1. Lecture 06: Linked Lists (2), Big O Notation
CISC220 Fall 2009 James Atlas
Lecture 06: Linked Lists (2),
Big O Notation

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

3. Collection
add(x)
remove(index)
member(x)
size()
first()

4. Single Linked List

5. Class Exercise
Implement add, member, first

6. 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

7. Example 2.14/2.16

8. 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)

9. 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(n2)

10. 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 25

11. Let’s count individual instructions
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
Simple Statement }
n2 executions

12. 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

13. 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 25
1 execution,
25 statements
= 25

14. 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 25
T(n) = total execution time as a function of n
T(n) = n2 + 5n + 25

15. Formal Big-O Definition
T(n) = n2 + 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)

16. n2 + 25n + 25 vs. 3n2

17. 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(n2)
Cubic -- O(n3)
Exponential -- O(2n)
Factorial -- O(n!)

18. Various Runtimes

19. Powers of n (Order of the polynomial)

20. Multiplicative Constants

21. Multiplicative Constants (cont’)

22. Dominant Terms (1)

23. Dominant Terms (2)

24. Dominant Terms (3)

25. Dominant Terms Comparison

26. Dominant Terms Comparison (cont’)

27. Class Exercise
Implement last on our Linked List

28. Group Exercise
Implement last in O(1)