Analysis of Algorithms

1. Analysis of Algorithms CPS212 Gordon College

2. Measuring the efficiency of algorithms • There are 2 algorithms: algo1 and algo2 that produce the same results. • How do you choose which algorithm to keep. • Algorithm Efficiency

3. Measuring the efficiency of algorithms • How do you determine an algorithm’s inherent efficiency? • Code them up and compare their running times on a lab machine. • How were they coded? We want to compare the algorithms - not their implementations • What computer do we use? What else is running? • What data is a fair test of an algorithm’s efficiency.

4. Measuring the efficiency of algorithms • Goal: • Measure the efficiency of an algorithm - independent of implementation, hardware, or data. • Solution: • Analyze and represent the number of operations the algorithm will perform as a function of input. • Example: • Copying an array with n elements requires x operations. What is x?

5. Growth rates • Algo1 requires n2 / 2 operations to solve a problem with input size n • Algo2 requires 5n + 10 operations Which is the better algorithm for the job?

6. Order of magnitude growth analysis • A function f(x) is O(g(x)) if and only if there exist 2 positive constants, c and n, such that | f(x) |≤ cg(x) for all x > n cg(x) f(x) number of operations size of input set

7. Order of magnitude growth analysis • Asymptotic growth - as the growth approaches infinity • An asymptote of a real-valued function y = f(x) is a curve which describes the behavior of f as either x or y tends to infinity. cg(x) f(x) number of operations size of input set

8. Order of magnitude growth analysis • Important points: • Focus only on the growth • Shape of g(x) is essential • As the input set grows large (ignore the shape for small x) cg(x) f(x) number of operations size of input set

9. Standard function shapes: constant • O(1) Examples? f(x) = cg(x) number of operations size of input set

10. Standard function shapes: linear • O(x) • Examples? cg(x) f(x)=ax+b number of operations size of input set

11. Standard function shapes: logarithmic • O(log x) (base 2) bc = a logba = c 23 = 8 log28 = 3 log(x*y) = log x + log y log(xa) = a log x Examples?

12. Standard function shapes: Polynomial • O(x2) Examples?

13. Polynomial vs. Linear

14. Polynomial vs. Linear Example: 50n + 20 & n2 At what point will n2 surpass 50n+20? n2 = 50n+20 Solve for x…quadratic formula n2 - 50n - 20 = 0 n = 101/2 = 50.5 n = -1/2

15. Standard function shapes: Exponential • O(cx) Examples? Hamiltonian Circuit Traveling Salesman Optimization problems Solution: Limit to small input sets Isolate special cases Find approximate solution (near optimal)

16. Complexity in action

17. Real Examples • Searching (sequential) • Unit of work: comparisons • Best case: O(1) [theta] • Worst case: O(n) [theta] • Average Case: O(n) [theta]

18. Real Examples • Sort (selection) • Unit of work: comparisons and exchanges • Best case: O(n2) [theta] • Worst case: O(n2) [theta] • Average Case: O(n2) [theta]

19. Real Examples • Search (binary) • Unit of work: comparisons • Best case: O(1) [theta] • Worst case: O(log n) [theta] • Average Case: O(log n) [theta]