By Herbert S. Wilf

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Extra info for Algorithms and Complexity (Internet edition, 1994)

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5 (1962), 10-15. 2 Quicksort This preliminary version won’t run, though. It looks like a recursive routine. It seems to call itself twice in order to get its job done. But it doesn’t. It calls something that’s just slightly different from itself in order to get its job done, and that won’t work. Observe the exact purpose of Quicksort, as described above. We are given an array of length n, and we want to sort it, all of it. Now look at the two ‘recursive calls,’ which really aren’t quite. The first one of them sorts the array to the left of xi .

1) The number 37 in the above list is in a very intriguing position. Every number that precedes it is smaller than it is and every number that follows it is larger than it is. What that means is that after sorting the list, the 37 will be in the same place it now occupies, the numbers to its left will have been sorted but will still be on its left, and the numbers on its right will have been sorted but will still be on its right. If we are fortunate enough to be given an array that has a ‘splitter,’ like 37, then we can (a) sort the numbers to the left of the splitter, and then (b) sort the numbers to the right of the splitter.

Describe the complement of the graph G in exercise 13 above. How many cliques does it have? 15. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, 3} a triangle? 16. Let H be a labeled graph of L vertices. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, . , L} equal to H? 17. Devise an algorithm that will decide if a given graph, of n vertices and m edges, does or does not contain a triangle, in time O(max(n2 , mn)).

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