# Algorithms and Complexity (Second edition) by Herbert S. Wilf By Herbert S. Wilf

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11) with x = 3. Therefore, the answer is (310 − 1)/2. Try to get used to the idea that a series in powers of x becomes a number if x is replaced by a number, and if we know a formula for the sum of the series, then we know the number that it becomes. Here are some more series to keep in your zoo. A parenthetical remark like ‘(|x| < 1)’ shows the set of values of x for which the series converges. ∞ X xk = ex = k=0 sin x = cos x = log 1 1−x = 1 1−x ∞ X xm m! 13) (−1)r x2r+1 (2r + 1)! 14) (−1)s x2s (2s)!

Vk } of vertices of G such that ∀i = 1, k − 1 : (vi , vi+1 ) ∈ E(G). 2 Did you realize that the number of people who shook hands an odd number of times yesterday is an even number of people? 6. Graphs 41 A graph is connected if there is a path between every pair of its vertices. A path P is simple if its vertices are all distinct, Hamiltonian if it is simple and visits every vertex of G exactly once, and Eulerian if it uses every edge of G exactly once. A subgraph of a graph G is a subset S of its vertices together with a subset of just those edges of G both of whose endpoints lie in S.

That is indeed a recursive call, because we can just change the ‘n’ to ‘i − 1’ and call Quicksort. 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[left] to x[right], inclusive, where left and right are input parameters.