1. Chapter 9: Searching, Sorting, and Algorithm Analysis
Starting Out with C++ Early Objects
Seventh Edition
by Tony Gaddis, Judy Walters, and Godfrey Muganda

2. Topics 9.1 Introduction to Search Algorithms
9.2 Searching an Array of Objects
9.3 Introduction to Sorting Algorithms
9.4 Sorting an Array of Objects
9.5 Sorting and Searching Vectors
9.6 Introduction to Analysis of Algorithms

3. 9.1 Introduction to Search Algorithms
Search: locate an item in a list (array, vector, etc.) of information
Two algorithms (methods) considered here:
Linear search
Binary search

4. Algorithm: Linear Search(position)
Given: search_key, N-element array List Answers the question: WHERE IS IT? for (int pos = 0; pos< N; pos++) if ( List [pos] == search_key ) return pos; return -1; Return position

5. Algorithm: Linear Search(found)
Given: search_key, N-element array List Answers the question: IS IS THERE? for (int pos = 0; pos<N; pos++) if (List [pos] == search_key ) return true; return false;

6. Linear Search Example Array numlist contains
Searching for the value 11, linear search examines 17, 23, 5, and 11
Searching for the key value 7, linear
search examines 17, 23, 5, 11, 2, 29, and 3, i.e., ALL values! 17 23 5 11 2 29 3
See pr9-01.cpp

7. Linear Search Tradeoffs
Benefits
Easy algorithm to understand
Array can be in any order
Disadvantage
Inefficient (slow): for array of N elements, examines N/2 elements on average for value that is found, N elements for value not found
1 million elements, expect 500,000 to be examined.

8. Binary Search Algorithm (e.g., finding name in phone book)
Divide a sorted array into three sections:
middle element
elements before the middle element
elements after of the middle element
If the middle element is the search key, done. Otherwise, go to step 1, using only the half of the array that may contain the correct value.
Continue steps 1 and 2 until either the value is found or there are no more elements to examine.

9. Binary Search (BS) Example
Array A contains
[0] [1] [2] [3] [4] [5] [6]
for key 11, BS examines the middle, 11 (A[3]), and then stops (success)
for key 3, BS examines 11(A[3]), then 3 (A[1]), then stops. (success)
for key 20, BS examines 11(A[3]), then 23 (A[5]), then 17 (A[4]), then stops (not success). 2 3 5 11 17 23 29
See pr9-02.cpp

10. Binary Search Function – Skeleton A Recursive Algorithm)
bool BinSearch(int key, int A[], int lo, int hi) {
//If no more elements to search, answer is false.
if ( lo > hi ) return false; // BRAKES!!
//Divide array into 3 parts: middle, [lo, mid-1], [mid+1, hi]
int mid = (lo + hi) / 2;
// Check middle element for match.
if ( A[mid] == key ) return true;
// Search the half of array that can contain the key.
if ( key < A[mid] ) return BinSearch(key, A, lo, mid-1);
else return BinSearch(key, A, mid+1, hi); }

11. Binary Search Tradeoffs
Benefit
Much more efficient than linear search
(For array of N elements, performs at most
log2N comparisons)
Disadvantage
Requires that array elements be sorted N 2 4 64
1024
4096
1,000,000
log2N 1 6 10 12 20

12. Binary Search Exercise
Take the test_binarySearch.cpp program from the public repository, and make the following changes to the code. Then observe what happens, learn the rules about sortedness and BRAKES to stop recursion.
Comment out the first if statement. Compile and run the program. If it hangs, hit CTR-C. Observe the output.
Change the array in the main so that is it NOT SORTED. Compile and run and describe what you observed.

13. Binary Search Exercise2
In the test_binarySearch.cpp program, revise the binSearch function so that it works for an array sorted in DESCENDING order. Use the test program to demonstrate that the revised function works correctly.
> OSsubmit VIDEOalgm test_binarySearch.cpp

14. 9.2 Searching an Array of Objects
Search algorithms are not limited to arrays of integers
When searching an array of objects or structures, the value being searched for is a member variable, not the entire object or structure
Member in object/structure: key field
Value used in search: search key
See pr9-03.cpp

15. 9.3 Introduction to Sorting Algorithms
Sort: arrange values into an order
Ascending – smallest value first
Descending – smallest value last
Two algorithms considered here
Selection sort
Bubble sort

16. Selection Sort Algorithm (multiple passes required)
Locate smallest element in array positions 0 to last, and swap (exchange) it with element in position 0.
Locate smallest element in array positions 1 to last, and exchange it with element in position 1.
Continue positions 2 to (last-1), until all elements are in order.
Easiest comparison based sort (to me)!!
See pr9-05.cpp

17. Selection Sort Example (pass 1)
Array List contains
[0] [1] [2] [3]
Smallest element, 2, at position 1;
Swap List[1] and List[0]. 11 2 29 3
Now in order 2 11 29 3

18. Selection Sort – Example (pass 2)
Smallest element in List[1:3] is 3, at position 3.
Swap List[1] and List[3]
Now in order 2 11 29 3
Now in order 2 3 29 11

19. Selection Sort – Example (pass 3)
Smallest element in List[2:3] is 11, at position 3.
Swap List[2] and List[3]
Now in order 2 3 29 11
Now in order 2 3 11 20

