Copyright © 2013 by John Wiley & Sons. All rights reserved. SORTING AND SEARCHING CHAPTER Slides by Rick Giles 14.

Chapter Goals  To study several sorting and searching algorithms  To appreciate that algorithms for the same task can differ widely in performance  To understand the big-oh notation  To estimate and compare the performance of algorithms  To write code to measure the running time of a program Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 2

Contents  Selection Sort  Profiling the Selection Sort Algorithm  Analyzing the Performance of the Selection Sort Algorithm  Merge Sort  Analyzing the Merge Sort Algorithm  Searching  Problem Solving: Estimating the Running Time of an Algorithm  Sorting and Searching in the Java Library Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 3

14.1 Selection Sort  Sorts an array by repeatedly finding the smallest element of the unsorted tail region and moving it to the front  Example: sorting an array of integers Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 4 11917512

Sorting an Array of Integers Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 5  Find the smallest and swap it with the first element  Find the next smallest. It is already in the correct place  Find the next smallest and swap it with first element of unsorted portion  Repeat  When the unsorted portion is of length 1, we are done 59171112 59171112 59111712 59111217 59111217

14.2 Profiling the Selection Sort Algorithm  Want to measure the time the algorithm takes to execute  Exclude the time the program takes to load  Exclude output time  Create a StopWatch class to measure execution time of an algorithm  It can start, stop and give elapsed time  Use System.currentTimeMillis method  Create a StopWatch object  Start the stopwatch just before the sort  Stop the stopwatch just after the sort  Read the elapsed time Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 12

Selection Sort on Various Array Sizes* Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 18 nMilliseconds 10,000786 20,0002,148 30,0004,796 40,0009,192 50,00013,321 60,00019,299 * Obtained with a 2GHz Pentium processor, Java 6, Linux

14.3 Analyzing the Performance of the Selection Sort Algorithm (1)  In an array of size n, count how many times an array element is visited  To find the smallest, visit n elements + 2 visits for the swap  To find the next smallest, visit (n - 1) elements + 2 visits for the swap  The last term is 2 elements visited to find the smallest + 2 visits for the swap Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 20

14.3 Analyzing the Performance of the Selection Sort Algorithm (2)  The number of visits:  n + 2 + (n - 1) + 2 + (n - 2) + 2 +...+ 2 + 2  This can be simplified to n 2 /2 + 5n/2 - 3  5n/2 - 3 is small compared to n 2 /2 — so ignore it  Also ignore the 1/2 — it cancels out when comparing ratios Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 21

14.3 Analyzing the Performance of the Selection Sort Algorithm (3)  The number of visits is of the order n 2  Using big-Oh notation: The number of visits is O(n 2 )  Multiplying the number of elements in an array by 2 multiplies the processing time by 4  Big-Oh notation “f(n) = O(g(n))” expresses that f grows no faster than g  To convert to big-Oh notation: Locate fastest-growing term, and ignore constant coefficient Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 22

14.4 Merge Sort  Sorts an array by  Cutting the array in half  Recursively sorting each half  Merging the sorted halves  Much more efficient than selection sort Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 23

14.5 Analyzing the Merge Sort Algorithm (1) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 31 nMerge Sort (milliseconds)Selection Sort (milliseconds) 10,00040786 20,000732,148 30,0001344,796 40,0001709,192 50,00019213,321 60,00020519,299

Analyzing the Merge Sort Algorithm (3) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 33  In an array of size n, count how many times an array element is visited  Assume n is a power of 2: n = 2 m  Calculate the number of visits to create the two sub-arrays and then merge the two sorted arrays  3 visits to merge each element or 3n visits  2n visits to create the two sub-arrays  total of 5n visits

Analyzing the Merge Sort Algorithm (4) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 34  Let T(n) denote the number of visits to sort an array of n elements then  T(n) = T(n/2) + T(n/2) + 5n or  T(n) = 2T(n/2) + 5n  The visits for an array of size n/2 is:  T(n/2) = 2T(n/4) + 5n/2  So T(n) = 2 × 2T(n/4) +5n + 5n  The visits for an array of size n/4 is:  T(n/4) = 2T(n/8) + 5n/4  So T(n) = 2 × 2 × 2T(n/8) + 5n + 5n + 5n

