Chapter 1 Algorithm Analysis

1. Chapter 1Algorithm Analysis ICS 202  Data Structures and Algorithms  Instructor : Da’ad Ahmad Albalawneh

2. Outline • Introduction • A Detailed Model of the Computer • The Basic Axioms • Example 1: Arithmetic Series Summation • Array Subscribing Operations • Example2: Horner’s Rule • Analyzing Recursive Methods • Example 3: Finding the Largest Element of an Array • Average Running Times • About Harmonic Numbers • Best-Case and Worst-Case Running Times • The Last Axiom • A Simplified Model of the Computer • Example 1: Geometric Series Summation • About Geometric Series Summation • Example 2: Computing Powers

3. 1. Introduction • What is an Algorithm? • “A step-by-step procedure for accomplishing some end” • Can be written in English, or using an appropriate mathematical formalism – programming language • Why do we analyze an algorithm and what can we analyze? • To learn more about the algorithm (some specifications) • To draw conclusions about how the implementation will perform • We can analyze • The running time of a program as a function of its inputs • The total or maximum memory space needed for program data • The total size of the program code • The complexity of the program • The robustness (powerful) of the program (how wll does it deal with unexpected inputs)

4. 1. Introduction • Factors affecting the running time of a program • The algorithm itself • The input data • The computer system used to run the program • The hardware (Processor, memory available, disk available, …) • The programming language • The language compiler • The computer operating system software

5. Java system Java Virtual Machine Physical Machine Java Program Java bytecode Machine code Java Compiler Java bytecode interpreter Hardware 2. A Detailed Model of the Computer • Detailed model of the running time performance of JAVA programs • The model is independent of the hardware and system software • Modeling the execution of a JAVA program on the “Java Virtual Machine”

6. The time required to fetch an operand from memory is a constant, and the time required to store a result in memory is a constant, y = x; has running time y = 1; has running time 2. A Detailed Model of the Computer: The Basic Axioms • Axiom 1 The two statements have the same running time because the constant 1 needs to be stored in memory

7. The times required to perform elementary operations, such as addition, subtraction, multiplication, division, and comparison, are all constants. These times are denoted by: respectively. 2. A Detailed Model of the Computer: The Basic Axioms • Axiom 2 • All of the simple operations can be accomplished in a fixed amount of time • The number of bits used to represent a value must be fixed • In Java, the number of bits range from 8 (byte) to 64 (long and double)

8. 2. A Detailed Model of the Computer: The Basic Axioms Example: By applying the axiom 1 and 2, the running time for the statement y = y + 1; is Because we need to fetch two operands, y and 1, add them, and, store the result back in y y+ = 1; ++y; y++; Have the same running time as y = y + 1;

9. The time required to call a method is a constant, and the time required to return from a method is a constant, 2. A Detailed Model of the Computer: The Basic Axioms • Axiom 3 • When a method is called: • The returned address must be saved • Any partially completed computations must also be saved • A new execution context (stack frame, …) must be allocated

10. Example: y = f(x); has a running time: Where is the running time of method f for input x. We need one store time to pass the parameter x to f and a store time to assign the result to y. 2. A Detailed Model of the Computer: The Basic Axioms • Axiom 4 The time required to pass an argument to method is the same as the time required to store a value in memory

11. Analyze the running time of a program to compute 2. A Detailed Model of the Computer: Example 1: Arithmetic Series Summation 1publicclass Example1 2 { 3publicstaticint sum (int n) 4 { 5int result = 0; 6for(int i = 1; i <= n; ++i) 7 result += i; 8return result; 9 } 10 }

12. 2. A Detailed Model of the Computer: Example 1: Arithmetic Series Summation

13. Axiom 5 The time required for the address calculation implied by an array subscribing operation, for example, a[i], is a constant, This time does not include the time to compute the subscript expression, nor does it include the time to access (i.e., fetch or store) the array element. 2. A Detailed Model of the Computer: Array Subscribing Operations • The elements of a one-dimensional array are stored in consecutive (sequential) memory locations • We need to know only the address of the first element of the array to determine the address of any other element

14. 2. A Detailed Model of the Computer: Array Subscribing Operations • Example: • y = a [i]; has a running time: • Three operand fetches: • The first to fetch a (the base address of the array) • The second to fetch i (the index into the array) • The third the fetch array element a[i]

15. 2. A Detailed Model of the Computer: Example 3: Finding the largest Element of an Array Analyze the running time of a program to find the largest element of an array of n non-negative integers, 1publicclass Example 2 { 3publicstaticint findMaximum (int [ ] a) 4 { 5int result = a [0]; 6 for(int i = 1; i <a.length; ++i) 7if(a[i]) > result) 8 result = a[i]; 9 return result; 10 } 11 }

16. 2. A Detailed Model of the Computer: Example 3: Finding the largest Element of an Array

17. 2. A Detailed Model of the Computer: Best-Case and Worst-Case Running Times • The worst case scenario occurs when line 8 is executed in every iteration of the loop. • The input array is ordered from smallest to largest

18. 2. A Detailed Model of the Computer: Best-Case and Worst-Case Running Times • The best case scenario occurs when line 8 is never executed. • The input array is ordered from largest to smallest

19. The time required to create a new object instance using the new operator is a constant, . This time does not include any time taken to initialize the object instance. Where is the running time of the Integer constructor 2. A Detailed Model of the Computer: The Last Axiom • Axiom 6 Example: Integer ref = new Integer (0); has a running time:

20. 3. A Simplified Model of the Computer • The performance analysis is easy to do but less accurate • All the arbitrary timing parameters of the detailed model are eliminated • All timing parameters are expressed in units of clock cycles , T =1 • To determine the running time of a program, we simply count the total number of cycles taken

21. Analyze the running time of a program to compute using Horner’s rule 3. A Simplified Model of the Computer: Example 2: Geometric Series Summation 1publicclass Example2 2 { 3publicstaticint geometricsum (int x, int n) 4 { 5int sum = 0; 6for(int i = 0; i <= n; ++i) 7 sum = sum * x +1; 8return sum; 9 } 10 }

22. 3. A Simplified Model of the Computer: Example 2: Geometric Series Summation

23. 3. A Simplified Model of the Computer: About Geometric Series Summation The series is a geometric series and the summation Is called the geometric series summation