20. Selection Sort Tradeoffs
Benefit- more efficient than Bubble Sort (later), due to fewer swaps/exchanges
Disadvantage – some consider it harder than Bubble Sort to understand (not me!!)
This is how YOU would sort a stack of numbered cards

21. Algorithm Skeleton / Analysis
Properties of Selection Sort:
Nested loop:
outer controls passes: for (int pass=0; pass<N-1; pass++)
inner finds position of smallest/largest in range [pass, N-1]
Need 2 functions:
void Swap( type &, type &);
int posMin(int A[ ], int lo, int hi); // subscript of smallest value in A[ lo:hi ].
Algorithm:
for (pass=0; pass<N-1; pass++) {
pos = posMin(A, pass, N-1); // Essentially the inner loop.
Swap(A[pos], A[pass]); }

22. Selection Sort Exercise
Take the test_binarySearch.cpp program, and convert it to test_selectionSort.cpp that implements the SelectionSort function and tests that it works. The main program must display the array BEFORE and AFTER sorting.
> OSsubmit VIDEOalgm test_selectionSort.cpp

23. Bubble Sort Algorithm
Compare 1st two elements and exchange (swap) them if they are out of order.
Move down one element and compare 2nd and 3rd elements. Exchange if necessary. Continue until end of array.
Pass through array again, repeating process and exchanging as necessary.
Repeat until a pass is made with no exchanges.
See pr9-04.cpp

24. Bubble Sort Example (first pass)
Array List contains
Compare values 17 and
23. In correct order, so
no swap (exchange).
Compare values 23 and
11. Not in correct order,
so swap (exchange) them. 17 23 5 11
Compare values 23
and 5. Out of order,
so swap (exchange)
them. 11 23 5 23

25. Bubble Sort Example (pass 2)
After first pass, array List contains 5 11 17 23
In order from previous pass
Compare values 17 and
5. Not in correct order,
so exchange them. 17 5 11
11. Not in correct order, 23

26. Bubble Sort Example (pass 3)
After second pass, array numlist3 contains
Compare values 5 and
11. In correct order, so
no swap. 5 11 17 23
In order from previous passes

27. Bubble Sort Tradeoffs Benefit Disadvantage MASTER IT!!
Easy to understand and implement
Disadvantage
Inefficiency makes it slow for large arrays
MASTER IT!!

28. Algorithm Skeleton / Analysis
Properties of Bubble Sort (MAX):
Nested loop:
outer controls passes: for (int pass=1; pass<N-1; pass++)
inner compares neighbors, swapping if out of order
Need 1 function:
void Swap( type &, type &);
Algorithm:
for (pass=1; pass<N-1; pass++) {
for (int k=0; k<N-pass; k++)
if (A[k] > A[k+1])
Swap(A[k], A[k+1]); }

29. Bubble Sort Exercise
Take your test_selectionSort program, and convert it to test_bubbleSort.cpp that implements the Bubble Sort function and tests that it works. The main program must display the array BEFORE and AFTER sorting.
> OSsubmit VIDEOalgm test_bubbleSort.cpp

30. 9.4 Sorting an Array of Objects
As with searching, arrays to be sorted can contain objects or structures
The key field determines how the structures or objects will be ordered
When exchanging contents of array elements, entire structures or objects must be exchanged, not just the key fields in the structures or objects
See pr9-06.cpp

31. 9.5 Sorting and Searching Vectors (LATER)
Sorting and searching algorithms can be applied to vectors as well as to arrays
Need slight modifications to functions to use vector arguments
vector <type> & used in prototype
No need to indicate vector size as functions can use size member function to calculate
See pr9-07.cpp

32. 9.6 Introduction to Analysis of Algorithms
Given two algorithms to solve a problem, what makes one better than the other?
Efficiency of an algorithm is measured by
space (computer memory used)
time (how long to execute the algorithm)
Analysis of algorithms is a more effective way to find efficiency than by using empirical data

33. Analysis of Algorithms: Terminology
Computational Problem: problem solved by an algorithm
Basic step: operation in the algorithm that executes in a constant amount of time
Examples of basic steps:
swap/exchange the contents of two variables
compare two values

34. Analysis of Algorithms: Terminology
Complexity of an algorithm: the number of basic steps required to execute the algorithm as a function of the input size N (N input values)
Worst-case complexity of an algorithm: number of basic steps for input of size N that requires the most work
Average case complexity function: the complexity for typical, average inputs of size N

35. Comparison of Algorithmic Complexity
Given algorithms F and G with complexity functions f(n) and g(n) for input of size n
If the ratio approaches a constant value as n gets large, F and G have equivalent efficiency
If the ratio gets larger as n gets large, algorithm G is more efficient than algorithm F
If the ratio approaches 0 as n gets large, algorithm F is more efficient than algorithm G

36. "Big O" Notation
Algorithm F is O(g(n)) ("F is big O of g") for some mathematical function g(n) if the ratio approaches a positive constant as n gets large
O(g(n)) defines a complexity class for the algorithm F
Increasing complexity class means faster rate of growth, less efficient algorithm

37. Chapter 9: Searching, Sorting, and Algorithm Analysis
Starting Out with C++ Early Objects
Seventh Edition
by Tony Gaddis, Judy Walters, and Godfrey Muganda