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X[n] of n numbers. We would like to rearrange these numbers as necessary so that they end up in nondecreasing order of size. This operation is called sorting the numbers. For instance, if we are given {9, 4, 7, 2, 1}, then we want our program to output the sorted array {1, 2, 4, 7, 9}. There are many methods of sorting, but we are going to concentrate on methods that rely on only two kinds of basic operations, called comparisons and interchanges. This means that our sorting routine is allowed to (a) pick up two numbers (‘keys’) from the array, compare them, and decide which is larger.

Then the performance of Quicksort can depend only on the sequence of size relationships in the input array and the choices of the random splitting elements. The actual numerical values that appear in the input array are not in themselves important, except that, to simplify the discussion we will assume that they are all different. The only thing that will matter, then, will be the set of outcomes of all of the paired comparisons of two elements that are done by the algorithm. Therefore, we will assume, for the purposes of analysis, that the entries of the input array are exactly the set of numbers 1, 2, .

Prove that the number of labeled graphs of n vertices all of whose vertices have even degree is equal to the number of all labeled graphs of n − 1 vertices. 29 Algorithms and Complexity Herbert S. Wilf University of Pennsylvania Philadelphia, PA 19104-6395 Copyright Notice Copyright 1994 by Herbert S. Wilf. This material may be reproduced for any educational purpose, multiple copies may be made for classes, etc. Charges, if any, for reproduced copies must be just enough to recover reasonable costs of reproduction.

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