Time Complexity Analysis Neil Tang 01192010 CS 223

Time Complexity Analysis Neil Tang 01/19/2010 CS 223 Advanced Data Structures and Algorithms 1

Outline Ø Algorithm and Time Complexity Ø Asymptotic Notations Ø Examples CS 223 Advanced Data Structures and Algorithms 2

Algorithm and Complexity Ø Algorithm: A clearly specified set of instructions to be followed to solve a problem. Ø Characteristics of an algorithm: - input - output - stop on any input Ø Time complexity: The number of operations required. Best vs. average vs. worst case complexity. Ø Space complexity: The amount of memory required. CS 223 Advanced Data Structures and Algorithms 3

Asymptotic Notations Ø T(N) = O(f(N)) if there exist positive constants c and n 0, s. t. T(N) cf(N) when N n 0 Ø T(N) = (g(N)) if there exist positive constants c and n 0, s. t. T(N) cg(N) when N n 0 Ø T(N) = (h(N)) iff T(N) = O(h(N)) and T(N) = (h(N)) Ø T(N) = o(p(N)) if T(N) = O(p(N)) and T(N) (p(N)) Ø O-notation is used to determine an upper bound on the order of growth of a function. Ø -notation is used to determine a lower bound on the order of growth of a function. CS 223 Advanced Data Structures and Algorithms 4

Properties Ø Rule 1: If T 1(N) = O(f(N)), T 2(N) = O(g(N)) - T 1(N) + T 2(N) = O(f(N)+g(N)) - T 1(N) * T 2(N) = O(f(N)*g(N)) Ø Rule 2: If T(N) is a polynomial of degree k, T(N) = (Nk). Ø Rule 3: logk. N = O(N) for any constant k. CS 223 Advanced Data Structures and Algorithms 5

Examples log. N = O(N), N = O(N 2), N 2 = O(N 3), N 3 = O(2 N); N 3 = (N 2); N 4 + 1000 N = (N 4); 1 + 2 +. . . + N = O(N 2). CS 223 Advanced Data Structures and Algorithms 6

Running Time The running time of algorithms for the Max Subsequence Sum problem (in sec) CS 223 Advanced Data Structures and Algorithms 7

Running Time CS 223 Advanced Data Structures and Algorithms 8

Running Time Calculation Ø The simple example in pp. 35 Ø Rule 1: for loop Ø Rule 2: Nested loops e. g. , for(…) Ø Rule 3: Consecutive statements e. g. , for(…) Ø Rule 4: If/else Ø Rule 5: Recursion: master method CS 223 Advanced Data Structures and Algorithms 9

The Max Subsequence Sum Problem Given integers A 1, A 2, …, AN, find the max value of the sum of a subsequence (return 0 if all integers are negative). CS 223 Advanced Data Structures and Algorithms 10

Algorithm 1 T(N) = Θ(N 3) CS 223 Advanced Data Structures and Algorithms 11

Algorithm 2 T(N) = O(N 2) CS 223 Advanced Data Structures and Algorithms 12

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Algorithm 3 T(N) = 2 T(N/2) + N T(N) = (Nlog. N) CS 223 Advanced Data Structures and Algorithms 14

Algorithm 4 T(N) = (N) CS 223 Advanced Data Structures and Algorithms 15

Binary Search Algorithm T(N) = T(N/2)+1 -> T(N) = (log. N) CS 223 Advanced Data Structures and Algorithms 16

Verify Your Analysis CS 223 Advanced Data Structures and Algorithms 17