1. Time Complexity Analysis Neil Tang 01/19/2010
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
Outline
Algorithm and Time Complexity
Asymptotic Notations
Examples
CS223 Advanced Data Structures and Algorithms

3. Algorithm and Complexity
Algorithm: A clearly specified set of instructions to be followed to solve a problem.
Characteristics of an algorithm:
- input
- output
- stop on any input
Time complexity: The number of operations required. Best vs. average vs. worst case complexity.
Space complexity: The amount of memory required.
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
Asymptotic Notations
T(N) = O(f(N)) if there exist positive constants c and n0, s.t. T(N)  cf(N) when N n0
T(N) = (g(N)) if there exist positive constants c and n0, s.t. T(N)  cg(N) when N n0
T(N) = (h(N)) iff T(N) = O(h(N)) and T(N) = (h(N))
T(N) = o(p(N)) if T(N) = O(p(N)) and T(N)  (p(N))
O-notation is used to determine an upper bound on the order of growth of a function.
-notation is used to determine a lower bound on the order of growth of a function.
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
Properties
Rule 1: If T1(N) = O(f(N)), T2(N) = O(g(N))
- T1(N) + T2(N) = O(f(N)+g(N))
- T1(N) * T2(N) = O(f(N)*g(N))
Rule 2: If T(N) is a polynomial of degree k, T(N) = (Nk).
Rule 3: logkN = O(N) for any constant k.
CS223 Advanced Data Structures and Algorithms

6. CS223 Advanced Data Structures and Algorithms
Examples
logN = O(N), N = O(N2), N2 = O(N3), N3 = O(2N);
N3 =  (N2); N N = (N4);
N = O(N2).
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Running Time
The running time of algorithms for the
Max Subsequence Sum problem (in sec)
CS223 Advanced Data Structures and Algorithms

8. CS223 Advanced Data Structures and Algorithms
Running Time
CS223 Advanced Data Structures and Algorithms

9. Running Time Calculation
The simple example in pp.35
Rule 1: for loop
Rule 2: Nested loops
e.g., for(…)
for(…)
Rule 3: Consecutive statements
Rule 4: If/else
Rule 5: Recursion: master method
CS223 Advanced Data Structures and Algorithms

10. The Max Subsequence Sum Problem
Given integers A1, A2, …, AN, find the max value of the sum of
a subsequence (return 0 if all integers are negative).
CS223 Advanced Data Structures and Algorithms

11. CS223 Advanced Data Structures and Algorithms
T(N) = Θ(N3)
CS223 Advanced Data Structures and Algorithms

12. CS223 Advanced Data Structures and Algorithms
T(N) = O(N2)
CS223 Advanced Data Structures and Algorithms

13. CS223 Advanced Data Structures and Algorithms

14. CS223 Advanced Data Structures and Algorithms
T(N) = 2T(N/2) + N
T(N) = (NlogN)
CS223 Advanced Data Structures and Algorithms

15. CS223 Advanced Data Structures and Algorithms
T(N) = (N)
CS223 Advanced Data Structures and Algorithms

16. Binary Search Algorithm
T(N) = T(N/2)+1 -> T(N) = (logN)
CS223 Advanced Data Structures and Algorithms

17. CS223 Advanced Data Structures and Algorithms
Verify Your Analysis
CS223 Advanced Data Structures and Algorithms