Searching is a task closely related to sorting. The basic idea is to quickly locate a specific value in an array or determine that the array doesn't contain the value. Although searching for a particular value in an array of simple data (e.g., characters or integers) has limited use, doing so is a good example of a searching algorithm. Searching for a specific element becomes more useful when the elements are objects, instances of structures or classes.
Imagine that we begin with a student structure that has three fields or members: (1) the student's name, (2) the student's identification number, and (3) the student's GPA. Next, we create an array of objects instantiated from the structure and initialize all of the fields in each object. Now, we can search for a specific student's ID, and if we find that ID, the search also returns the student's name and GPA. This operation is called an associative search and the structure field used for the search, ID in this example, is called the key.
Binary search is a simple but effective searching algorithm often used with arrays. However, it has one requirement: we must sort the array before we can apply the binary search algorithm. The basic operation is to divide the array in half and ask, "Is the element I'm looking for in the top half or the bottom half?" Ignore the half of the array that doesn't contain the element, and then repeat the process until the algorithm locates the correct element or determines that the element is not in the array. The binary search doesn't change the array; instead, it performs its operations through three variables that index into the array.
(a)
(b)
(c)
(d)
(e)
Binary search. The binary search algorithm divides an array into two sub-arrays, determines if the key or target element is in the upper or lower sub-array, and then repeats. At some point, the sub-array is small enough that the key's location is determined or that the key is not in the array. All partitioning steps (dividing the array into sub-arrays) are done through three index variables:
top: the index location of the top of the current sub-array
mid: the index location of the middle of the current sub-array
bottom: the index location of the bottom of the current sub-array
Illustrated are the steps binary search takes to locate 'Q' (the key or target) in a simple array of characters:
The original, unsorted array.
The array is sorted, top and bottom are initialized, and the first value of mid is calculated.
The key is in the bottom half of the array, so top is updated and a new value for mid is calculated.
The key is in the top half of the array, so bottom is updated and a new value for mid is calculated.
In this specific example, all three indexes are equal; it is not a requirement that top and bottom point to the key, only that mid does.
Binary Search: Simple Data
#include <iostream>
using namespace std;
void ssort(char* data, int size);
int binsearch(char key, char* data, int size);
int main()
{
char data[] = { 'S','E','E','T','H','E','Q','U','I', // (a)
'C','K','R','E','D','F','O','X' };
int size = sizeof(data) / sizeof(char); // (b)
ssort(data, size); // (c)
int index = binsearch('Q', data, size); // (d)
if (index >= 0) // (e)
cout << "key found at index = " << index << endl;
else
cout << "key not in array" << endl;
return 0;
}
A driver to test and demonstrate binary search.
Test data
The array size; NOTE: sizeof does NOT work with function arguments - that is, we can't calculate the size of the data array in the ssort function (see Figure 5)
Call the selection sort function (see the previous section); binary search only works with sorted arrays
Call binary search; the function returns the index in the array where the key is located or -1 if the key is not in the array
Testing the return value to determine what message to print
int binsearch(char key, char* data, int size)
{
int top = 0; // (a)
int bottom = size - 1; // (b)
do // (c)
{
int mid = (top + bottom) / 2; // (d)
if (key < data[mid])
bottom = mid - 1; // (e)
else if (key > data[mid])
top = mid + 1; // (f)
else
return mid; // (g)
} while (top <= bottom); // (h)
return -1; // (i)
}
A C++ implementation of binary search.
Initialize the top index
Initialize the bottom index
Repeat until the key found or it is determined that it isn't in the array
Calculate an index in the middle of the sub-array
If the key is in the top half of the sub-array, discard the bottom half
If the key is in the bottom half of the sub-array, discard the top half
If the key is located at the mid-point, return mid
Binary search ends without finding the key when the top and bottom indexes cross
Return -1 (an invalid array index) to denote that the key was not found
#include <iostream>
using namespace std;
struct student
{ char *name;
int id;
double gpa;
};
void ssort(student* data, int size);
int binsearch(int key, student* data, int size);
int main()
{
student students[] = {
{ "Dilbert", 123, 3.5 },
{ "Wally", 456, 2.0 },
{ "Alice", 987, 3.9 },
{ "Asok", 730, 3.8 },
{ "Catbert", 501, 3.0 },
{ "Pointy Haired Boss", 666, 1.0 },
{ "Dogbert", 111, 4.0 }
};
int size = sizeof(students) / sizeof(student);
ssort(students, size); // array must be sorted for binary search
int index = binsearch(501, students, size); // call binary search
if (index >= 0)
cout << "key found at index = " << index << endl;
else
cout << "key not in array" << endl;
return 0;
}
int binsearch(int key, student* data, int size)
{
int top = 0;
int bottom = size - 1;
do
{
int mid = (top + bottom) / 2;
if (key < data[mid].id)
bottom = mid - 1; // search top half of sub-array
else if (key > data[mid].id)
top = mid + 1; // search bottom half of sub-array
else
return mid; // key found
} while (top <= bottom);
return -1; // key not in the data array
}
Searching for objects based on one field. This version of the program searches in an array of objects instantiated from a structure. The array is ordered by the student's ID (see version 4 of selection sort in the previous section). The program searches for the student with ID number 501.
