CHAPTER 2: Analysis of Algorithms

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Presentation transcript:

CHAPTER 2: Analysis of Algorithms Java Software Structures: Designing and Using Data Structures Third Edition John Lewis & Joseph Chase

Chapter Objectives Discuss the goals of software development with respect to efficiency Introduce the concept of algorithm analysis Explore the concept of asymptotic complexity Compare various growth functions

Analysis of Algorithms An aspect of software quality is the efficient use of resources, including the CPU Algorithm analysis is a core computing topic It gives us a basis to compare the efficiency of algorithms Example: which sorting algorithm is more efficient?

Growth Functions Analysis is defined in general terms, based on: the problem size (ex: number of items to sort) key operation (ex: comparison of two values) A growth function shows the relationship between the size of the problem (n) and the time it takes to solve the problem t(n) = 15n2 + 45 n

Growth Functions

Growth Functions It's not usually necessary to know the exact growth function The key issue is the asymptotic complexity of the function – how it grows as n increases Determined by the dominant term in the growth function This is referred to as the order of the algorithm We often use Big-Oh notation to specify the order, such as O(n2)

Some growth functions and their asymptotic complexity

Increase in problem size with a ten-fold increase in processor speed

Comparison of typical growth functions for small values of N

Comparison of typical growth functions for large values of N

Analyzing Loop Execution A loop executes a certain number of times (say n) Thus the complexity of a loop is n times the complexity of the body of the loop When loops are nested, the body of the outer loop includes the complexity of the inner loop

Analyzing Loop Execution The following loop is O(n) because the loop executes n times and the body of the loop is O(1): for (int i=0; i<n; i++) { x = x + 1; }

Analyzing Loop Execution The following loop is O(n2) because the loop executes n times and the body of the loop, including a nested loop, is O(n): for (int i=0; i<n; i++) { x = x + 1; for (int j=0; j<n; j++) y = y - 1; }

Analyzing Method Calls To analyze method calls, we simply replace the method call with the order of the body of the method A call to the following method is O(1) public void printsum(int count) { sum = count*(count+1)/2; System.out.println(sum); }