Analyzing the Merge Sort Algorithm (5) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 35  Repeating the process k times:  T(n) = 2 k T(n/2 k ) +5nk  Since n = 2 m, when k=m: T(n) = 2 m T(n/2 m ) +5nm  T(n) = nT(1) +5nm  T(n) = n + 5nlog 2 (n)

Analyzing the Merge Sort Algorithm (6) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 36  To establish growth order  Drop the lower-order term n  Drop the constant factor 5  Drop the base of the logarithm since all logarithms are related by a constant factor  We are left with n log(n)  Using big-Oh notation: Number of visits is O(nlog(n))

Merge Sort vs Selection Sort Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 37  Selection sort is an O(n 2 ) algorithm  Merge sort is an O(nlog(n)) algorithm  The nlog(n) function grows much more slowly than n 2

14.6 Searching  Linear search: Examines all values in an array until it finds a match or reaches the end  Also called sequential search  Number of visits for a linear search of an array of n elements:  The average search visits n/2 elements  The maximum visits is n  A linear search locates a value in an array in O(n) steps Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 38

Binary Search (1)  Locates a value in a sorted array by  Determining whether the value occurs in the first or second half  Then repeating the search in one of the halves Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 42

Analyzing Binary Search (1)  Count the number of visits to search a sorted array of size n  We visit one element (the middle element) then search either the left or right subarray  Thus: T(n) = T(n/2) + 1  Using the same equation, T(n/2) = T(n/4) + 1  Substituting into the original equation: T(n) = T(n/4) + 2  This generalizes to: T(n) = T(n/2 k ) + k Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 46

Analyzing Binary Search (2)  Assume n is a power of 2, n = 2m, where m = log 2 (n)  Then: T(n) = 1 + log 2 (n)  Binary search is an O(log(n)) algorithm Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 47

14.7 Problem Solving: Estimating the Running Time of an Algorithm  Differentiating between O(nlog(n)) and O(n 2 ) has great practical implications  Being able to estimate running times of other algorithms is an important skill  Will practice estimating the running times of array algorithms Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 48

Linear Time (1)  An algorithm to count how many elements of an array have a particular value: int count = 0; for (int i = 0; i < a.length; i++) { if (a[i] == value) { count++; } } Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 49

Linear Time (2)  To visualize the pattern of array element visits:  Imagine the array as a sequence of light bulbs  As the i th element gets visited, imagine the i th light bulb lighting up  When each visit involves a fixed number of actions, the running time is n times a constant, or O(n) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 50

Linear Time (3)  When you don’t always run to the end of the array; e.g.: boolean found = false; for (int i = 0; !found && i < a.length; i++) { if (a[i] == value) { found = true; } }  Loop can stop in middle:  Still O(n), because in some cases the match may be at the very end of the array Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 51

Quadratic Time (1)  What if we do a lot of work with each visit?  Example: find the most frequent element in an array  Obvious for small array  For larger array?  Put counts in a second array of same length  Take the maximum of the counts, 3, look up where the 3 occurs in the counts, and find the corresponding value, 7 Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 52

Quadratic Time (2)  First estimate how long it takes to compute the counts for (int i = 0; i < a.length; i++) { counts[i] = Count how often a[i] occurs in a }  Still visit each element once, but work per visit is much larger  Each counting action is O(n)  When we do O(n) work in each step, total running time is O(n 2 ) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 53

Quadratic Time (3)  Algorithm has three phases: 1.Compute all counts 2.Compute the maximum 3.Find the maximum in the counts  First phase is O(n 2 ), second and third are each O(n)  The big-Oh running time for doing several steps in a row is the largest of the big-Oh times for each step  Thus algorithm for finding the most frequent element is O(n 2 ) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 54

Logarithmic Time (1)  An algorithm that cuts the size of work in half in each step runs in O(log(n)) time  E.g. binary search, merge sort  To improve our algorithm for finding the most frequent element, first sort the array:  Sorting costs O(n log(n)) time  If we can complete the algorithm in O(n) time, have a better algorithm that the pervious O(n 2 ) algorithm Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 55

Logarithmic Time (2)  While traversing the sorted array:  As long as you find a value that is equal to its predecessor, increment a counter  When you find a different value, save the counter and start counting anew Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 56

Logarithmic Time (3) int count = 0; for (int i = 0; i < a.length; i++) { count++; if (i == a.length - 1 || a[i] != a[i + 1]) { counts[i] = count; count = 0; } Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 57

Logarithmic Time (4)  Constant amount of work per iteration, even though it visits two elements:  2n is still O(n)  Entire algorithm is now O(log(n)) Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 58

