By Herbert S. Wilf
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Additional resources for Algorithms and Complexity (Internet edition, 1994)
11), and is ∼ 2n log n (n → ∞). Quicksort is, on average, a very quick sorting method, even though its worst case requires a quadratic amount of labor. 2 1. Write out an array of 10 numbers that contains no splitter. Write out an array of 10 numbers that contains 10 splitters. 2. Write a program that does the following. Given a positive integer n. Choose 100 random permutations of [1, 2, . , n],* and count how many of the 100 had at least one splitter. Execute your program for n = 5, 6, . , 12 and tabulate the results.
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.
Hence, in the case of graph algorithms, the ‘easiness’ vs. ‘hardness’ judgment is not altered if we base the distinction on polynomials in n itself, rather than on polynomials in the number of bits of input data. Hence, with a clear conscience, we are going to estimate the running time or complexity of graph algorithms in terms of the number of vertices of the graph that is input. Now let’s do this for algorithm maxset1 above. The first step is to find out if G has any edges. To do this we simply have to look at the input data.