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The second recursive call is the problem. It wants to sort a portion of the array that doesn’t begin at the beginning of the array. The routine Quicksort as written so far doesn’t have enough flexibility to do that. So we will have to give it some more parameters. Instead of trying to sort all of the given array, we will write a routine that sorts only the portion of the given array x that extends from x[lef t] to x[right], inclusive, where lef t and right are input parameters. This leads us to the second version of the routine: procedure qksort(x:array; lef t, right:integer); {sorts the subarray x[left], .

Procedure split correctly splits the array x around the chosen value T . Proof: We claim that as the loop in lines 7 and 8 is repeatedly executed for j := lef t + 1 to right, the following three assertions will always be true just after each execution of lines 7, 8: (a) x[lef t] = T and (b) x[r] < T for all lef t < r ≤ i and (c) x[r] ≥ T for all i < r ≤ j Fig. 1 illustrates the claim. Fig. 1: Conditions (a), (b), (c) To see this, observe first that (a), (b), (c) are surely true at the beginning, when j = lef t + 1.

Show that a tree is a bipartite graph. 2. Find the chromatic number of the n-cycle. 3. Describe how you would find out, on a computer, if a given graph G is bipartite. 4. Given a positive integer K. Find two different graphs each of whose chromatic numbers is K. 5. Exactly how many labeled graphs of n vertices and E edges are there? 6. In how many labeled graphs of n vertices do vertices {1, 2, 3} form an independent set? 7. How many cliques does an n-cycle have? 8. True or false: a Hamilton circuit is an induced cycle in a graph.

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