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By Herbert S. Wilf

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Let’s state a preliminary version of the recursive procedure as follows (look carefully for how the procedure handles the trivial case where n=1). procedure quicksortprelim(x : an array of n numbers); {sorts the array x into nondecreasing order} if n ≥ 2 then permute the array elements so as to create a splitter; let x[i] be the splitter that was just created; quicksortprelim(the subarray x1, . . , xi−1) in place; quicksortprelim(the subarray xi+1, . . {quicksortprelim} * C. A. R. Hoare, Comp.

We will also average over all such random choices of the splitting elements. Therefore, when we speak of the function F (n), the average complexity of Quicksort, we are speaking of the average number of pairwise comparisons of array entries that are made by Quicksort, where the averaging 35 Chapter 2: Recursive Algorithms 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.

Write out an algorithm that will change the vertex adjacency matrix of a graph G to the vertex adjacency matrix of the graph G/{e}, where e is a given edge of G. 10. 12) is the smaller of the two? 11. Let α(G) be the size of the largest independent set of vertices of a graph G, let χ(G) be its chromatic number, and let n = |V (G)|. Show that, for every G, α(G) ≥ n/χ(G). 4 Fast matrix multiplication Everybody knows how to multiply two 2 × 2 matrices. 1) then, ‘of course,’ 2 ci,j = ai,k bk,j (i, j = 1, 2).

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