1. 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

2. 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

3. 4.8 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.

4. Finding an element in an array
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
index of 100 is 6
index of 35 is NOT FOUND

5. Takes array, search key, and array size.
// Fig. 4.19: fig04_19.cpp
// Linear search of an array.
#include <iostream> 4
using std::cout;
using std::cin;
using std::endl; 8
int linearSearch( const int [], int, int ); // prototype 10
11 int main()
12 {
const int arraySize = 100; // size of array a
int a[ arraySize ]; // create array a
int searchKey; // value to locate in a 16
for ( int i = 0; i < arraySize; i++ ) // create some data
a[ i ] = 2 * i; 19
cout << "Enter integer search key: ";
cin >> searchKey; 22
// attempt to locate searchKey in array a
int element = linearSearch( a, searchKey, arraySize ); 25
fig04_19.cpp (1 of 2)
Takes array, search key, and array size.

6. fig04_19.cpp (2 of 2) 26 // display results 27 if ( element != -1 )
cout << "Found value in element " << element << endl;
else
cout << "Value not found" << endl; 31
return 0; // indicates successful termination 33
34 } // end main 35
36 // 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 {
for ( int j = 0; j < sizeOfArray; j++ ) 42
if ( array[ j ] == key ) // if found,
return j; // return location of key 45
return -1; // key not found 47
48 } // end function linearSearch
fig04_19.cpp (2 of 2)

7. 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
fig04_19.cpp output (1 of 1)

8. Linear Search Analysis
In the worst case, how many elements have to be compared?
int linearSearch( const int array[], int key, int sizeOfArray )
40 {
for ( int j = 0; j < sizeOfArray; j++ ) 42
if ( array[ j ] == key ) // if found,
return j; // return location of key 45
return -1; // key not found 47
48 } // end function linearSearch

9. 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

10. 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

11. Better Linear Search for Sorted Array
“Early termination” ---
int linearSearchSorted( const int array[], int key, int sizeOfArray )
40 {
int j = 0;
while ( j < sizeOfArray-1 && key < array[j])
j++ ; 42
if ( array[ j ] == key ) // if found,
return j; // return location of key 45
return -1; // key not found 47
48 } // end function linearSearch

12. How to use a telephone book

13. To find the name “Randy Jackson”
you would not start with Aardvark
and continue until you hit the name...

14. start
Input item;
set lower index to zero
set upper index to size-1 no
While lower index < upper index
Return -1
Calculate midpoint
yes
Item == midpoint?
Return index no
found/not found
Loop until
yes
Set lower index to midpoint + 1
Item > midpoint
Set upper index to midpoint-1 no

15. 1 3 4 7 29 45 69 100 134 156 200 Example 1: Looking for the number 140
high
low
low
Midpoint
high
Midpoint
high
low
Midpoint
low
high
How many comparisons? 3

16. 1 3 4 7 29 45 69 100 134 156 200 Example 1: Looking for the number 7
low
Midpoint
high
Midpoint
high
low
Midpoint
low
high
How many comparisons? 3

17. fig04_20.cpp (1 of 6) 1 // Fig. 4.20: fig04_20.cpp
// Binary search of an array.
#include <iostream> 4
using std::cout;
using std::cin;
using std::endl; 8
#include <iomanip> 10
11 using std::setw; 12
13 // function prototypes
14 int binarySearch( const int [], int, int, int, int );
15 void printHeader( int );
16 void printRow( const int [], int, int, int, int ); 17
18 int main()
19 {
const int arraySize = 15; // size of array a
int a[ arraySize ]; // create array a
int key; // value to locate in a 23
for ( int i = 0; i < arraySize; i++ ) // create some data
a[ i ] = 2 * i; 26
fig04_20.cpp (1 of 6)

18. 27 cout << "Enter a number between 0 and 28: ";
cin >> key; 29
printHeader( arraySize ); 31
// search for key in array a
int result =
binarySearch( a, key, 0, arraySize - 1, arraySize ); 35
// display results
if ( result != -1 )
cout << '\n' << key << " found in array element "
<< result << endl;
else
cout << '\n' << key << " not found" << endl; 42
return 0; // indicates successful termination 44
45 } // end main 46
fig04_20.cpp (2 of 6)

19. Determine middle element
47 // function to perform binary search of an array
48 int binarySearch( const int b[], int searchKey, int low,
int high, int size )
50 {
int middle; 52
// loop until low subscript is greater than high subscript
while ( low <= high ) { 55
// determine middle element of subarray being searched
middle = ( low + high ) / 2; 58
// display subarray used in this loop iteration
printRow( b, low, middle, high, size ); 61
fig04_20.cpp (3 of 6)
Determine middle element

20. Use the rule of binary search: If key equals middle, match
54 while ( low <= high ) { 55
// determine middle element of subarray being searched
middle = ( low + high ) / 2; 58
// display subarray used in this loop iteration
printRow( b, low, middle, high, size );
// if searchKey matches middle element, return middle
if ( searchKey == b[ middle ] ) // match
return middle; 65
else 67
// if searchKey less than middle element,
// set new high element
if ( searchKey < b[ middle ] )
high = middle - 1; // search low end of array 72
// if searchKey greater than middle element,
// set new low element
else
low = middle + 1; // search high end of array } 78
return -1; // searchKey not found 80
81 } // 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.

