Computer Science 101 Efficiency and Complexity Analysis

Things to Desire in an Algorithm Correctness Maintainability Efficiency

Measuring Efficiency Empirical - use clock to get actual running times for different inputs Problems: –Different machines have different running times –Some running times are so long that it is impractical to check them

Measuring Efficiency Analytical - use pencil and paper to determine the abstract amount of work that a program does for different inputs Advantages: –Machine independent –Can predict running times of programs that are impractical to run

Complexity Analysis Pick an instruction that will run most often in the code Determine the number of times this instruction will be executed as a function of the size of the input data Focus on abstract units of work, not actual running time

Example: Search for Largest We focus on the comparison ( > ) inside the loop and ignore the other instructions. for a list of length 1, zero comparisons for a list of length 2, one comparison. for list of length N, N - 1 comparisons set Largest 1 set Current to 2 while Current <= N do if A(Current) > A(Largest) then set Largest to Current increment Current

Big-O Notation Big-O notation expresses the amount of work a program does as a function of the size of the input O(n) stands for order of magnitude n, or order of n for short Search for the largest is O(n), where n is the size of the input

Common Orders of Magnitude ConstantO(k) LogarithmicO(log 2 n) LinearO(n) QuadraticO(n 2 ) ExponentialO(k n )

Common Orders of Magnitude n O(log 2 n) O(n) O(n 2 )O(2 n ) digits!

Recall that search for a value required exactly 3N + 3 steps in the worst case As N gets very large, the difference between N and N + K becomes negligible (where K is a constant) As N gets very large, the difference between N and N / K or N * K also becomes negligible Use the highest degree term in a polynomial and drop the others (N 2 – 2N + 2)  N 2 Approximations

Example Approximations n O(n) O(n) + 2 O(n 2 ) O(n 2 ) + n

Analysis of Selection Sort The main loop runs approximately N times Thus, the search for largest algorithm runs N times The loop within the ith run of search for largest runs N - i times

Analysis of Selection Sort Overall, the number of comparisons performed by the search for the largest value within selection sort is N N N = (N 2 – N)/2  N 2 Selection sort is a quadratic algorithm

Other Algorithms (Worst Cases) Search Search for largestO(?) Sequential search for a target valueO(?) Binary search for a target valueO(?) Sorting Bubble sortO(?) Quick sortO(?)