 2003 Prentice Hall, Inc. All rights reserved Sorting Arrays Sorting data –Important computing application –Virtually every organization must sort some data Massive amounts must be sorted Bubble sort (sinking sort) –Several passes through the array –Successive pairs of elements are compared If increasing order (or identical), no change If decreasing order, elements exchanged –Repeat these steps for every element

 2003 Prentice Hall, Inc. All rights reserved. 2 Simple Sorting Bubble sort As we scan the list swap elements out of order. After the first scan, the largest element will be at the end of the list. Keep scanning the list until all of the elements are in the correct place. Bubble sort – because the small elements bubble up to the top…..

 2003 Prentice Hall, Inc. All rights reserved Sorting Arrays Example: –Go left to right, and exchange elements as necessary One pass for each element –Original: –Pass 1: (elements exchanged) –Pass 2: –Pass 3: (no changes needed) –Pass 4: –Pass 5: –Small elements "bubble" to the top (like 2 in this example)

 2003 Prentice Hall, Inc. All rights reserved. Bubble Sort Put smaller first No change Put smaller first

 2003 Prentice Hall, Inc. All rights reserved. Bubble Sort Begin again and put smaller first No change Put smaller first

 2003 Prentice Hall, Inc. All rights reserved. Bubble Sort Example 2 Begin --Put smaller first Put smaller first No change Put smaller first

 2003 Prentice Hall, Inc. All rights reserved. Bubble Sort – Example 2 Begin again and put smaller first No change Put smaller first

 2003 Prentice Hall, Inc. All rights reserved. Bubble Sort – Example 2 Begin again -- no change Swap – put smaller first Begin Again -- swap Sorted list

 2003 Prentice Hall, Inc. All rights reserved. 9 Let’s build up the algorithm – bubble sort outer loop – array scans, each scan starts from the first element of the list until _________ inner loop compare adjacent elements and swap next outer loop

 2003 Prentice Hall, Inc. All rights reserved Sorting Arrays Swapping variables int x = 3, y = 4; y = x; x = y; What happened? –Both x and y are 3! –Need a temporary variable Solution int x = 3, y = 4, temp = 0; temp = x; // temp gets 3 x = y; // x gets 4 y = temp; // y gets 3

 2003 Prentice Hall, Inc. All rights reserved. Outline 11 fig04_16.cpp (1 of 3) 1 // Fig. 4.16: fig04_16.cpp 2 // This program sorts an array's values into ascending order. 3 #include 4 5 using std::cout; 6 using std::endl; 7 8 #include 9 10 using std::setw; int main() 13 { 14 const int arraySize = 10; // size of array a 15 int a[ arraySize ] = { 2, 6, 4, 8, 10, 12, 89, 68, 45, 37 }; 16 int hold; // temporary location used to swap array elements cout << "Data items in original order\n"; // output original array 21 for ( int i = 0; i < arraySize; i++ ) 22 cout << setw( 4 ) << a[ i ]; 23

 2003 Prentice Hall, Inc. All rights reserved. Outline 12 fig04_16.cpp (2 of 3) 24 // bubble sort 25 // loop to control number of passes 26 for ( int pass = 0; pass < arraySize - 1; pass++ ) // loop to control number of comparisons per pass 29 for ( int j = 0; j < arraySize - 1; j++ ) // compare side-by-side elements and swap them if 32 // first element is greater than second element 33 if ( a[ j ] > a[ j + 1 ] ) { 34 hold = a[ j ]; 35 a[ j ] = a[ j + 1 ]; 36 a[ j + 1 ] = hold; } // end if 39 Do a pass for each element in the array. If the element on the left (index j ) is larger than the element on the right (index j + 1 ), then we swap them. Remember the need of a temp variable.

 2003 Prentice Hall, Inc. All rights reserved. Outline 13 fig04_16.cpp (3 of 3) fig04_16.cpp output (1 of 1) 40 cout << "\nData items in ascending order\n"; // output sorted array 43 for ( int k = 0; k < arraySize; k++ ) 44 cout << setw( 4 ) << a[ k ]; cout << endl; return 0; // indicates successful termination } // end main Data items in original order Data items in ascending order

 2003 Prentice Hall, Inc. All rights reserved. 14 Can we improve the algorithm? In the first example, we did not have to keep scanning the list since the list was sorted after the “2 nd ” scan…… Check to see if any swaps were performed on the previous inner loop. If none were performed do not scan the list anymore since it is sorted WE CAN END THE ALGORITHM EARLY: EARLY TERMINATION how can we accomplish this?

