1. Data Structures & Algorithms CHAPTER 4 Searching Ms. Manal Al-Asmari

2. Searching Algorithms Searching is the process of determining whether or not a given value exists in a data structure or a storage media. We will study two searching algorithms: – Linear Search – Binary Search Ms. Manal Al-Asmari

3. Linear Search The linear (or sequential) search algorithm on an array is: –Start from beginning of an array/list and continues until the item is found or the entire array/list has been searched. –Sequentially scan the array, comparing each array item with the searched value. –If a match is found; return the index of the matched element; otherwise return –1. Note: linear search can be applied to both sorted and unsorted arrays. Ms. Manal Al-Asmari

4. Linear Search int seqSearch(int[] list, int listLength, int searchItem) { int loc; boolean found = false; for (loc = 0; loc < listLength; loc++) if (list[loc] == searchItem) { found = true; break; } if (found) return loc; else return -1; } Ms. Manal Al-Asmari

5. Linear Search Benefits – Easy to understand – Array can be in any order Ms. Manal Al-Asmari

6. Linear Search Disadvantages – Inefficient for array of N elements Examines N/2 elements on average for value in array, N elements for value not in array Ms. Manal Al-Asmari

7. Binary Search Binary search looks for an item in an array/list using divide and conquer strategy Ms. Manal Al-Asmari

8. Binary Search –Binary search algorithm assumes that the items in the array being searched is sorted –The algorithm begins at the middle of the array in a binary search –If the item for which we are searching is less than the item in the middle, we know that the item won’t be in the second half of the array –Once again we examine the “middle” element –The process continues with each comparison cutting in half the portion of the array where the item might be Ms. Manal Al-Asmari

9. Binary Search Ms. Manal Al-Asmari Binary search uses a recursive method to search an array to find a specified value The array must be a sorted array: a[0]≤a[1]≤a[2]≤... ≤ a[finalIndex] If the value is found, its index is returned If the value is not found, -1 is returned Note: Each execution of the recursive method reduces the search space by about a half

10. Execution of Binary Search Ms. Manal Al-Asmari

11. Execution of Binary Search

12. Binary Search Algorithm public static int binarySearch(int[] list, int listLength,int searchItem){ int first = 0; int last = listLength - 1; int mid; boolean found = false; while (first <= last){ mid = (first + last) / 2; if (list[mid] == searchItem) found = true; else if (list[mid] > searchItem) last = mid - 1; else first = mid + 1;} if (found) return mid; else return –1; } Ms. Manal Al-Asmari

13. 1.There is no infinite recursion On each recursive call, the value of first is increased, or the value of last is decreased If the chain of recursive calls does not end in some other way, then eventually the method will be called with first larger than last 2.Each stopping case performs the correct action for that case If first > last, there are no array elements between a[first] and a[last], so key is not in this segment of the array, and result is correctly set to -1 If key == a[mid], result is correctly set to mid Ms. Manal Al-Asmari Key Points in Binary Search

14. Ms. Manal Al-Asmari Key Points in Binary Search 3.For each of the cases that involve recursion, if all recursive calls perform their actions correctly, then the entire case performs correctly If key < a[mid], then key must be one of the elements a[first] through a[mid-1], or it is not in the array The method should then search only those elements, which it does The recursive call is correct, therefore the entire action is correct

15. Ms. Manal Al-Asmari Key Points in Binary Search If key > a[mid], then key must be one of the elements a[mid+1] through a[last], or it is not in the array The method should then search only those elements, which it does The recursive call is correct, therefore the entire action is correct The method search passes all three tests: Therefore, it is a good recursive method definition

16. Ms. Manal Al-Asmari Efficiency of Binary Search The binary search algorithm is extremely fast compared to an algorithm that tries all array elements in order –About half the array is eliminated from consideration right at the start –Then a quarter of the array, then an eighth of the array, and so forth

17. Sequential Search Analysis The statements in the for loop are repeated several times For each iteration of the loop, the search item is compared with an element in the list When analyzing a search algorithm, you count the number of comparisons Suppose that L is a list of length n The number of key comparisons depends on where ”in the list” the search item is located Ms. Manal Al-Asmari

18. Sequential Search Analysis (continued) Best case The item is the first element of the list You make only one key comparison Worst case The item is the last element of the list You make n key comparisons What is the average case ? Ms. Manal Al-Asmari

19. Sequential Search Analysis (continued) To determine the average case Consider all possible cases Find the number of comparisons for each case Add them and divide by the number of cases Ms. Manal Al-Asmari

20. On average, a successful sequential search searches half the list Ms. Manal Al-Asmari Sequential Search Analysis (continued) Average Case:

21. Ms. Manal Al-Asmari