Intro to Analysis of Algorithms

Algorithm “A sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.” Named for Al Khwarizmi, who laid out basic procedures for arithmetic functions. (Read about him!)

Analysis of Algorithms Correctness Generality Optimality Simplicity Time Efficiency Space Efficiency

Measuring Efficiency What is basic unit to measure input size? (n) What is basic unit of resource? – Time: basic unit operation – Space: memory units Best, worst, or average case? Find its efficiency class

Why do we care? Let’s look at Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, …

Fibonacci Sequence We want to compute the nth number in the sequence. (F 3 = 2, for example.)

This definition can be translated directly into code – a recursive method. How many additions does it take to compute F n ?

Which is better? function fib2(n) create an array f[0...n] f[0] = 0, f[1] = 1 for i = 2...n: f[i] = f[i-1] + f[i-2] return f[n] function fib1(n) if n = 0: return 0 if n = 1: return 1 return fib1(n-1) + fib1(n-2)

Consider calculating F 200. The fib1 method takes over steps. Computers can do several billion instructions per second. Suppose we have a supercomputer that does 40 trillion instructions per second.

Consider calculating F 200. The fib1 method takes over steps. Computers can do several billion instructions per second. Suppose we have a supercomputer that does 40 trillion instructions per second. Even on this machine, fib1(200) takes at least 2 92 seconds, or centuries, long after the expected end of our sun!!!

Consider calculating F 200. The fib1 method takes over steps. Computers can do several billion instructions per second. Suppose we have a supercomputer that does 40 trillion instructions per second. Even on this machine, fib1(200) takes at least 2 92 seconds, or centuries, long after the expected end of our sun!!! But, fib2(200) would take less than a billionth of a second to compute!!!

10 6 instructions/sec, runtimes N O(log N)O(N)O(N log N)O(N 2 ) , , min 100, hr 1,000, day 1,000,000, min18.3 hr318 centuries

Basics of Efficiency Big-oh – upper bound on the efficiency class Efficiency classes don’t worry with constants A cubic is worse than a quadratic, quartic worse than cubic… Getting big-oh analysis of non-recursive code is pretty easy

The algorithm is O(3n+2), which is O(n). We only care about the efficiency class. Why? At some point, every parabola (n 2 ) overtakes any line (n). We only really care about large input.

Know the shapes constant logarithmic linear quadratic exponential