 2003 Prentice Hall, Inc. All rights reserved. Outline // bubble sort 25 int flag = 1; 26 for ( int pass = 0; (pass < arraySize – 1) && flag; pass++ ) 27 flag = 0; 28 // loop to control number of comparisons per pass 29 for ( int j = 0; j < arraySize - 1; j++ ) // compare side-by-side elements and swap them if 32 // first element is greater than second element 33 if ( a[ j ] > a[ j + 1 ] ) { 34 hold = a[ j ]; 35 a[ j ] = a[ j + 1 ]; 36 a[ j + 1 ] = hold; 37 flag = 1; //set flag since swap occurred } // end if 39 Bubble sort with early termination Possible early termination

 2003 Prentice Hall, Inc. All rights reserved. Outline // bubble sort 25 // loop to control number of passes 26 for ( int pass = 0; pass < arraySize - 1; pass++ ) // loop to control number of comparisons per pass 29 for ( int j = 0; j < arraySize - 1; j++ ) // compare side-by-side elements and swap them if 32 // first element is greater than second element 33 if ( a[ j ] > a[ j + 1 ] ) { 34 hold = a[ j ]; 35 a[ j ] = a[ j + 1 ]; 36 a[ j + 1 ] = hold; } // end if 39 How many times does the outer loop execute in the worst case? How many times is the swap performed (inner loop) in the worst case?

 2003 Prentice Hall, Inc. All rights reserved. 17 ADT LIST So far we have seen the following operations on a “list”: Adding an element Deleting an element Sorting the list for display Computing statistics on a list of numeric values Last operation  Finding an element in a list

 2003 Prentice Hall, Inc. All rights reserved. 18 Finding an element is important: FOR DELETION: usually you must “find” the element before deleting it…. FOR INSERTION: must find the place to insert an element (if not in alphabetic order)…. TO PRINT: i.e., find students name & print grade

 2003 Prentice Hall, Inc. All rights reserved Searching Arrays: Linear Search and Search array for a key value Linear search –Key value is the value to be searched for. It is usually inputted by the user. –Compare each element of array with key value Start at one end, go to other If the element is found, return the index number of the array. Remember --- we do not usually need to return the value, we know the value since it is what we were searching for. We need to know the POSITION of the value, i.e., its index.

 2003 Prentice Hall, Inc. All rights reserved. 20 SEARCHING Finding an element in an array Example: Given an array which contains a list of integers, find the index of a particular integer index of 24 is 1 index of 100 is 6 index of 35 is NOT FOUND

 2003 Prentice Hall, Inc. All rights reserved. Outline 21 fig04_19.cpp (1 of 2) 1 // Fig. 4.19: fig04_19.cpp 2 // Linear search of an array. 3 #include 4 5 using std::cout; 6 using std::cin; 7 using std::endl; 8 9 int linearSearch( const int [], int, int ); // prototype int main() 12 { 13 const int arraySize = 100; // size of array a 14 int a[ arraySize ]; // create array a 15 int searchKey; // value to locate in a for ( int i = 0; i < arraySize; i++ ) // create some data 18 a[ i ] = 2 * i; cout << "Enter integer search key: "; 21 cin >> searchKey; // attempt to locate searchKey in array a 24 int element = linearSearch( a, searchKey, arraySize ); 25 Takes array, search key, and array size.

