Sorting Algorithms. Array Data is being arranged in ascending order using the bubble sort algorithm. #1 #2 #3 #4 #5 #6 #7

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1 Sorting Algorithms One of the fundamental problems of computer science is ordering a list of items. There s a plethora of solutions to this problem, known as sorting algorithms. Some sorting algorithms are simple and intuitive, such as the bubble sort. Sorting data (i.e the arrangement of data into some particular order such as ascending or descending ) is an important computing application. Bubble(Sinking) sort algorithm This technique is called bubble sort or sinking sort because while considering sorting an array into ascending order small values bubble their way upward to the top of the array like air bubbles rising in water. Algorithm: ¾ It makes several passes through the unsorted array. ¾ On each pass, successive pair of elements are compared. ¾ If the higher indexed element is less than the lower indexed element, values are swapped. Array Data is being arranged in ascending order using the bubble sort algorithm. #1 #2 #3 #4 #5 #6 #7 Element Data st pass nd pass rd pass #-- comparison #1 elements at the positions 1 and 2 compared and if required swapped similarly #2 compares elements at the positions 2 and 3

2 #3 #7 compare successive elements. The first two data items (27 and 63) are compared and the smaller one placed on the left hand side. The second and third items (63 and 1) are then compared and the smaller one placed on the left and so on. After all the data has been passed through once, the largest data item (72) will have "bubbled" through to the end of the list. At the end of the second pass, the second largest data item (64) will be in the second last position. For n data items, the process continues for n-1 passes, or until no exchanges are made in a single pass. Observations: ¾ The largest element appears at the largest index at the end of the pass ¾ n passes are not required (n is the number of elements in the array). ¾ As the number of passes increase the array being processed tends to the sorted list. ¾ After the first pass, the last and the second last element need not be compared. A Function Bubble sort is designed and the unsorted array is fed to this function. Function Prototype Void bubblesort (float [], int); Function definition Void bubble sort (float [], int m) for(int i = 1; i<n; ++i) // keeps track of the number of passes through the unsorted // array. for(int j = 0; j<n-i; j++)//keeps track of the number of comparisons //in each pass the number of comparisons decrease by // the variable i. if(a[j]>a[j+1]) swap(a[j], a[j+1]);

3 function Swap void swap(float &x, float &y ) float temp = x; //value of x is saved in a variable temp x = y; // value of x is overwritten by the value of y. y = temp; // value saved in temp is copied to y. Searching arrays There are two searching techniques ¾ Linear search. ¾ Binary search. Most of the times, a programmer will be working with large amounts of data stored in the arrays. It may be necessary to determine whether any array contains a value that matches a key value. The process of finding a particular element of an array is called searching. Linear Search Linear search compares each element of array with the search key and determines whether the value is in the array. It works well with small arrays or for unsorted arrays. However, for large arrays, linear searching is inefficient. int Data[6]; Key_value = 8 Key_value is compared to all the elements of the array to check its existence in the array, if the value is found in the array, its index is returned.

4 Binary Search Binary search algorithm is very efficient algorithm. This technique can only be used if the array is sorted. Algorithm: The binary search algorithm eliminates one-half of the elements in the array being searched after each comparison. This algorithm locates the middle element of the array and compares it with the search key. If they are equal the search key is found and array index of the element is returned. Otherwise,the problem is reduced to searching one half of the array. If the search key is less then the middle element of the array, the first half of the array is searched,otherwise the second half of the array is searched. In this way the array is bisected, creating sub arrays(parts of the main array) and the middle element is checked again.the search continues till the search key is equal to the middle element of a subarray or until the subarray consists of element not equal to the search key.(i.e. search key not found). The maximum number of comparisons needed for the binary search of any sorted array can be determined by finding the first power of 2 greater than the number of elements in the array Key_value = 8 Data[low] Search Data[middle] Data[high] key searchkey <Data[middle] Data[low] Search key Data[high]

5 searchkey >Data[middle] Data[low] Search key Data[high] pseudocode middle=(low+high)/2.0; if(data[middle] = searchkey) return middle; else if (searchkey <Data[middle]) high=middle 1; else low=middle+1; //search low end of the array //search high end of the array Consider the function for the binary search algorithm (iterative version) int binarysearch( int b[],int searchkey, int low, int high, int size) int middle; while (low <=high) middle = (low + high)/2; if(searchkey == b[middle]) return(middle); else if (searchkey <b[middle]) high = middle - 1; else low = middle + 1; return(-1); This function receives four arguments an integer array b, an integer searchkey, low array index and a high array index.

6 References : [1] C++ How To Program, Deitel and Deitel.