简单语言算法的概率分析？

Question

Could anyone explain the Probabilistic Analysis of Algorithm in simple words? Couldn't find much info in Wiki page.

Answer 1

Suppose you want to determine the expected runtime of a comparison-based sort algorithm. Here's how you would approach this problem:

• Define a distribution over all possible inputs to the algorithm
  • For example, say that all N! permutations of ordering N distinct numbers are equally likely (1/N!)
• Determine the runtime for each possible input
• Compute the expected runtime by summing up over all possible inputs the probability of the input times the runtime for the input
  • That is, E[Runtime] = sum _i=1,...,N! { P(i) * Runtime(i) } = sum _i=1,...,N! { Runtime(i) } / N!

For example, if you do this with the quicksort algorithm (with naive choice of pivot), you will find that the expected runtime of quicksort is O(N log N) while the worst-case runtime is O(N ² ).

Answer 2

I found a great presentation which explains this in simplistic terms

http://www.lsi.upc.edu/~conrado/research/talks/survey-Stellenbosch.pdf

IMHO, it boils down to "Given an algorithm and "typical input" what would be the performance of a randomized algorithm"

where, typical input really depends on for what purpose you are evaluating the algorithm, you can provide a uniform or a random distribution for input value and measure performance characteristics

Answer 3

In simple language:

If you want to cook a steak, you'd know that the size of your steak will change the time needed on the grill for it to be ready. The raw steak is your input, the grill is your function (ie alg.) and the cooked one is your output.

Now, say that I ask you: "how much time does it take your grill to grill a steak on average?" how would you go about that? Probabilistic Analysis suggests that all possible sizes of steaks could come your way.Therefore, this method produces average running time .This method will provide you a number that represents this situation, so you could perhaps compare different grills. Hope that helps!