 2003 Prentice Hall, Inc. All rights reserved. Outline 22 fig04_19.cpp (2 of 2) 26 // display results 27 if ( element != -1 ) 28 cout << "Found value in element " << element << endl; 29 else 30 cout << "Value not found" << endl; return 0; // indicates successful termination } // end main // compare key to every element of array until location is 37 // found or until end of array is reached; return subscript of 38 // element if key or -1 if key not found 39 int linearSearch( const int array[], int key, int sizeOfArray ) 40 { 41 for ( int j = 0; j < sizeOfArray; j++ ) if ( array[ j ] == key ) // if found, 44 return j; // return location of key return -1; // key not found } // end function linearSearch

 2003 Prentice Hall, Inc. All rights reserved. Outline 23 fig04_19.cpp output (1 of 1) Enter integer search key: 36 Found value in element 18 Enter integer search key: 37 Value not found

 2003 Prentice Hall, Inc. All rights reserved. 24 Linear Search Analysis In the worst case, how many elements have to be compared? int linearSearch( const int array[], int key, int sizeOfArray ) 40 { 41 for ( int j = 0; j < sizeOfArray; j++ ) if ( array[ j ] == key ) // if found, 44 return j; // return location of key return -1; // key not found } // end function linearSearch

 2003 Prentice Hall, Inc. All rights reserved. 25 Analysis IF an element is not in the array, all of the elements in the array have to be checked to determine that a particular element is not there. FOR EXAMPLE: if there are 10 elements in the array, 10 comparisons have to be made in the worst case. Therefore, if there are n elements in the array... n comparisons have to be made in the worst case

 2003 Prentice Hall, Inc. All rights reserved. 26 What if the array were already sorted? Search for an element in a sorted array Do we have to check all of the elements if we know something about the order of the array? No -- we can search until we know the element cannot appear anymore, i.e. array[j] > target. However in the worst case the # of comparisons is still the number of elements in the array, I.e. we have to check all of the elements in the array

 2003 Prentice Hall, Inc. All rights reserved. 27 Better Linear Search for Sorted Array “Early termination” --- int linearSearchSorted( const int array[], int key, int sizeOfArray ) 40 { 41 int j = 0; 42 while ( j < sizeOfArray-1 && key < array[j]) 43 j++ ; if ( array[ j ] == key ) // if found, 44 return j; // return location of key return -1; // key not found } // end function linearSearch

 2003 Prentice Hall, Inc. All rights reserved. 28 How to use a telephone book

 2003 Prentice Hall, Inc. All rights reserved. 29 To find the name “Randy Jackson” you would not start with Aardvark and continue until you hit the name...

 2003 Prentice Hall, Inc. All rights reserved. 30 start Input item; set lower index to zero set upper index to size-1 While lower index < upper indexReturn -1 Calculate midpoint Item == midpoint? Return index Item > midpoint Set upper index to midpoint-1 no yes Set lower index to midpoint + 1 no yes no Loop until found/not found

 2003 Prentice Hall, Inc. All rights reserved Midpoint low high Example 1: Looking for the number 140 Midpoint high low low Midpoint high highlow How many comparisons?3

 2003 Prentice Hall, Inc. All rights reserved Midpoint low high Example 1: Looking for the number 7 Midpoint high low low Midpoint high How many comparisons? 3

 2003 Prentice Hall, Inc. All rights reserved. Outline 33 fig04_20.cpp (1 of 6) 1 // Fig. 4.20: fig04_20.cpp 2 // Binary search of an array. 3 #include 4 5 using std::cout; 6 using std::cin; 7 using std::endl; 8 9 #include using std::setw; // function prototypes 14 int binarySearch( const int [], int, int, int, int ); 15 void printHeader( int ); 16 void printRow( const int [], int, int, int, int ); int main() 19 { 20 const int arraySize = 15; // size of array a 21 int a[ arraySize ]; // create array a 22 int key; // value to locate in a for ( int i = 0; i < arraySize; i++ ) // create some data 25 a[ i ] = 2 * i; 26

 2003 Prentice Hall, Inc. All rights reserved. Outline 34 fig04_20.cpp (2 of 6) 27 cout << "Enter a number between 0 and 28: "; 28 cin >> key; printHeader( arraySize ); // search for key in array a 33 int result = 34 binarySearch( a, key, 0, arraySize - 1, arraySize ); // display results 37 if ( result != -1 ) 38 cout << '\n' << key << " found in array element " 39 << result << endl; 40 else 41 cout << '\n' << key << " not found" << endl; return 0; // indicates successful termination } // end main 46

