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 n 2 / 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 size of input set number of operations cg(x) f(x)

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. size of input set number of operations cg(x) f(x)

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) size of input set number of operations cg(x) f(x)

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

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

11. Standard function shapes: logarithmic O(log x) (base 2) b c = a log b a = c 2 3 = 8 log 2 8 = 3 log(x*y) = log x + log y log(x a ) = a log x Examples?

12. Standard function shapes: Polynomial O(x 2 ) Examples?

13. Polynomial vs. Linear

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

15. Standard function shapes: Exponential O(c x ) 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(n 2 ) [theta] Worst case: O(n 2 ) [theta] Average Case: O(n 2 ) [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]