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**Sample text**

2 Quicksort is done first of all over all n! of the possible input orderings of the array elements, and second, for each such input ordering, we average also over all sequences of choices of the splitting elements. Now let’s consider the behavior of the function F (n). What we are going to show is that F (n) = O(n log n). The labor that F (n) estimates has two components. First there are the pairwise comparisons involved in choosing a splitting element and rearranging the array about the chosen splitting value.

We will see in Chapter 5 that this question is typical of a breed of problems that are the main subject of that chapter, and are perhaps the most (in-)famous unsolved problems in theoretical computer science. 1) it is easy to decide if a graph has an Eulerian path or circuit. Next we’d like to discuss graph coloring, surely one of the prettier parts of graph theory. Suppose that there are K colors available to us, and that we are presented with a graph G. 6 Graphs the two endpoints of e are different.

Hence, suppose it is true for all graphs of V vertices and fewer than E edges, and let G have V vertices and E edges. 4) implies that P (K; G) is a polynomial of the required degree V because G − {e} has fewer edges than G does, so its chromatic polynomial is a polynomial of degree V . G/{e} has fewer vertices than G has, and so P (K; G/{e}) is a polynomial of lower degree. The claim is proved, by induction. 4) gives a recursive algorithm for computing the chromatic polynomial of a graph G, since the two graphs that appear on the right are both ‘smaller’ than G, one in the sense that it has fewer edges than G has, and the other in that it has fewer vertices.