Searching algorithm

Searching algorithm

• 1.
  Linear and Binary SearchingAlgorithms
• 2.
  Search To search anelement in a given array, it can be done in two ways Linear search and Binary search.
• 3.
  Linear Search Step througharray of records, one at a time. Look for record with matching key. It compares the element with all the other elements given in the list and if the element is matched it returns the value index else it return -1
• 4.
  Linear Search Analysis Numberof operations depends on n, the number of entries in the list.
• 5.
  Worst Case Timefor Linear Search For an array of n elements, the worst case time for serial search requires n array accesses: O(n). Consider cases where we must loop over all n records: desired record appears in the last position of the array desired record does not appear in the array at all
• 6.
  Average Case forSerial Search Assumptions: 1. All keys are equally likely in a search 2. We always search for a key that is in the array Example:  We have an array of 10 records.  If search for the first record, then it requires 1 array access; if the second, then 2 array accesses. etc. The average of all these searches is: (1+2+3+4+5+6+7+8+9+10)/10 = 5.5
• 7.
  Average Case Timefor Serial Search Generalize for array size n. Expression for average-case running time: (1+2+…+n)/n = n(n+1)/2n = (n+1)/2 Therefore, average case time complexity for serial search is O(n).
• 8.
  Binary Search Perhaps wecan do better than O(n) in the average case? Assume that we are give an array of records that is sorted. For instance: an array of records with integer keys sorted from smallest to largest (e.g., ID numbers), or an array of records with string keys sorted in alphabetical order (e.g., names).
• 9.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ]
• 10.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Find approximate midpoint
• 11.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Is 7 = midpoint key? NO.
• 12.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Is 7 < midpoint key? YES.
• 13.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Search for the target in the area before midpoint.
• 14.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Find approximate midpoint
• 15.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Target = key of midpoint? NO.
• 16.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Target < key of midpoint? NO.
• 17.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Target > key of midpoint? YES.
• 18.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Search for the target in the area after midpoint.
• 19.
  Binary Search [ 0] [ 1 ] Example: sorted array of integer keys. Target=7. 3 6 7 11 32 33 53 [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Find approximate midpoint. Is target = midpoint key? YES.
• 20.
  Binary Search: Analysis Worstcase complexity? What is the maximum depth of recursive calls in binary search as function of n? Each level in the recursion, we split the array in half (divide by two). Therefore maximum recursion depth is floor(log2n) and worst case = O(log2n). Average case is also = O(log2n).
• 21.
  Summary Linear search: averagecase O(n) Binary search: average case O(log2n)