21. fig04_20.cpp (5 of 6) 82 83 // print header for output
84 void printHeader( int size )
85 {
cout << "\nSubscripts:\n"; 87
// output column heads
for ( int j = 0; j < size; j++ )
cout << setw( 3 ) << j << ' '; 91
cout << '\n'; // start new line of output 93
// output line of - characters
for ( int k = 1; k <= 4 * size; k++ )
cout << '-'; 97
cout << endl; // start new line of output 99
100 } // end function printHeader
101
fig04_20.cpp (5 of 6)

22. 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,
int high, int size )
106 {
// loop through entire array
for ( int m = 0; m < size; m++ )
109
// display spaces if outside current subarray range
if ( m < low || m > high )
cout << " ";
113
// display middle element marked with a *
else
116
if ( m == mid ) // mark middle value
cout << setw( 3 ) << b[ m ] << '*';
119
// display other elements in subarray
else
cout << setw( 3 ) << b[ m ] << ' ';
123
cout << endl; // start new line of output
125
126 } // end function printRow
fig04_20.cpp (6 of 6)

23. fig04_20.cpp output (1 of 2) Enter a number between 0 and 28: 6
Enter a number between 0 and 28: 6
Subscripts: * *
6 found in array element 3
Enter a number between 0 and 28: 25 *
24 26* 28
24*
25 not found
fig04_20.cpp output (1 of 2)

24. fig04_20.cpp output (2 of 2) Enter a number between 0 and 28: 8
Subscripts: * *
8 10* 12 8*
8 found in array element 4
fig04_20.cpp output (2 of 2)

25. SEARCH ALGORITHM BEST CASE WORST CASE
Sorted array:
SEARCH ALGORITHM BEST CASE WORST CASE
Entire array
Linear Search 1 1
Binary Search
The number of times you can divide the array by 2
= log2(#elems_in_array)
*******

26. What does log2 mean? Z Z 22 2x Z 23 Z elements
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x is the number of times I can keep splitting the list until there is only 1 element left
xxxxxxxxxxxxxxxxxxxxxxxx
Z/2 elements
xxxxxxxxxxx Z Z 2x
=1?
Log2 z = x 22
Z/2/2 elements
xxxxxx Z
Z/2/2/2 elements 23

27. Powers of 2
Below are tables for the power 2x x 2x 1 2 3 4 5 6 7 8 9 10 11 12 13 16 32 64
128
256
512
1024
2048
4096
8192 14 15 16 17 18 19 20 21 22
16384
32768
65536
131072
262144
524288

28. SEARCH ALGORITHM BEST CASE WORST CASEn
Linear Search 1 1
Log2(n)
Binary Search
For an ORDERED array of n elements

29. SEARCH ALGORITHM BEST CASE WORST CASE
What about an unsorted array?
SEARCH ALGORITHM BEST CASE WORST CASE 1
Linear Search n
NOT APPLICABLE
Binary Search

30. 4.7 Case Study: Computing Mean, Median and Mode Using Arrays
Average (sum/number of elements)
Median
Number in middle of sorted list
1, 2, 3, 4, 5 (3 is median)
If even number of elements, take average of middle two
Mode
Number that occurs most often
1, 1, 1, 2, 3, 3, 4, 5 (1 is mode)

31. fig04_17.cpp (1 of 8) 1 // Fig. 4.17: fig04_17.cpp
// This program introduces the topic of survey data analysis.
// It computes the mean, median, and mode of the data.
#include <iostream> 5
using std::cout;
using std::endl;
using std::fixed;
using std::showpoint; 10
11 #include <iomanip> 12
13 using std::setw;
14 using std::setprecision; 15
16 void mean( const int [], int );
17 void median( int [], int );
18 void mode( int [], int [], int );
19 void bubbleSort( int[], int );
20 void printArray( const int[], int ); 21
22 int main()
23 {
const int responseSize = 99; // size of array responses 25
fig04_17.cpp (1 of 8)

32. 26 int frequency[ 10 ] = { 0 }; // initialize array frequency27
// initialize array responses
int response[ responseSize ] =
{ 6, 7, 8, 9, 8, 7, 8, 9, 8, 9,
, 8, 9, 5, 9, 8, 7, 8, 7, 8,
, 7, 8, 9, 3, 9, 8, 7, 8, 7,
, 8, 9, 8, 9, 8, 9, 7, 8, 9,
, 7, 8, 7, 8, 7, 9, 8, 9, 2,
, 8, 9, 8, 9, 8, 9, 7, 5, 3,
, 6, 7, 2, 5, 3, 9, 4, 6, 4,
, 8, 9, 6, 8, 7, 8, 9, 7, 8,
, 4, 4, 2, 5, 3, 8, 7, 5, 6,
, 5, 6, 1, 6, 5, 7, 8, 7 }; 40
// process responses
mean( response, responseSize );
median( response, responseSize );
mode( frequency, response, responseSize ); 45
return 0; // indicates successful termination 47
48 } // end main 49
fig04_17.cpp (2 of 8)

