1. Algorithms & Data Structures CS112
Spring 2012
Lecture 4
Syed Muhammad Raza

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

3. Linear Search: O(n)
 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.

4. Linear Search
bool LinSearch(double x[ ], int n, double item)
{
for(int i=0;i<n;i++)
{
if(x[i]==item)
{
return true;
}
else
{
return false;
}
}
return false;
}

5. Linear Search Tradeoffs
 Benefits
 Easy to understand
 Array can be in any order
 Disadvantages
 Inefficient for array of N elements
Examines N/2 elements on average for value in array, N
elements for value not in array

6. Binary Search: O(log2 n)
 Binary search looks for an item in an array/list using
divide and conquer strategy

7. 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

8. 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

9. Pseudocode for Binary Search

10. Execution of Binary Search

11. Execution of Binary Search

12. Key Points in Binary Search
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

13. 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

14. 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

15. 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

16. 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

17. Binary Search

18. Algorithms & Data Structures CSC-112
Fall 2011
Lecture 5
Syed Muhammad Raza

19. Sorting Algorithms
 Sorting is the process of rearranging your data elements/Item in
ascending or descending order
 Unsorted Data
4 3 2 7 1 6 5 8 9
 Sorted Data (Ascending)
1 2 3 4 5 6 7 8 9
 Sorted Data (Descending)
9 8 7 6 5 4 3 2 1

20. Sorting Algorithms
 They are many
 Bubble Sort
 Selection Sort
 Insertion Sort
 Shell sort
 Comb Sort
 Merge Sort
 Heap Sort
 Quick Sort
 Counting Sort
 Bucket Sort
 Radix Sort
 Distribution Sort
 Time Sort
Source: Wikipedia

21. Bubble Sort
 Compares Adjacent Items and Exchanges Them if They are Out of
Order
 When You Order Successive Pairs of Elements, the Largest Element
Bubbles to the Top(end) of the Array
 Bubble Sort (Usually) Requires Several Passes Through the Array

22. Bubble Sort
Pass 1 Pass 2
29 10 14 37 13 10 14 29 13 37
10 29 14 37 13 10 14 29 13 37
10 14 29 37 13 10 14 29 13 37
10 14 29 37 13 10 14 13 29 37
10 14 29 13 37

23. Selection Sort
 To Sort an Array into Ascending Order, First Search for the Largest Element
 Because You Want the Largest Element to be in the Last Position of the
Array, You Swap the Last Item With the Largest Item to be in the Last Position of
the Array, You Swap the Last Item with the Largest Item, Even if These Items
Appear to be Identical
 Now, Ignoring the Last (Largest) Item of the Array, search Rest of the Array For
Its Largest Item and Swap it With Its Last Item, Which is the Next-to-Last Item in
the original Array
 You Continue Until You Have Selected and Swapped N-1 of the N Items in the
Array
 The Remaining Item, Which is Now in the First Position of the Array, is in its
Proper Order

24. Selection Sort
Initial Array 29 10 14 37 13
After1st Swap 29 10 14 13 37
After2nd Swap 13 10 14 29 37
After3rd Swap 13 10 14 29 37
After4th Swap 10 13 14 29 37

25. Insertion Sort
 Divide Array Into Sorted Section At Front (Initially Empty), Unsorted
Section At End
 Step Iteratively Through Array, Moving Each Element To Proper
Place In Sorted Section
Sorted Unsorted
….. …..
0 i N-1
After i Iterations

26. Insertion Sort
Initial Array 29 10 14 37 13 Copy 10
29 29 14 37 13 Shift 29
10 29 14 37 13 Insert 10, Copy 14
10 29 29 37 13 Shift 29
Insert 14; Copy 37
10 14 29 37 13
Insert 37 on Itself
10 14 29 37 13 Copy 13
10 14 14 29 37 Shift 14, 29, 37
Sorted Array 10 13 14 29 37 Insert 13