CSC 380: Design and Analysis of Algorithms Dr. Curry Guinn

Quick Info Dr. Curry Guinn CIS 2015 guinnc@uncw.edu www.uncw.edu/people/guinnc 962-7937 Office Hours: MTF: 10:00am-11:00m and by appointment

Today Determining O(n) using common code examples Some examples of mathematical analysis Blackboard Quiz 3 due Tuesday, Jan 29, 11:59pm On Chapter 2, Appendix A, Appendix B Homework due Sunday, Feb 03

Program loop runtimes for (int i = 0; i < n; i += c) // O(n) statement(s); Adding to the loop counter means that the loop runtime grows linearly when compared to its maximum value n. Loop executes its body exactly n / c times. for j in range(0, n, c): // O(n) statement(s) Or if myList contains n elements for item in myList: // O(n)

More loop runtimes Nesting loops multiplies their runtimes. for (int i = 0; i < n; i += c) { //O(n2) for (int j = 0; j < n; i += c) { statement; } } for j in range(0, n, c): // O(n2) for k in range(0, n, c): statement(s) Or if myList contains n elements for item in myList: // O(n2) for item in myList:

The loop maximum is n2, so the runtime is quadratic. for (int i = 0; i < n * n; i += c) // O(n2) statement(s); The loop maximum is n2, so the runtime is quadratic. Loop executes its body exactly (n2 / c) times. for j in range(0, n*n, c): // O(n2) statement(s)

for (int i = 1; i <= n; i *= c) // O(log n) statement(s); Multiplying the loop counter means that the maximum value n must grow exponentially to linearly increase the loop runtime; therefore, it is logarithmic. Loop executes its body exactly logc n times. j = 1 while j <= n: // O(log n) statement(s) j *= c

for (int i = n; i >= 1; i /= c) // O(log n) statement(s); Multiplying the loop counter means that the maximum value n must grow exponentially to linearly increase the loop runtime; therefore, it is logarithmic. Loop executes its body exactly logc n times. while n >= 1: // O(log n) statement(s) n /= c

Loops in sequence add together their runtimes, which means the loop set with the larger runtime dominates. for (int i = 0; i < n; i += c) { // O(n) statement; } // O(nlog n) for (int i = 0; i < n; i += c) { for (int j = 0; j < n; i *= c) { } }

Types of runtime analysis Express the running time as f(N), where N is the size of the input worst case: your enemy gets to pick the input average case: need to assume a probability distribution on the inputs However, even with input size N, cost of an algorithm could vary on different input.

Time efficiency of nonrecursive algorithms General Plan for Analysis Decide on parameter n indicating input size Identify algorithm’s basic operation Determine worst, average, and best cases for input of size n Set up a sum for the number of times the basic operation is executed Simplify the sum using standard formulas and rules (see Appendix A) A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Useful summation formulas and rules liu1 = 1+1+ ⋯ +1 = u - l + 1 In particular, 1in1 = n - 1 + 1 = n  (n) 1in i = 1+2+ ⋯ +n = n(n+1)/2  n2/2  (n2) 1in i2 = 12+22+ ⋯ +n2 = n(n+1)(2n+1)/6  n3/3  (n3) 0in ai = 1 + a + ⋯ + an = (an+1 - 1)/(a - 1) for any a  1 In particular, 0in 2i = 20 + 21 + ⋯ + 2n = 2n+1 - 1  (2n ) (ai ± bi ) = ai ± bi cai = cai liuai = limai + m+1iuai A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example 1: Sequential search Worst case Best case Average case A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example 2: Maximum element A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example 3: Element uniqueness problem A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example 4: Matrix multiplication A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example 5: Gaussian elimination Algorithm GaussianElimination(A[0..n-1,0..n]) //Implements Gaussian elimination of an n-by-(n+1) matrix A for i  0 to n - 2 do for j  i + 1 to n - 1 do for k  i to n do A[j,k]  A[j,k] - A[i,k]  A[j,i] / A[i,i] Find the efficiency class and a constant factor improvement. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

Example 6: Counting binary digits It cannot be investigated the way the previous examples are. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

For Next Class, Wednesday Blackboard Quiz 3 due Tuesday, Jan 29, 11:59pm On Chapter 2, Appendix A, Appendix B Homework due Sunday, Feb 03