33. 50 // calculate average of all response values
51 void mean( const int answer[], int arraySize )
52 {
int total = 0; 54
cout << "********\n Mean\n********\n"; 56
// total response values
for ( int i = 0; i < arraySize; i++ )
total += answer[ i ]; 60
// format and output results
cout << fixed << setprecision( 4 ); 63
cout << "The mean is the average value of the data\n"
<< "items. The mean is equal to the total of\n"
<< "all the data items divided by the number\n"
<< "of data items (" << arraySize
<< "). The mean value for\nthis run is: "
<< total << " / " << arraySize << " = "
<< static_cast< double >( total ) / arraySize
<< "\n\n"; 72
73 } // end function mean 74
fig04_17.cpp (3 of 8)
We cast to a double to get decimal points for the average (instead of an integer).

34. 75 // sort array and determine median element's value
76 void median( int answer[], int size )
77 {
cout << "\n********\n Median\n********\n"
<< "The unsorted array of responses is"; 80
printArray( answer, size ); // output unsorted array 82
bubbleSort( answer, size ); // sort array 84
cout << "\n\nThe sorted array is";
printArray( answer, size ); // output sorted array 87
// display median element
cout << "\n\nThe median is element " << size / 2
<< " of\nthe sorted " << size
<< " element array.\nFor this run the median is "
<< answer[ size / 2 ] << "\n\n"; 93
94 } // end function median 95
fig04_17.cpp (4 of 8)
Sort array by passing it to a function. This keeps the program modular.

35. fig04_17.cpp (5 of 8) 96 // determine most frequent response
97 void mode( int freq[], int answer[], int size )
98 {
int largest = 0; // represents largest frequency
int modeValue = 0; // represents most frequent response
101
cout << "\n********\n Mode\n********\n";
103
// initialize frequencies to 0
for ( int i = 1; i <= 9; i++ )
freq[ i ] = 0;
107
// summarize frequencies
for ( int j = 0; j < size; j++ )
freq[ answer[ j ] ];
111
// output headers for result columns
cout << "Response" << setw( 11 ) << "Frequency"
<< setw( 19 ) << "Histogram\n\n" << setw( 55 )
<< " \n" << setw( 56 )
<< " \n\n";
117
fig04_17.cpp (5 of 8)

36. // output results
for ( int rating = 1; rating <= 9; rating++ ) {
cout << setw( 8 ) << rating << setw( 11 )
<< freq[ rating ] << " ";
122
// keep track of mode value and largest fequency value
if ( freq[ rating ] > largest ) {
largest = freq[ rating ];
modeValue = rating;
127
} // end if
129
// output histogram bar representing frequency value
for ( int k = 1; k <= freq[ rating ]; k++ )
cout << '*';
133
cout << '\n'; // begin new line of output
135
} // end outer for
137
// display the mode value
cout << "The mode is the most frequent value.\n"
<< "For this run the mode is " << modeValue
<< " which occurred " << largest << " times." << endl;
142
143 } // end function mode
The mode is the value that occurs most often (has the highest value in freq).
fig04_17.cpp (6 of 8)

37. 144
145 // function that sorts an array with bubble sort algorithm
146 void bubbleSort( int a[], int size )
147 {
int hold; // temporary location used to swap elements
149
// loop to control number of passes
for ( int pass = 1; pass < size; pass++ )
152
// loop to control number of comparisons per pass
for ( int j = 0; j < size - 1; j++ )
155
// swap elements if out of order
if ( a[ j ] > a[ j + 1 ] ) {
hold = a[ j ];
a[ j ] = a[ j + 1 ];
a[ j + 1 ] = hold;
161
} // end if
163
164 } // end function bubbleSort
165
fig04_17.cpp (7 of 8)

38. fig04_17.cpp (8 of 8) 166 // output array contents (20 values per row)
167 void printArray( const int a[], int size )
168 {
for ( int i = 0; i < size; i++ ) {
170
if ( i % 20 == 0 ) // begin new line every 20 values
cout << endl;
173
cout << setw( 2 ) << a[ i ];
175
} // end for
177
178 } // end function printArray
fig04_17.cpp (8 of 8)

39. fig04_17.cpp output (1 of 2) ******** Mean
The mean is the average value of the data
items. The mean is equal to the total of
all the data items divided by the number
of data items (99). The mean value for
this run is: 681 / 99 =
Median
The unsorted array of responses is
The sorted array is
The median is element 49 of
the sorted 99 element array.
For this run the median is 7
fig04_17.cpp output (1 of 2)

40. fig04_17.cpp output (2 of 2) ******** Mode
Response Frequency Histogram *
***
****
*****
********
*********
***********************
***************************
*******************
The mode is the most frequent value.
For this run the mode is 8 which occurred 27 times.
fig04_17.cpp output (2 of 2)