Data Structures and Algorithm - Week 9 - Search Algorithms
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This document discusses search algorithms, including linear search and binary search. It covers: 1) Linear search sequentially scans an array to find a match, working for both sorted and unsorted arrays. 2) Binary search uses recursion to search a sorted array, eliminating half the search space each iteration, making it faster than linear search. 3) Implementation examples of linear and binary search in Java are provided, along with analysis of the time efficiency of each method.
![Linear Search
• The algorithm translates to the following Java method:
public static int linearSearch(Object[] array,
Object key)
{
for(int k = 0; k < array.length; k++)
if(array[k].equals(key))
return k;
return -1;
}
Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-
4](https://image.slidesharecdn.com/week9-190319081228/85/Data-Structures-and-Algorithm-Week-9-Search-Algorithms-4-320.jpg)
![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
Lecture series for Data Structures and
Algorithms, Data Science and Analytics,
Thai-Nichi Institute of Technology
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![Checking the search Method
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
Lecture series for Data Structures and
Algorithms, Data Science and Analytics,
Thai-Nichi Institute of Technology
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![Checking the search Method
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
Lecture series for Data Structures and
Algorithms, Data Science and Analytics,
Thai-Nichi Institute of Technology
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![Checking the search Method
• 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
Lecture series for Data Structures and
Algorithms, Data Science and Analytics,
Thai-Nichi Institute of Technology
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Data Structures and Algorithm - Week 9 - Search Algorithms
- 1. Data Structures and Algorithms Week 9: Search Algorithms Ferdin Joe John Joseph, PhD Faculty of Information Technology Thai-Nichi Institute of Technology, Bangkok
- 2. Week 9 • Linear Search • Binary Search • Implementation in Java Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 2
- 3. Linear Search • Searching is the process of determining whether or not a given value exists in a data structure or a storage media. • We discuss two searching methods on one-dimensional arrays: linear search and binary search. • The linear (or sequential) search algorithm on an array is: • 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. Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai- 3
- 4. Linear Search • The algorithm translates to the following Java method: public static int linearSearch(Object[] array, Object key) { for(int k = 0; k < array.length; k++) if(array[k].equals(key)) return k; return -1; } Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai- 4
- 5. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 5
- 6. Binary Search • An algorithm to solve this task looks at the middle of the array or array segment first • If the value looked for is smaller than the value in the middle of the array • Then the second half of the array or array segment can be ignored • This strategy is then applied to the first half of the array or array segment Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 6
- 7. Binary Search • If the value looked for is larger than the value in the middle of the array or array segment • Then the first half of the array or array segment can be ignored • This strategy is then applied to the second half of the array or array segment • If the value looked for is at the middle of the array or array segment, then it has been found • If the entire array (or array segment) has been searched in this way without finding the value, then it is not in the array Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 7
- 8. Pseudocode for Binary Search Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 8
- 9. Recursive Method for Binary Search Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 9
- 10. Execution of the Method search (Part 1 of 2) Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 10
- 11. Execution of the Method search (Part 1 of 2) Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 11
- 12. Checking the search Method 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 12
- 13. Checking the search Method 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 13
- 14. Checking the search Method 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 14
- 15. Checking the search Method • 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 15
- 16. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 16
- 17. Efficiency of Binary Search • Given an array with 1,000 elements, the binary search will only need to compare about 10 array elements to the key value, as compared to an average of 500 for a serial search algorithm • The binary search algorithm has a worst-case running time that is logarithmic: O(log n) • A serial search algorithm is linear: O(n) • If desired, the recursive version of the method search can be converted to an iterative version that will run more efficiently Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 17
- 18. Iterative Version of Binary Search (Part 1 of 2) Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 18
- 19. Iterative Version of Binary Search (Part 2 of 2) Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 19
- 20. Next Week Sorting Algorithms Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 20