Probabilistic Analysis of Algorithms

Rather than analyzing the worst case performance of algorithms, one can investigate their performance on typical instances of a given size. This is the approach we investigate in this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size? Sometimes, there is a natural answer to this question. For example, in developing an algorithm which is typically efficent for an NP-complete optimization problems on graphs, we might assume that an vertex input is equally likely to be any of the labelled graphs with vertices. This allows us to exploit any property which holds on almost all such graphs when developing the algorithm. There is no such obvious choice of a typical input to an algorithm which sorts numbers for, e.g., it is not clear how big we want to permit the to become. One of many possible approaches is to impose the condition that each number is drawn uniformly from . Another is to note that in analyzing our algorithm, we may not need to know the values of the variables but simply their relative sizes. We can then perform our analysis assuming that the are a random permutation of "! $# !% ! with each permutation equally likely. More generally, we will choose some probability distribution on the inputs of a given size and analyze the performance of our algorithm when applied to a random input drawn from this distribution. Now, in general, probability distributions are complicated objects which must be formally described and analyzed using much messy measure theory. Fortunately, we will be concerned only with relatively simple distributions which will be much easier to deal with. We often consider finite distributions in which our probability space is a finite set & , and for each