Data Structures and Algorithm Analysis - Computer Science at ...

70 Chap. 3 Algorithm Analysissimplifying rule (4) it has cost c 3 i. The outer for loop is executed n times,but each time the cost of the inner loop is different because it costs c 3 i withi changing each time. You should see that for the first execution of the outerloop, i is 1. For the second execution of the outer loop, i is 2. Each timethrough the outer loop, i becomes one greater, until the last time throughthe loop when i = n. Thus, the total cost of the loop is c 3 times the sum ofthe integers 1 through n. From Equation 2.1, we know thatn∑ n(n + 1)i = ,2i=1which is Θ(n 2 ). By simplifying rule (3), Θ(c 1 + c 2 n + c 3 n 2 ) is simplyΘ(n 2 ).Example 3.12 Compare the asymptotic analysis for the following twocode fragments:sum1 = 0;for (i=1; i