 2003 Prentice Hall, Inc. All rights reserved. Outline 35 fig04_20.cpp (3 of 6) 47 // function to perform binary search of an array 48 int binarySearch( const int b[], int searchKey, int low, 49 int high, int size ) 50 { 51 int middle; // loop until low subscript is greater than high subscript 54 while ( low <= high ) { // determine middle element of subarray being searched 57 middle = ( low + high ) / 2; // display subarray used in this loop iteration 60 printRow( b, low, middle, high, size ); 61 Determine middle element

 2003 Prentice Hall, Inc. All rights reserved. Outline // if searchKey matches middle element, return middle 63 if ( searchKey == b[ middle ] ) // match 64 return middle; else // if searchKey less than middle element, 69 // set new high element 70 if ( searchKey < b[ middle ] ) 71 high = middle - 1; // search low end of array // if searchKey greater than middle element, 74 // set new low element 75 else 76 low = middle + 1; // search high end of array 77 } return -1; // searchKey not found } // end function binarySearch Use the rule of binary search: If key equals middle, match If less, search low end If greater, search high end Loop sets low, middle and high dynamically. If searching the high end, the new low is the element above the middle. while ( low <= high ) { // determine middle element of subarray being searched 57 middle = ( low + high ) / 2; // display subarray used in this loop iteration 60 printRow( b, low, middle, high, size );

 2003 Prentice Hall, Inc. All rights reserved. Outline 37 fig04_20.cpp (5 of 6) // print header for output 84 void printHeader( int size ) 85 { 86 cout << "\nSubscripts:\n"; // output column heads 89 for ( int j = 0; j < size; j++ ) 90 cout << setw( 3 ) << j << ' '; cout << '\n'; // start new line of output // output line of - characters 95 for ( int k = 1; k <= 4 * size; k++ ) 96 cout << '-'; cout << endl; // start new line of output } // end function printHeader 101

 2003 Prentice Hall, Inc. All rights reserved. Outline 38 fig04_20.cpp (6 of 6) 102 // print one row of output showing the current 103 // part of the array being processed 104 void printRow( const int b[], int low, int mid, 105 int high, int size ) 106 { 107 // loop through entire array 108 for ( int m = 0; m < size; m++ ) // display spaces if outside current subarray range 111 if ( m high ) 112 cout << " "; // display middle element marked with a * 115 else if ( m == mid ) // mark middle value 118 cout << setw( 3 ) << b[ m ] << '*'; // display other elements in subarray 121 else 122 cout << setw( 3 ) << b[ m ] << ' '; cout << endl; // start new line of output } // end function printRow

 2003 Prentice Hall, Inc. All rights reserved. Outline 39 fig04_20.cpp output (1 of 2) Enter a number between 0 and 28: 6 Subscripts: * * found in array element 3 Enter a number between 0 and 28: 25 Subscripts: * * * 28 24* 25 not found

 2003 Prentice Hall, Inc. All rights reserved. Outline 40 fig04_20.cpp output (2 of 2) Enter a number between 0 and 28: 8 Subscripts: * * * 12 8* 8 found in array element 4

 2003 Prentice Hall, Inc. All rights reserved. 41 SEARCH ALGORITHM BEST CASE WORST CASE Linear Search Entire array 1 Binary Search 1 The number of times you can divide the array by 2 = log 2 (#elems_in_array) ******* Sorted array:

 2003 Prentice Hall, Inc. All rights reserved. 42 What does log 2 mean? xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Z elements xxxxxxxxxxxxxxxxxxxxxxxx Z/2 elements xxxxxxxxxxx Z/2/2 elements xxxxxx Z/2/2/2 elements Z 2 Z 2323 Z 2x2x =1? Log 2 z = x x is the number of times I can keep splitting the list until there is only 1 element left

 2003 Prentice Hall, Inc. All rights reserved. 43 Powers of 2 Below are tables for the power 2 x x2xx2x

 2003 Prentice Hall, Inc. All rights reserved. 44 SEARCH ALGORITHM BEST CASE WORST CASE Linear Search n 1 Binary Search 1 Log 2 (n) For an ORDERED array of n elements

 2003 Prentice Hall, Inc. All rights reserved. 45 SEARCH ALGORITHM BEST CASE WORST CASE Linear Searchn 1 Binary Search What about an unsorted array? NOT APPLICABLE