14.8 Sorting and Searching in the Java Library  When you write Java programs, you don’t have to implement your own sorting algorithms  Arrays and Collections classes provide sorting and searching methods Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 59

Sorting  The Arrays class contains static sort methods  To sort an array of integers: int[] a =... ; Arrays.sort(a);  That sort method uses the Quicksort algorithm (see Special Topic 14.3)  To sort an array of Arrays List use Collections.sort method: ArrayList names =... ; Collections.sort(names);  That sort method uses the merge sort algorithm Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 60

Binary Search  Arrays and Collections classes contains static binarySearch methods  These methods implement the binary search algorithm, with a useful enhancement:  If the value is not found in the array, return -k -1, where k is the position before which the element should be inserted  E.g. int[] a = { 1, 4, 9 }; int v = 7; int pos = Arrays.binarySearch(a, v); // Returns –3; v should be inserted before position 2 Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 61

Comparing Objects (1)  Arrays and ArrayLists classes provide sort and binarySearch methods for objects  These methods cannot know how to compare arbitrary objects  Methods require that the objects belong to a class that implements the Comparable interface with a single method: public interface Comparable { int compareTo(Object otherObject); }  These methods cannot know how to compare arbitrary objects Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 62

Comparing Objects (2)  Call a.compareTo(b) must return a negative number if a should come before b, 0 if a and b are the same, and a positive number otherwise  Several classes in the standard Java Library implement the Comparable interface  E.g. String, Date Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 63

Comparing Objects (3)  You can implement the Comparable interface for your classes as well  E.g. to sort a collection of countries by area, class Country can implement this interface and supply a compareTo method: Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 64

Comparing Objects (3)  You can implement the Comparable interface for your classes as well  E.g. to sort a collection of countries by area, class Country can implement this interface and supply a compareTo method: public class Country implements Comparable { public int compareTo(Object otherObject) { Country other = (Country) otherObject; if (area < other.area) { return -1; } else if (area == other.area) { return 0; } else { return 1; } } Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 65

Summary: Selection Sort Algorithm  The selection sort algorithm sorts an array by repeatedly finding the smallest element of the unsorted tail region and moving it to the front. Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 66

Summary: Measuring the Running Time of an Algorithm  To measure the running time of a method, get the current time immediately before and after the method call. Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 67

Summary: Big-Oh Notation  Computer scientists use the big-Oh notation to describe the growth rate of a function.  Selection sort is an O(n 2 ) algorithm. Doubling the data set means a fourfold increase in processing time.  Insertion sort is an O(n 2 ) algorithm. Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 68

Summary: Contrasting Running Times of Merge Sort and Selection Sort Algorithms  Merge sort is an O(n log(n)) algorithm. The n log(n) function grows much more slowly than n 2. Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 69

Summary: Running Times of Linear Search and Binary Search Algorithms  A linear search examines all values in an array until it finds a match or reaches the end.  A linear search locates a value in an array in O(n) steps.  A binary search locates a value in a sorted array by determining whether the value occurs in the first or second half, then repeating the search in one of the halves.  A binary search locates a value in a sorted array in O(log(n)) steps. Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 70

Summary: Practice Developing Big-Oh Estimates of Algorithms  A loop with n iterations has O(n) running time if each step consists of a fixed number of actions.  A loop with n iterations has O(n 2 ) running time if each step takes O(n) time.  The big-Oh running time for doing several steps in a row is the largest of the big-Oh times for each step.  A loop with n iterations has O(n 2 ) running time if the i th step takes O(i) time.  An algorithm that cuts the size of work in half in each step runs in O(log(n)) time. Copyright © vby John Wiley & Sons. All rights reserved. Page 71

Summary: Use the Java Library Methods for Sorting and Searching Data  The Arrays class implements a sorting method that you should use for your Java programs.  The Collections class contains a sort method that can sort array lists.  The sort methods of the Arrays and Collections classes sort objects of classes that implement the Comparable interface. Copyright © 2013 by John Wiley & Sons. All rights reserved. Page 72

Copyright © 2013 by John Wiley & Sons. All rights reserved. SORTING AND SEARCHING CHAPTER Slides by Rick Giles 14.

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Copyright © 2013 by John Wiley & Sons. All rights reserved. SORTING AND SEARCHING CHAPTER Slides by Rick Giles 14.

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