary over £20000.

Example 2 Design an algorithm for finding all the factors of a positive integer. For example, in the case of the integer 12, your algorithm should report the values 1, 2, 3, 4, 6 and November 2015Brookshear Ch 5 review problems 113

Example 3 The Euclidean algorithm finds the greatest common divisor of two positive integers x and y by the following process: As long as x and y are both not zero divide the larger number by the smaller. Replace the larger value with the remainder. When x or y has the value 0 the value of the other variable is the GCD. Produce a pseudo code version of the algorithm 17 November 2015Brookshear Section 5.24

Example 4 Write a pseudo code algorithm to carry out the following task: input: a 1-dimensional array A of integers output: the integer -1 if A[i] ≥ A[i+1] for 0 ≤ i ≤ Length[A]-2, otherwise the least integer j such that A[j] < A[j+1]. 17 November 2015Birkbeck College5

Example 5 Design an algorithm that tests two one dimensional arrays of the same length to see if the two arrays contain the same entries in the same order. 17 November 2017Birkbeck College6

17 November 2015Birkbeck College7 Example 6 Design an algorithm that lists all possible rearrangements of the symbols in a string of three distinct characters.

17 November 2015Birkbeck College8 Example 7 Design an algorithm that when given an arrangement of the digits 0,1,2,3,4,5,6,7,8,9 rearranges the digits so that the new arrangement represents the next larger value that can be represented by these digits (or reports that no such arrangement exists if no rearrangement produces a larger value). Thus would produce

Sorted and Unsorted Lists In a sorted list it is known that the elements are in a predefined order, eg. {-1, 4, 26, 30, 31} In an unsorted list it is not known if the elements are in a predefined order, eg. {3, 0, -6, 2, 1} and {-1, 4, 26, 30, 31} Sorting is an important task. See structures 24 November 2015Brookshear Section 5.49

24 November 2015Brookshear Section Sketch 1 of Insertion Sort Task: sort a list L. Suppose the section L[0], …, L[N] is sorted Sort the section L[0],…L[N], L[N+1] by inserting L[N+1] into the correct place in L[0], … L[N] Set N = N+1 and keep going until the whole list is sorted

24 November 2015Brookshear, Section Sketch 2 of Insertion Sort  The symbol || means list concatenation.  Assume L=L1 || {e} || L2 with L1 sorted (e is the pivot entry)  Sort L1 || {e} to give the list L3  Form the new list L3 || L2  Notice L3 is i) sorted, and ii) strictly larger than L1

24 November 2015Brookshear, Section Example of Insertion Sort  At an intermediate stage, L = {2, 6, 8, 4, 5, 7} L1 = {2, 6, 8} e = 4 L2 = {5, 7}  The position L[3] is the pivot, 4 is the pivot entry.  The pivot entry is stored in the variable e, thus the pivot L[3] can be overwritten without losing information

Example Continued L = {2, 6, 8,, 5, 7} The position is called the hole. 4 < 8 therefore move 8 one place right to obtain L = {2, 6,, 8, 5, 7} 4 < 6 therefore move 6 one place right to obtain L = {2,, 6, 8, 5, 7} 4 > 2 therefore copy 4 into the hole to obtain L = {2, 4, 6, 8, 5, 7} 24 November 2015Birkbeck College, U. London13

24 November 2015Brookshear, Section Insertion Sort def insertionSort(L): N = 1 while (N < Length[L]): e = L[N] hole = N while (there is an entry immediately preceding hole and that entry is greater than e): copy the entry to L[hole] hole = hole-1 L[hole] = e N = N+1 return L

24 November 2015Brookshear, Section Sketch of Bubble Sort Task: sort a list L If a pair L[N], L[N+1] are in the wrong order then interchange them Problem: how to choose the pairs such that the algorithm terminates with a sorted list? Answer: make sure low entries move quickly towards the beginning of L

Example of Bubble Sort L = {6, 2, 8, 4, 5, 7} L[4] < L[5]: no change L[3] < L[4]: no change L[2] > L[3]: interchange L[2] and L[3] to obtain L = {6, 2, 4, 8, 5, 7} L[1] < L[2]: no change L[0] > L[1]: interchange L[0] and L[1] to obtain L = {2, 6, 4, 8, 5, 7} Note: the least entry, 2, is in the correct place. 24 November 2015Birkbeck College, U. London16

24 November 2015Brookshear, Section Bubble Sort 1 def bubbleSort(L): N = 0 while (N < Length(L)-1): bubble(section of L from index N to index Length(L)-1) N = N+1 return L

24 November 2015Brookshear, Section Bubble Sort 2 def bubble(L): i = Length(L)-2 while(i >= 0): if(L[i] > L[i+1]): interchange L[i] and L[i+1] i = i-1 return L

24 November 2015compare Brookshear, Section Algorithm Efficiency for Searching  Searching a list of length n Unsorted list: average number of comparisons=n/2. Sorted list: average number of comparisons is of order Log 2 (n).  If n is large then Log 2 (n) is much less than n/2. E.g. Log 2 (1,030,576)=20.

24 November 2015Brookshear, Section Search Example: solving the eight-puzzle An unsolved eight puzzle A solved eight puzzle Number of configurations = 9x8x…x1 = 362,880

24 November 2015Brookshear, Section Possible Moves

24 November 2015Brookshear, Section Search Strategy Build the tree of all possible moves, breadth first. Stop when the solved eight puzzle is found. Assumption: a solution exists. Problem: low efficiency (why?)