Ms. Manal Al-Asmari

Data Structures & Algorithms

CHAPTER 4Searching

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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

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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.

Linear Searchint 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;elsereturn -1;}

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Linear Search Benefits

– Easy to understand– Array can be in any order

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Linear Search Disadvantages

– Inefficient for array of N elementsExamines N/2 elements on average for value in array, N elements for value not in array

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Binary Search

Binary search looks for an item in an array/list using divide and conquer strategy

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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

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Binary Search

• 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

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Execution of Binary Search

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Execution of Binary Search

Binary Search Algorithmpublic 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;elseif (list[mid] > searchItem)last = mid - 1;elsefirst = mid + 1;}if (found)return mid;elsereturn –1;}

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Ms. Manal Al-Asmari

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

Key Points in Binary Search

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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

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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

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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

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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

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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 ?

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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

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On average, a successful sequential search searches half the list

Sequential Search Analysis (continued)

Average Case:

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THE END