Asymptotic Notation (O, Ω,  )

1. Asymptotic Notation (O, Ω, ) • Describes the behavior of the time or space complexity for large instance characteristics • Common asymptotic functions • 1 (constant), log n, n (linear) • n log n, n2 , n3 • 2n ( exponential), n! • where n usually refer to the number of instance of data (data的個數)

2. Common asymptotic functions

3. Common asymptotic functions

4. Common asymptotic functions

5. Big-O notation • The big O notation provides an upper bound for the function f • Definition: • f(n) = O(g(n)) iff positive constants c and n0 exist such that f(n)  c g(n) for all n, nn0 • e.g. f(n) = 10n2 + 4n + 2 then, f(n) = O(n2), or O(n3), O(n4), …

6. Ω(Omega) notation • The Ω notation provides a lower bound for the function f • Definition: • f(n) = Ω(g(n)) iff positive constants c and n0 exist such that f(n)  c g(n) for all n, nn0 • e.g. f(n) = 10n2 + 4n + 2 then, f(n) = Ω(n2), or Ω(n), Ω(1)

7.  notation • The  notation is used when the function f can be bounded both from above and below by the same function g • Definition: • f(n) = (g(n)) iff positive constants c1and c2, and an n0 exist such that c1g(n) f(n) c2g(n) for all n, nn0 • e.g. f(n) = 10n2 + 4n + 2 then, f(n) = Ω(n2), and f(n)=O(n2), therefore, f(n)=  (n2)

8. Practical Complexities 109 instructions/second

9. Impractical Complexities 109instructions/second

10. Faster Computer Vs Better Algorithm Algorithmic improvement more useful than hardware improvement. E.g. 2n to n3

11. Amortized Complexity of Task Sequence • Suppose that a sequence of n tasks is performed. • The worst-case cost of a task is cwc. • Let ci be the (actual) cost of the ith task in this sequence. • So, ci <=cwc, 1 <= i <= n. • n *cwc is an upper bound on the cost of the sequence. • j *cwc is an upper bound on the cost of the first j tasks.

12. Task Sequence • Let cavg be the average cost of a task in this sequence. • So, cavg = Sci/n. • n *cavg is the cost of the sequence. • j *cavg is not an upper bound on the cost of the first j tasks. • Usually, determining cavg is quite hard.

13. Amortized complexity • At times, a better upper bound than j *cwc or n *cwc on sequence cost is obtained using amortized complexity. • The purpose of analyzing amortized complexity is to get tighter upper bound of the actual cost

14. Amortized Complexity • The amortized complexity of a task is the amount you charge the task. • The conventional way to bound the cost of doing a task n times is to use one of the expressions • n*(worst-case cost of task) • S(worst-case costof task i) • The amortized complexity way to bound the cost of doing a task n times is to use one of the expressions • n*(amortized cost of task) • S(amortized cost of task i)

15. Amortized Complexity • The amortized complexity/cost of individual tasks in any task sequence must satisfy: S(actual costof task i) <= S(amortized cost of task i) • So, we can use S(amortized cost of task i) as a bound on the actual complexity of the task sequence.

16. Amortized Complexity • The amortized complexity of a task may bear no direct relationship to the actual complexity of the task.

17. Amortized Complexity • In conventional complexity analysis, each task is charged an amount that is >= its cost. S(actual costof task i) <= S(worst-case cost of task i) • In amortized analysis, some tasks may be charged an amount that is < their cost. S(actual costof task i) <= S(amortized cost of task i)

18. Amortized complexity example: • suppose that a sequence of insert/delete operations: i1, i2, d1, i3, i4, i5, i6, i7, i8, d2, i9 actual cost of each insert is 1 and d1 is 8, d2 is 12. So, the total cost is 29. • In convention, for the upper bound of the cost of the sequence of tasks, we will consider the worst case of each operations; i.e., the cost of insert is 1 and the cost of delete is 12. Therefore, the upper bound of the cost of the sequence of tasks is 9*1+12*2 = 33

19. Amortized complexity • In amortization scheme we charge the actual cost of some operations to others. • The amortized cost of each insert could be 2, d1 and d2 become 6. total is 9*2+6*2=30, which is less than worst case upper bound 33.

20. Amortized Complexity • more examples • the amortized complexity of a sequence of Union and Find operations (will be discuss in chapter 8) • the amortized complexity of a sequence of inserts and deletes of binomial heaps