Data Structures Using C++ 2E The Big-O Notation and Array-Based Lists

Algorithm Analysis: The Big-O Notation Analyze algorithm after design Example 50 packages delivered to 50 different houses 50 houses one mile apart, in the same area FIGURE 1-1 Gift shop and each dot representing a house Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Example (cont’d.) Driver picks up all 50 packages Drives one mile to first house, delivers first package Drives another mile, delivers second package Drives another mile, delivers third package, and so on Distance driven to deliver packages 1+1+1+… +1 = 50 miles Total distance traveled: 50 + 50 = 100 miles FIGURE 1-2 Package delivering scheme Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Example (cont’d.) Similar route to deliver another set of 50 packages Driver picks up first package, drives one mile to the first house, delivers package, returns to the shop Driver picks up second package, drives two miles, delivers second package, returns to the shop Total distance traveled 2 * (1+2+3+…+50) = 2550 miles FIGURE 1-3 Another package delivery scheme Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Example (cont’d.) n packages to deliver to n houses, each one mile apart First scheme: total distance traveled 1+1+1+… +n = 2n miles Function of n Second scheme: total distance traveled 2 * (1+2+3+…+n) = 2*(n(n+1) / 2) = n2+n Function of n2 Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Analyzing an algorithm Count number of operations performed Not affected by computer speed TABLE 1-1 Various values of n, 2n, n2, and n2 + n Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Example 1-1 Illustrates fixed number of executed operations 1 operation 2 operations 1 operation 1 operation 1 operation 3 operations Total of 9 operations Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation Example 1-2 Illustrates dominant operations 2 operations 1 operation 1 operation 1 operation N+1 operations 2N operations N operations N operations 3 operations 1 operation 2 operations 1 operation 3 operation If the while loop executes N times then: 2+1+1+1+5*N + 1 + 3 + 1 + (2 ) + 3 = 5N+(15 ) Data Structures Using C++ 2E

For(i=1; i<=5; i++) Example For(j=1;j<=5; j++) { } Finally: 147 1 op 5 op 6 op For(i=1; i<=5; i++) For(j=1;j<=5; j++) { Cout<<“*”; Sum=sum+j; } For(i=1; i<=n; i++) For(j=1; j<=n; j++) { cout<<“*”; Sum=sum+j; } 30 op 25 op 5 op 25 op 2n+2+ 2n2+2n+ 3n2 = 5n2+ 4n+2 25 op 25 op Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Search algorithm n: represents list size f(n): count function Number of comparisons in search algorithm c: units of computer time to execute one operation cf(n): computer time to execute f(n) operations Constant c depends computer speed (varies) f(n): number of basic operations (constant) Determine algorithm efficiency Knowing how function f(n) grows as problem size grows Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) TABLE 1-2 Growth rates of various functions Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) TABLE 1-3 Time for f(n) instructions on a computer that executes 1 billion instructions per second Figure 1-4 Growth rate of functions in Table 1-3 Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) Notation useful in describing algorithm behavior Shows how a function f(n) grows as n increases without bound Asymptotic Study of the function f as n becomes larger and larger without bound Examples of functions g(n)=n2 (no linear term) f(n)=n2 + 4n + 20 Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) As n becomes larger and larger Term 4n + 20 in f(n) becomes insignificant Term n2 becomes dominant term TABLE 1-4 Growth rate of n2 and n2 + 4n + 20n Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) If function complexity can be described by complexity of a quadratic function without the linear term We say the function is of O(n2) or Big-O of n2 Let f and g be real-valued functions Assume f and g nonnegative For all real numbers n, f(n) >= 0 and g(n) >= 0 f(n) is Big-O of g(n): written f(n) = O(g(n)) If there exists positive constants c and n0 such that f(n) <= cg(n) for all n >= n0 Data Structures Using C++ 2E

Algorithm Analysis: The Big-O Notation (cont’d.) TABLE 1-5 Some Big-O functions that appear in algorithm analysis Data Structures Using C++ 2E

Array-Based Lists List Length of a list Collection of elements of same type Length of a list Number of elements in the list Many operations may be performed on a list Store a list in the computer’s memory Using an array Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Three variables needed to maintain and process a list in an array The array holding the list elements A variable to store the length of the list Number of list elements currently in the array A variable to store array size Maximum number of elements that can be stored in the array Desirable to develop generic code Used to implement any type of list in a program Make use of templates Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Define class implementing list as an abstract data type (ADT) FIGURE 3-29 UML class diagram of the class arrayListType Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Definitions of functions isEmpty, isFull, listSize and maxListSize Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Template print (outputs the elements of the list) and template isItemAtEqual Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Template insertAt Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Template insertEnd and template removeAt Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Template replaceAt and template clearList Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Definition of the constructor and the destructor Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Copy constructor Called when object passed as a (value) parameter to a function Called when object declared and initialized using the value of another object of the same type Copies the data members of the actual object into the corresponding data members of the formal parameter and the object being created Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Copy constructor (cont’d.) Definition Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Overloading the assignment operator Definition of the function template Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Searching for an element Linear search example: determining if 27 is in the list Definition of the function template FIGURE 3-32 List of seven elements Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Inserting an element Data Structures Using C++ 2E

Array-Based Lists (cont’d.) Removing an element Data Structures Using C++ 2E

Array-Based Lists (cont’d.) TABLE 3-1 Time complexity of list operations Data Structures Using C++ 2E