By Herbert S. Wilf

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N in some order. There are n! possible orders in which these elements might appear, so we are considering the action of Quicksort on just these n! inputs. Second, for each particular one of these inputs, the choices of the splitting elements will be made by choosing, at random, one of the entries of the array at each step of the recursion. We will also average over all such random choices of the splitting elements. Therefore, when we speak of the function F (n), the average complexity of Quicksort, we are speaking of the average number of pairwise comparisons of array entries that are made by Quicksort, where the averaging 35 Chapter 2: Recursive Algorithms is done first of all over all n!

Surprisingly, it was found that the coloring could be done using only red, blue, yellow and green. It was noticed over 100 years ago that no matter how complicated a map is drawn, and no matter how many countries are involved, it seems to be possible to color the countries in such a way that (a) every pair of countries that have a common stretch of border have different colors and (b) no more than four colors are used in the entire map. It was then conjectured that four colors are always sufficient for the proper coloring of the countries of any map at all.

Settling this conjecture turned out to be a very hard problem. It was finally solved in 1976 by K. Appel and W. Haken* by means of an extraordinary proof with two main ingredients. First they showed how to reduce the general problem to only a finite number of cases, by a mathematical argument. Then, since the ‘finite number’ was over 1800, they settled all of those cases with quite a lengthy computer calculation. So now we have the ‘Four Color Theorem,’ which asserts that no matter how we carve up the plane or the sphere into countries, we will always be able to color those countries with at most four colors so that countries with a common frontier are colored differently.

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Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf


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