CISC220 Spring 2010 James Atlas Lecture 06: Linked Lists (2), Big O Notation

Objectives for Today Introduce Big-O notation Reading - K+W Chap , 2.6

Collection add(x) remove(x) member(x) size() first()

Single Linked List

Class Exercise Implement add, member, first

Efficiency of Algorithms An operation for our ADT can be thought of as a “problem” An algorithm solves the “problem” –A series of steps –Each step has a cost Time Space –Efficiency is a measurement of this cost

Example 2.14/2.16

Big-O notation Describes the relationship between input size and execution time If we double the number of inputs, n, and the execution time is approximately doubled –Linear growth rate –Growth rate has an order of n –O(n)

Example 2.14/2.16 Analysis 2.14 –execution time proportional to x_length –O(n) 2.16 –execution time proportional to (x_length) 2 –O(n 2 )

Let’s count individual instructions for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { Simple Statement } for (int k = 0; k < n; k++) { Simple Statement 1 Simple Statement 2 Simple Statement 3 Simple Statement 4 Simple Statement 5 } Simple Statement 1 Simple Statement 2... Simple Statement 25

Let’s count individual instructions for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { Simple Statement } n 2 executions

Let’s count individual instructions for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { Simple Statement } for (int k = 0; k < n; k++) { Simple Statement 1 Simple Statement 2 Simple Statement 3 Simple Statement 4 Simple Statement 5 } n executions, 5 statements per = 5n

Let’s count individual instructions for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { Simple Statement } for (int k = 0; k < n; k++) { Simple Statement 1 Simple Statement 2 Simple Statement 3 Simple Statement 4 Simple Statement 5 } Simple Statement 1 Simple Statement 2... Simple Statement 25 1 execution, 25 statements = 25

Let’s count individual instructions for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { Simple Statement } for (int k = 0; k < n; k++) { Simple Statement 1 Simple Statement 2 Simple Statement 3 Simple Statement 4 Simple Statement 5 } Simple Statement 1 Simple Statement 2... Simple Statement 25 T(n) = total execution time as a function of n T(n) = n 2 + 5n + 25

Formal Big-O Definition T(n) = n 2 + 5n + 25 T(n) = O(f(n)) means that there exists a function, f(n), that for sufficiently large n and some constant c: cf(n)  T(n)

n n + 25 vs. 3n 2

Common Big-O Runtimes Constant -- O(1) Logarithmic -- O(log n) Fractional -- O(sqrt n) Linear -- O(n) Log-Linear-- O(n log n) Quadratic -- O(n 2 ) Cubic -- O(n 3 ) Exponential -- O(2 n ) Factorial -- O(n!)

Various Runtimes

Powers of n (Order of the polynomial)

Multiplicative Constants

Multiplicative Constants (cont’)

Dominant Terms (1)

Dominant Terms (2)

Dominant Terms (3)

Dominant Terms Comparison

Dominant Terms Comparison (cont’)

Class Exercise Implement last on our Linked List

Group Exercise Implement last in O(1)