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By Herbert S. Wilf

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Additional resources for Algorithms and Complexity (Internet edition, 1994)

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2 Quicksort procedure qksort(x:array; left, right:integer); {sorts the subarray x[left], . . {Quicksort} Now let’s consider the complexity of Quicksort. How long does it take to sort an array? Well, the amount of time will depend on exactly which array we happen to be sorting, and furthermore it will depend on how lucky we are with our random choices of splitting elements. If we want to see Quicksort at its worst, suppose we have a really unlucky day, and that the random choice of the splitter element happens to be the smallest element in the array.

An insurance company that wants to alphabetize its list of 5,000,000 policyholders will gratefully notice the difference between n2 = 25, 000, 000, 000, 000 comparisons and n log n = 77, 124, 740 comparisons. If we choose as our unit of complexity the number of swaps of position, then the running time may depend strongly on the input array. ). If we average over all n! possible arrangements of the input data, assuming that the keys are distinct, then it is not hard to see that the average number of swaps that slowsort needs is Θ(n2 ).

We claim that the number of such colorings is equal to the number of all colorings of a certain new graph G/{e}, whose construction we now describe: The vertices of G/{e} consist of the vertices of G other than v or w and one new vertex that we will call ‘vw’ (so G/{e} will have one less vertex than G has). Now we describe the edges of G/{e}. First, if a and b are two vertices of G/{e} neither of which is the new vertex ‘vw’, then (a, b) is an edge of G/{e} if and only if it is an edge of G. Second, (vw, b) is an edge of G/{e} if and only if either (v, b) or (w, b) (or both) is an edge of G.

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